Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition (Advances in Design and Control)

Advances in Design and Control SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines.

Editor-in-Chief Ralph C. Smith, North Carolina State University

Editorial Board Athanasios C. Antoulas, Rice University Siva Banda, Air Force Research Laboratory Belinda A. Batten, Oregon State University John Betts, The Boeing Company (retired) Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, Davis Kirsten Morris, University of Waterloo Richard Murray, California Institute of Technology Ekkehard Sachs, University of Trier

Series Volumes Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based Approach ¸ Adaptive Control Tutorial Ioannou, Petros and Fidan, Barıs, Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

Shapes and Geometries

Metrics, Analysis, Differential Calculus, and Optimization Second Edition

M. C. Delfour Université de Montréal Montréal, Québec Canada

J.-P. Zolésio

National Center for Scientific Research (CNRS) and National Institute for Research in Computer Science and Control (INRIA) Sophia Antipolis France

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2011 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. The research of the first author was supported by the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the National Sciences and Engineering Research Council of Canada; and the FQRNT program of the Ministère de l’Éducation du Québec. Library of Congress Cataloging-in-Publication Data Delfour, Michel C., 1943Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C. Delfour, J.-P. Zolésio. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-898719-36-9 (hardcover : alk. paper) 1. Shape theory (Topology) I. Zolésio, J.-P. II. Title. QA612.7.D45 2011 514’.24--dc22 2010028846

is a registered trademark.

j

This book is dedicated to Alice, Jeanne, Jean, and Roger

Contents List of Figures Preface 1 Objectives and Scope of the Book . 2 Overview of the Second Edition . . 3 Intended Audience . . . . . . . . . 4 Acknowledgments . . . . . . . . . .

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1 Introduction: Examples, Background, and Perspectives 1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Geometry as a Variable . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the Introductory Chapter . . . . . . . . . . . . . . 2 A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . . 6 Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . . 6.1 Free Interface between Two Materials . . . . . . . . . . . . . 6.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 7 Design of a Thermal Diffuser . . . . . . . . . . . . . . . . . . . . . . 7.1 Description of the Physical Problem . . . . . . . . . . . . . . 7.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 7.3 Reformulation of the Problem . . . . . . . . . . . . . . . . . . 7.4 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 7.5 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . . 8.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 8.2 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 9 A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . . 9.1 Automatic Image Processing . . . . . . . . . . . . . . . . . . 9.2 Image Smoothing/Filtering by Convolution and Edge Detectors 9.2.1 Construction of the Convolution of I . . . . . . . . 9.2.2 Space-Frequency Uncertainty Relationship . . . . . 9.2.3 Laplacian Detector . . . . . . . . . . . . . . . . . . . vii

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Objective Functions Defined on the Whole Edge . . . . . . . 9.3.1 Eulerian Shape Semiderivative . . . . . . . . . . . . 9.3.2 From Local to Global Conditions on the Edge . . . 9.4 Snakes, Geodesic Active Contours, and Level Sets . . . . . . 9.4.1 Objective Functions Defined on the Contours . . . . 9.4.2 Snakes and Geodesic Active Contours . . . . . . . . 9.4.3 Level Set Method . . . . . . . . . . . . . . . . . . . 9.4.4 Velocity Carried by the Normal . . . . . . . . . . . 9.4.5 Extension of the Level Set Equations . . . . . . . . 9.5 Objective Function Defined on the Whole Image . . . . . . . 9.5.1 Tikhonov Regularization/Smoothing . . . . . . . . . 9.5.2 Objective Function of Mumford and Shah . . . . . . 9.5.3 Relaxation of the (N − 1)-Hausdorff Measure . . . . 9.5.4 Relaxation to BV-, H s -, and SBV-Functions . . . . 9.5.5 Cracked Sets and Density Perimeter . . . . . . . . . Shapes and Geometries: Background and Perspectives . . . . . . . . 10.1 Parametrize Geometries by Functions or Functions by Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Shape Analysis in Mechanics and Mathematics . . . . . . . . 10.3 Characteristic Functions: Surface Measure and Geometric Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Distance Functions: Smoothness, Normal, and Curvatures . . 10.5 Shape Optimization: Compliance Analysis and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Shape Calculus and Tangential Differential Calculus . . . . . 10.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . . Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . . 11.1 Geometries Parametrized by Functions . . . . . . . . . . . . . 11.2 Functions Parametrized by Geometries . . . . . . . . . . . . . 11.3 Shape Continuity and Optimization . . . . . . . . . . . . . . 11.4 Derivatives, Shape and Tangential Differential Calculuses, and Derivatives under State Constraints . . . . . . . . . . . . . .

2 Classical Descriptions of Geometries and Their Properties 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Abelian Group Structures on Subsets of a Fixed Holdall D . 2.2.1 First Abelian Group Structure on (P(D),
) . . . . 2.2.2 Second Abelian Group Structure on (P(D), ) . . . 2.3 Connected Space, Path-Connected Space, and Geodesic Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bouligand’s Contingent Cone, Dual Cone, and Normal Cone 2.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Sets Sets 5.1 5.2 5.3 5.4

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2.5.2 The Space W0m,p (Ω) . . . . . . . . . . . . . . . . . . 61 2.5.3 Embedding of H01 (Ω) into H01 (D) . . . . . . . . . . 62 2.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 63 Spaces of Continuous and Differentiable Functions . . . . . . 63 2.6.1 Continuous and C k Functions . . . . . . . . . . . . 63 2.6.2 H¨ older (C 0,
) and Lipschitz (C 0,1 ) Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 65 2.6.4 Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance 66 Locally Described by an Homeomorphism or a Diffeomorphism 67 Sets of Classes C k and C k,
. . . . . . . . . . . . . . . . . . . 67 Boundary Integral, Canonical Density, and Hausdorff Measures 70 3.2.1 Boundary Integral for Sets of Class C 1 . . . . . . . 70 3.2.2 Integral on Submanifolds . . . . . . . . . . . . . . . 71 3.2.3 Hausdorff Measures . . . . . . . . . . . . . . . . . . 72 Fundamental Forms and Principal Curvatures . . . . . . . . . 73 Globally Described by the Level Sets of a Function . . . . . . . 75 Locally Described by the Epigraph of a Function . . . . . . . . 78 Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions . . . . . . . . . . . 79 olderian/Lipschitzian Sets . . . . 87 Local C k,
-Epigraphs and H¨ Local C k,
-Epigraphs and Sets of Class C k,
. . . . . . . . . . 89 Locally Lipschitzian Sets: Some Examples and Properties . . 92 5.4.1 Examples and Continuous Linear Extensions . . . . 92 5.4.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . 93 5.4.3 Boundary Measure and Integral for Lipschitzian Sets 94 5.4.4 Geodesic Distance in a Domain and in Its Boundary 97 5.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100 Locally Described by a Geometric Property . . . . . . . . . . . 101 Definitions and Main Results . . . . . . . . . . . . . . . . . . 102 Equivalence of Geometric Segment and C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Equivalence of the Uniform Fat Segment and the Equi-C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 109 Uniform Cone/Cusp Properties and H¨ olderian/Lipschitzian Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.1 Uniform Cone Property and Lipschitzian Sets . . . 114 6.4.2 Uniform Cusp Property and H¨ olderian Sets . . . . . 115 Hausdorff Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generic Constructions of Micheletti . . . . . . . . . . . . . 2.1 Space F(Θ) of Transformations of RN . . . . . . . 2.2 Diffeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

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Contents Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . . Courant Metric on the Quotient Group F(Θ)/G . . . . . . . Assumptions for B k (RN , RN ), C k (RN , RN ), and C0k (RN , RN ) 2.5.1 Checking the Assumptions . . . . . . . . . . . . . . 2.5.2 Perturbations of the Identity and Tangent Space . . 2.6 Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) . . . . . . 2.6.1 Checking the Assumptions . . . . . . . . . . . . . . 2.6.2 Perturbations of the Identity and Tangent Space . . Generalization to All Homeomorphisms and C k -Diffeomorphisms . .

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4 Transformations Generated by Velocities 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Metrics on Transformations Generated by Velocities . . . . . . . 2.1 Subgroup GΘ of Transformations Generated by Velocities 2.2 Complete Metrics on GΘ and Geodesics . . . . . . . . . . 2.3 Constructions of Azencott and Trouv´e . . . . . . . . . . . 3 Semiderivatives via Transformations Generated by Velocities . . 3.1 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gateaux and Hadamard Semiderivatives . . . . . . . . . . 3.3 Examples of Families of Transformations of Domains . . . 3.3.1 C ∞ -Domains . . . . . . . . . . . . . . . . . . . . 3.3.2 C k -Domains . . . . . . . . . . . . . . . . . . . . 3.3.3 Cartesian Graphs . . . . . . . . . . . . . . . . . 3.3.4 Polar Coordinates and Star-Shaped Domains . . 3.3.5 Level Sets . . . . . . . . . . . . . . . . . . . . . . 4 Unconstrained Families of Domains . . . . . . . . . . . . . . . . . 4.1 Equivalence between Velocities and Transformations . . . 4.2 Perturbations of the Identity . . . . . . . . . . . . . . . . 4.3 Equivalence for Special Families of Velocities . . . . . . . 5 Constrained Families of Domains . . . . . . . . . . . . . . . . . . 5.1 Equivalence between Velocities and Transformations . . . 5.2 Transformation of Condition (V2D ) into a Linear Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Continuity of Shape Functions along Velocity Flows . . . . . . . 5 Metrics via Characteristic Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Abelian Group Structure on Measurable Characteristic Functions 2.1 Group Structure on Xµ (RN ) . . . . . . . . . . . . . . . . 2.2 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Complete Metric for Characteristic Functions in Lp -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lebesgue Measurable Characteristic Functions . . . . . . . . . . 3.1 Strong Topologies and C ∞ -Approximations . . . . . . . . 3.2 Weak Topologies and Microstructures . . . . . . . . . . . 3.3 Nice or Measure Theoretic Representative . . . . . . . . .

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xi 3.4 The Family of Convex Sets . . . . . . . . . . . . . . . . . . . 3.5 Sobolev Spaces for Measurable Domains . . . . . . . . . . . . Some Compliance Problems with Two Materials . . . . . . . . . . . 4.1 Transmission Problem and Compliance . . . . . . . . . . . . 4.2 The Original Problem of C´ea and Malanowski . . . . . . . . . 4.3 Relaxation and Homogenization . . . . . . . . . . . . . . . . Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . . 6.1 Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . . 6.2 Decomposition of the Integral along Level Sets . . . . . . . . 6.3 Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces . . . . . . . . . . . . . . . . . . . 6.4 Compactness and Uniform Cone Property . . . . . . . . . . . Existence for the Bernoulli Free Boundary Problem . . . . . . . . . . 7.1 An Example: Elementary Modeling of the Water Wave . . . 7.2 Existence for a Class of Free Boundary Problems . . . . . . . 7.3 Weak Solutions of Some Generic Free Boundary Problems . . 7.3.1 Problem without Constraint . . . . . . . . . . . . . 7.3.2 Constraint on the Measure of the Domain Ω . . . . 7.4 Weak Existence with Surface Tension . . . . . . . . . . . . .

6 Metrics via Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . . 2.1 Family of Distance Functions Cd (D) . . . . . . . . . . . . . . 2.2 Pomp´eiu–Hausdorff Metric on Cd (D) . . . . . . . . . . . . . . 2.3 Uniform Complementary Metric Topology and Cdc (D) . . . . c 2.4 Families Cdc (E; D) and Cd,loc (E; D) . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . . 4 W 1,p -Metric Topology and Characteristic Functions . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . . 5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Reach and Federer’s Sets of Positive Reach . . . . . . . . . . . . . . 6.1 Definitions and Main Properties . . . . . . . . . . . . . . . . 6.2 C k -Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Compact Family of Sets with Uniform Positive Reach . . . 7 Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . . 8.1 Convex Sets and Properties of dA . . . . . . . . . . . . . . . . 8.2 Semiconvexity and BV Character of dA . . . . . . . . . . . . 8.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 8.4 Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cd,loc

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Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 324 9.1 Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 325 9.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Family of Oriented Distance Functions Cb (D) . . . . . . 2.2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . . 4 W 1,p (D)-Metric Topology and the Family Cb0 (D) . . . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Boundary of Bounded and Locally Bounded Curvature . . . . . . . . 5.1 Examples and Limit of Tubular Norms as h Goes to Zero . . 6 Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . . 7 Federer’s Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . . 7.1 Approximation by Dilated Sets/Tubular Neighborhoods . . . 7.2 Boundaries with Positive Reach . . . . . . . . . . . . . . . . . 8 Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . . 9 Sobolev or W m,p Domains . . . . . . . . . . . . . . . . . . . . . . . . 10 Characterization of Convex and Semiconvex Sets . . . . . . . . . . . 10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . . 10.2 Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . . 11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . . 11.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . . 11.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . . 13 Compactness and Uniform Fat Segment Property . . . . . . . . . . . 13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Equivalent Conditions on the Local Graph Functions . . . . . 14 Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . . 14.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . . 15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . . 16 A Variation of the Image Segmentation Problem of Mumford and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 16.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . . 16.2.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . 16.2.2 Another Compactness Theorem . . . . . . . . . . . 16.2.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . 16.3 Existence of a Cracked Set with Minimum Density Perimeter

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xiii 16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization 1 Introduction and Generic Examples . . . . . . . . . . . . . . . . . . . 1.1 First Generic Example . . . . . . . . . . . . . . . . . . . . . . 1.2 Second Generic Example . . . . . . . . . . . . . . . . . . . . . 1.3 Third Generic Example . . . . . . . . . . . . . . . . . . . . . 1.4 Fourth Generic Example . . . . . . . . . . . . . . . . . . . . . 2 Upper Semicontinuity and Maximization of the First Eigenvalue . . 3 Continuity of the Transmission Problem . . . . . . . . . . . . . . . . 4 Continuity of the Homogeneous Dirichlet Boundary Value Problem . 4.1 Classical, Relaxed, and Overrelaxed Problems . . . . . . . . . 4.2 Classical Dirichlet Boundary Value Problem . . . . . . . . . . 4.3 Overrelaxed Dirichlet Boundary Value Problem . . . . . . . . 4.3.1 Approximation by Transmission Problems . . . . . . 4.3.2 Continuity with Respect to X(D) in the Lp (D)-Topology . . . . . . . . . . . . . . . . . . . . 4.4 Relaxed Dirichlet Boundary Value Problem . . . . . . . . . . 5 Continuity of the Homogeneous Neumann Boundary Value Problem 6 Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . . 6.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . 6.2 Quasi-continuous Representative and H 1 -Functions . . . . . . 6.3 Transport of Sets of Zero Capacity . . . . . . . . . . . . . . . 7 Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . . 7.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . 7.2 Continuity and Optimization over L(D, r, O, λ) . . . . . . . . 7.2.1 Continuity of the Classical Homogeneous Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . 7.2.2 Minimization/Maximization of the First Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . 8 Continuity under Capacity Constraints . . . . . . . . . . . . . . . . . 9 Compact Families Oc,r (D) and Lc,r (O, D) . . . . . . . . . . . . . . . 9.1 Compact Family Oc,r (D) . . . . . . . . . . . . . . . . . . . . 9.2 Compact Family Lc,r (O, D) and Thick Set Property . . . . . 9.3 Maximizing the Eigenvalue λA (Ω) . . . . . . . . . . . . . . . 9.4 State Constrained Minimization Problems . . . . . . . . . . . 9.5 Examples with a Constraint on the Gradient . . . . . . . . .

409 409 411 411 411 412 412 417 418 418 421 423 423

9 Shape and Tangential Differential Calculuses 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Differentiation in Topological Vector Spaces 2.1 Definitions of Semiderivatives and Derivatives . 2.2 Derivatives in Normed Vector Spaces . . . . . . 2.3 Locally Lipschitz Functions . . . . . . . . . . . 2.4 Chain Rule for Semiderivatives . . . . . . . . .

457 457 458 458 461 465 465

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

424 425 426 429 429 431 432 434 434 437 437 438 440 447 447 450 452 453 454

xiv

Contents

3

4

5

6

2.5 Semiderivatives of Convex Functions . . . . . . . . . . . 2.6 Hadamard Semiderivative and Velocity Method . . . . . First-Order Shape Semiderivatives and Derivatives . . . . . . . 3.1 Eulerian and Hadamard Semiderivatives . . . . . . . . . 3.2 Hadamard Semidifferentiability and Courant Metric Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Perturbations of the Identity and Gateaux and Fr´echet Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Shape Gradient and Structure Theorem . . . . . . . . . Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . . 4.1 Basic Formula for Domain Integrals . . . . . . . . . . . 4.2 Basic Formula for Boundary Integrals . . . . . . . . . . 4.3 Examples of Shape Derivatives . . . . . . . . . . . . . . 4.3.1 Volume of Ω and Surface Area of Γ . . . . . . 4.3.2 H 1 (Ω)-Norm . . . . . . . . . . . . . . . . . . . 4.3.3 Normal Derivative . . . . . . . . . . . . . . . . Elements of Tangential Calculus . . . . . . . . . . . . . . . . . 5.1 Intrinsic Definition of the Tangential Gradient . . . . . 5.2 First-Order Derivatives . . . . . . . . . . . . . . . . . . 5.3 Second-Order Derivatives . . . . . . . . . . . . . . . . . 5.4 A Few Useful Formulae and the Chain Rule . . . . . . . 5.5 The Stokes and Green Formulae . . . . . . . . . . . . . 5.6 Relation between Tangential and Covariant Derivatives 5.7 Back to the Example of Section 4.3.3 . . . . . . . . . . . Second-Order Semiderivative and Shape Hessian . . . . . . . . 6.1 Second-Order Derivative of the Domain Integral . . . . 6.2 Basic Formula for Domain Integrals . . . . . . . . . . . 6.3 Nonautonomous Case . . . . . . . . . . . . . . . . . . . 6.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . 6.5 Decomposition of d2 J(Ω; V (0), W (0)) . . . . . . . . . .

. . . .

. . . .

. . . .

467 469 471 471

. . . 476 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

10 Shape Gradients under a State Equation Constraint 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 An Illustrative Example and a Shape Variational Principle 2.2 Function Space Parametrization . . . . . . . . . . . . . . . 2.3 Differentiability of a Minimum with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Application of the Theorem . . . . . . . . . . . . . . . . . . 2.5 Domain and Boundary Integral Expressions of the Shape Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Transport of H0k (Ω) by W k,∞ -Transformations of RN . . . 4.2 Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . . 4.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

476 479 482 482 484 486 486 487 488 491 492 495 496 497 498 498 501 501 502 504 505 510 515

. . . .

519 519 521 521 522

. 523 . 526 . . . . . .

530 532 535 536 537 546

Contents 5

6

xv Saddle 5.1 5.2 5.3 5.4

Point Formulation and Function Space Parametrization . . An Illustrative Example . . . . . . . . . . . . . . . . . . . . Saddle Point Formulation . . . . . . . . . . . . . . . . . . . Function Space Parametrization . . . . . . . . . . . . . . . Differentiability of a Saddle Point with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application of the Theorem . . . . . . . . . . . . . . . . . . 5.6 Domain and Boundary Expressions for the Shape Gradient Multipliers and Function Space Embedding . . . . . . . . . . . . . 6.1 The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . 6.2 A Saddle Point Formulation of the State Equation . . . . . 6.3 Saddle Point Expression of the Objective Function . . . . . 6.4 Verification of the Assumptions of Theorem 5.1 . . . . . . .

. . . .

551 551 552 553

. . . . . . . .

555 559 561 562 562 563 564 566

Elements of Bibliography

571

Index of Notation

615

Index

619

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6

Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Column of height one and cross section area A under the load . Triangulation and basis function associated with node Mi . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . . Heat spreading scheme for high-power solid-state devices. . . . . (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω;
. . . . . . . . . . . . . . . . . . . . . (C) Surface D generating Ω. 1.7 Volume Ω and its cross section. . . . . . . . . . . . . . . . . . . .
and its generating surface A. . . . . . . . . . . . . . . 1.8 Volume Ω 1.9 Image I of objects and their segmentation in the frame D. . . . . 1.10 Image I containing black curves or cracks in the frame D. . . . . 1.11 Example of a two-dimensional strongly cracked set. . . . . . . . . 1.12 Example of a surface with facets associated with a ball. . . . . .

. . . . .

. . . . .

4 5 8 11 14

. . . . . . .

. . . . . . .

15 19 20 22 22 35 37 68 79 91 92

2.8

Diffeomorphism gx from U (x) to B. . . . . . . . . . . . . . . . . . . Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . . Domain Ω0 and its image T (Ω0 ) spiraling around the origin. . . . . . Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin. . . . Examples of arbitrary and axially symmetrical O around the direction d = Ax (0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . . The cone x + Ax C(λ, ω) in the direction Ax eN . . . . . . . . . . . . . Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . .

4.1

Transport of Ω by the velocity field V . . . . . . . . . . . . . . . . . . 171

2.1 2.2 2.3 2.4 2.5 2.6 2.7

5.1 5.2 5.3 5.4 5.5 5.6


. . . . . . . . . . . . . Smiling sun Ω and expressionless sun Ω. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 . . . . . . . . . . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . The function f (x, y) = 56 (1 − |x| − |y|)6 . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of section 4.1. . . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of C´ea and Malanowski. . . . . . . . . xvii

. . . . . . . . . . . . =1 . . . =1 . . .

109 114 118 119

220 227 228 234 235 239

xviii

List of Figures

5.7

The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1 6.2 6.3 6.4 6.5

Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). Nonuniqueness of the exterior normal. . . . . . . . . . . . . Vertical stripes of Example 4.1. . . . . . . . . . . . . . . . . ∇dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . Set of critical points of A. . . . . . . . . . . . . . . . . . . .

. . . . .

280 286 293 301 318

7.1 7.2

∇bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . . W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R2 with uniformly bounded density perimeter to a set with empty interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of a two-dimensional strongly cracked set. . . . . . . . . . . The two-dimensional strongly cracked set of Figure 7.3 in an open frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two open components Ω1 and Ω2 of the open domain Ω for N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

356

7.3 7.4 7.5

. . . . .

. . . . .

. . . . .

. . . . .

387 396 400 406

Preface

1

Objectives and Scope of the Book

The objective of this book is to give a comprehensive presentation of mathematical constructions and tools that can be used to study problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. In that context, a good analytical framework and good modeling techniques must be able to handle the occurrence of singular behaviors whenever they are compatible with the mechanics or the physics of the problems at hand. In some optimization problems, the natural intuitive notion of a geometric domain undergoes mutations into relaxed entities such as microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still makes sense mathematically. This book covers the basic mathematical ideas, constructions, and methods that come from different fields of mathematical activities and areas of applications that have often evolved in parallel directions. The scope of research is frighteningly broad because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, and constrained optimization, with applications to classical mechanics of continuous media such as fluid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary problems, and image processing. Innovative modeling or new issues raised in some applications force a new look at the fundamentals of well-established mathematical areas such as geometry, to relax basic notions of volume, perimeter, and curvature or boundary value problems, and to find suitable relaxations of solutions. In that spirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion of volume to the one of measure on an equivalence class of measurable sets in 1907. He was followed in that endeavor in the early 1950s by the celebrated work of E. De Giorgi, who used the relaxed notion of perimeter defined on the class of Caccioppoli sets to solve Plateau’s problem of minimal surfaces. The material that is pertinent to the study of geometric objects and the entities and functions that are defined on them would necessitate an encyclopedic investment to bring together the basic theories and their fields of applications. This objective is obviously beyond the scope of a single book and two authors. The xix

xx

Preface

coverage of this book is more modest. Yet, it contains most of the important fundamentals at this stage of evolution of this expanding field. Even if shape analysis and optimization have undergone considerable and important developments on the theoretical and numerical fronts, there are still cultural barriers between areas of applications and between theories. The whole field is extremely active, and the best is yet to come with fundamental structures and tools beginning to emerge. It is hoped that this book will help to build new bridges and stimulate cross-fertilization of ideas and methods.

2

Overview of the Second Edition

The second edition is almost a new book. All chapters from the first edition have been updated and, in most cases, considerably enriched with new material. Many chapters or parts of chapters have been completely rewritten following the developments in the field over the past 10 years. The book went from 9 to 10 chapters with a more elaborate sectioning of each chapter in order to produce a much more detailed table of contents. This makes it easier to find specific material. A series of illustrative generic examples has been added right at the beginning of the introductory Chapter 1 to motivate the reader and illustrate the basic dilemma: parametrize geometries by functions or functions by geometries? This is followed by the big picture: a section on background and perspectives and a more detailed presentation of the second edition. The former Chapter 2 has been split into Chapter 2 on the classical descriptions and properties of domains and sets and a new Chapter 3, where the important material on Courant metrics and the generic constructions of A. M. Micheletti have been reorganized and expanded. Basic definitions and material have been added and regrouped at the beginning of Chapter 2: Abelian group structure on subsets of a set, connected and path-connected spaces, function spaces, tangent and dual cones, and geodesic distance. The coverage of domains that verify some segment property and have a local epigraph representation has been considerably expanded, and Lipschitzian (graph) domains are now dealt with as a special case. The new Chapter 3 on domains and submanifolds that are the image of a fixed set considerably expands the material of the first edition by bringing up the general assumptions behind the generic constructions of A. M. Micheletti that lead to the Courant metrics on the quotient space of families of transformations by subgroups of isometries such as identities, rotations, translations, or flips. The general results apply to a broad range of groups of transformations of the Euclidean space and to arbitrary closed subgroups. New complete metrics on the whole spaces of homeomorphisms and C k -diffeomorphisms are also introduced to extend classical results for transformations of compact manifolds to general unbounded closed sets and open sets that are crack-free. This material is central in classical mechanics and physics and in modern applications such as imaging and detection. The former Chapter 7 on transformations versus flows of velocities has been moved right after the Courant metrics as Chapter 4 and considerably expanded. It now specializes the results of Chapter 3 to spaces of transformations that are

2. Overview of the Second Edition

xxi

generated by the flow of a velocity field over a generic time interval. One important motivation is to introduce a notion of semiderivatives as well as a tractable criterion for continuity with respect to Courant metrics. Another motivation for the velocity point of view is the general framework of R. Azencott and A. Trouv´e starting in 1994 with applications in imaging. They construct complete metrics in relation with geodesic paths in spaces of diffeomorphisms generated by a velocity field. The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and 5 on distance and oriented distance functions have become Chapters 5, 6, and 7. Those chapters have been renamed Metrics Generated by . . . in order to emphasize one of the main thrusts of the book: the construction of complete metrics on shapes and geometries.1 Those chapters emphasize the function analytic description of sets and domains: construction of metric topologies and characterization of compact families of sets or submanifolds in the Euclidean space. In that context, we are now dealing with equivalence classes of sets that may or may not have an invariant open or closed representative in the class. For instance, they include Lebesgue measurable sets and Federer’s sets of positive reach. Many of the classical properties of sets can be recovered from the smoothness or function analytic properties of those functions. The former Chapter 6 on optimization of shape functions has been completely rewritten and expanded as Chapter 8 on shape continuity and optimization. With meaningful metric topologies, we can now speak of continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc., compact families of sets, and existence of optimal geometries. The chapter concentrates on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is defined as an infimum over a family of functions over a fixed domain or set such as the eigenvalue problems. We first characterize the continuity of the transmission problem and the upper semicontinuity of the first eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of definition since they require different constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial differential equations. An introduction is also given to the concepts and results from capacity theory from which very general families of sets stable with respect to boundary conditions can be constructed. Note that some material has been moved from one chapter to another. For instance, section 7 on the continuity of the Dirichlet boundary problem in the former Chapter 3 has been merged with the content of the former Chapter 4 in the new Chapter 8. The former Chapters 8 and 9 have become Chapters 9 and 10. They are devoted to a modern version of the shape calculus, an introduction to the tangential differential calculus, and the shape derivatives under a state equation constraint. In Chapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries. Those spaces are nonlinear and nonconvex. However, several of them have a group 1 This is in line with current trends in the literature such as in the work of the 2009 Abel Prize ´moli and G. winner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Me Sapiro [1] to identify objects up to an isometry.

xxii

Preface

structure and, in some cases, it is possible to construct C 1 -paths in the group from velocity fields. This leads to the notion of Eulerian semiderivative that is somehow the analogue of a derivative on a smooth manifold. In fact, two types of semiderivatives are of interest: the weaker Gateaux style semiderivative and the stronger Hadamard style semiderivative. In the latter case, the classical chain rule is still available even for nondifferentiable functions. In order to prepare the ground for shape derivatives, an enriched self-contained review of the pertinent material on semiderivatives and derivatives in topological vector spaces is provided. The important Chapter 10 concentrates on two generic examples often encountered in shape optimization. The first one is associated with the so-called compliance problems, where the shape functional is itself the minimum of a domain-dependent energy functional. The special feature of such functionals is that the adjoint state coincides with the state. This obviously leads to considerable simplifications in the analysis. In that case, it will be shown that theorems on the differentiability of the minimum of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the minimizer is not unique. The second one will deal with shape functionals that can be expressed as the saddle point of some appropriate Lagrangian. As in the first example, theorems on the differentiability of the saddle point of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the differentiation of the state equation with respect to the domain is particularly advantageous in shape problems.

3

Intended Audience

The targeted audience is applied mathematicians and advanced engineers and scientists, but the book is also suitable for a broader audience of mathematicians as a relatively well-structured initiation to shape analysis and calculus techniques. Some of the chapters are fairly self-contained and of independent interest. They can be used as lecture notes for a mini-course. The material at the beginning of each chapter is accessible to a broad audience, while the latter sections may sometimes require more mathematical maturity. Thus the book can be used as a graduate text as well as a reference book. It complements existing books that emphasize specific mechanical or engineering applications or numerical methods. It can be considered ´sio [9], Introduction a companion to the book of J. Soko
lowski and J.-P. Zole to Shape Optimization, published in 1992. Earlier versions of parts of this book have been used as lecture notes in graduate courses at the Universit´e de Montr´eal in 1986–1987, 1993–1994, 1995–1996, and 1997–1998 and at international meetings, workshops, or schools: S´eminaire de Math´ematiques Sup´erieures on Shape Optimization and Free Boundaries (Montr´eal, Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis (K´enitra, Morocco, December 1993), course of the COMETT MATARI European Program on Shape Optimization and Mutational Equations (Sophia-Antipolis, France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

4. Acknowledgments

xxiii

Interfaces and Transitions (Banff, Canada, August 6–18, 1995), and CIME course on Optimal Design (Troia, Portugal, June 1998).

4

Acknowledgments

The first author is pleased to acknowledge the support of the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the constant support of the National Sciences and Engineering Research Council of Canada; ´ and the FQRNT program of the Minist`ere de l’Education du Qu´ebec. Many thanks also to Louise Letendre and Andr´e Montpetit of the Centre de Recherches Math´ematiques, who provided their technical support, experience, and talent over the extended period of gestation of this book. Michel Delfour Jean-Paul Zol´esio August 13, 2009

Chapter 1

Introduction: Examples, Background, and Perspectives 1 1.1

Orientation Geometry as a Variable

The central object of this book1 is the geometry as a variable. As in the theory of functions of real variables, we need a differential calculus, spaces of geometries, evolution equations, and other familiar concepts in analysis when the variable is no longer a scalar, a vector, or a function, but is a geometric domain. This is motivated by many important problems in science and engineering that involve the geometry as a modeling, design, or control variable. In general the geometric objects we shall consider will not be parametrized or structured. Yet we are not starting from scratch, and several building blocks are already available from many fields: geometric measure theory, physics of continuous media, free boundary problems, the parametrization of geometries by functions, the set derivative as the inverse of the integral, the parametrization of functions by geometries, the Pomp´eiu–Hausdorff metric, and so on. As is often the case in mathematics, spaces of geometries and notions of derivatives with respect to the geometry are built from well-established elements of functional analysis and differential calculus. There are many ways to structure families of geometries. For instance, a domain can be made variable by considering 1 The numbering of equations, theorems, lemmas, corollaries, definitions, examples, and remarks is by chapter. When a reference to another chapter is necessary it is always followed by the words in Chapter and the number of the chapter. For instance, “equation (2.18) in Chapter 9.” The text of theorems, lemmas, and corollaries is slanted; the text of definitions, examples, and remarks is normal shape and ended by a square . This makes it possible to aesthetically emphasize certain words especially in definitions. The bibliography is by author in alphabetical order. For each author or group of coauthors, there is a numbering in square brackets starting with [1]. A ´ [3] and a reference to an reference to an item by a single author is of the form J. Dieudonne item with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae or statements are used in some chapters for two distinct purposes. First, they emphasize certain important definitions, results, or identities; second, in long proofs of some theorems, lemmas, or corollaries, they isolate key intermediary results which will be necessary to more easily follow the subsequent steps of the proof.

1

2

Chapter 1. Introduction: Examples, Background, and Perspectives

the images of a fixed domain by a family of diffeomorphisms that belong to some function space over a fixed domain. This naturally occurs in physics and mechanics, where the deformations of a continuous body or medium are smooth, or in the numerical analysis of optimal design problems when working on a fixed grid. This construction naturally leads to a group structure induced by the composition of the diffeomorphisms. The underlying spaces are no longer topological vector spaces but groups that can be endowed with a nice complete metric space structure by introducing the Courant metric. The practitioner might or might not want to use the underlying mathematical structure associated with his or her constructions, but it is there and it contains information that might guide the theory and influence the choice of the numerical methods used in the solution of the problem at hand. The parametrization of a fixed domain by a fixed family of diffeomorphisms obviously limits the family of variable domains. The topology of the images is similar to the topology of the fixed domain. Singularities that were not already present there cannot be created in the images. Other constructions make it possible to considerably enlarge the family of variable geometries and possibly open the doors to pathological geometries that are no longer open sets with a nice boundary. Instead of parametrizing the domains by functions or diffeomorphisms, certain families of functions can be parametrized by sets. A single function completely specifies a set or at least an equivalence class of sets. This includes the distance functions and the characteristic function, but also the support function from convex analysis. Perhaps the best known example of that construction is the Pomp´eiu–Hausdorff metric topology. This is a very weak topology that does not preserve the volume of a set. When the volume, the perimeter, or the curvatures are important, such functions must be able to yield relaxed definitions of volume, perimeter, or curvatures. The characteristic function that preserves the volume has many applications. It played a fundamental role in the integration theory of Henri Lebesgue at the beginning of the 20th century. It was also used in the 1950s by E. De Giorgi to define a relaxed notion of perimeter in the theory of minimal surfaces. Another technique that has been used successfully in free or moving boundary problems, such as motion by mean curvature, shock waves, or detonation theory, is the use of level sets of a function to describe a free or moving boundary. Such functions are often the solution of a system of partial differential equations. This is another way to build new tools from functional analysis. The choice of families of function parametrized sets or of families of set parametrized functions, or other appropriate constructions, is obviously problem dependent, much like the choice of function spaces of solutions in the theory of partial differential equations or optimization problems. This is one aspect of the geometry as a variable. Another aspect is to build the equivalent of a differential calculus and the computational and analytical tools that are essential in the characterization and computation of geometries. Again, we are not starting from scratch and many building blocks are already available, but many questions and issues remain open. This book aims at covering a small but fundamental part of that program. We had to make difficult choices and refer the reader to appropriate books and references for background material such as geometric measure theory and specialized topics such as homogenization theory and microstructures which are available in excellent

2. A Simple One-Dimensional Example

3

books in English. It was unfortunately not possible to include references to the considerable literature on numerical methods, free and moving boundary problems, and optimization.

1.2

Outline of the Introductory Chapter

We first give a series of generic examples where the shape or the geometry is the modeling, control, or optimization variable. They will be used in the subsequent chapters to illustrate the many ways such problems can be formulated. The first example is the celebrated problem of the optimal shape of a column formulated by Lagrange in 1770 to prevent buckling. The extremization of the eigenvalues has also received considerable attention in the engineering literature. The free interface between two regions with different physical or mechanical properties is another generic problem that can lead in some cases to a mixing or a microstructure. Two typical problems arising from applications to condition the thermal environment of satellites are described in sections 7 and 8. The first one is the design of a thermal diffuser of minimal weight subject to an inequality constraint on the output thermal power flux. The second one is the design of a thermal radiator to effectively radiate large amounts of thermal power to space. The geometry is a volume of revolution around an axis that is completely specified by its height and the function which specifies its lateral boundary. Finally, we give a glimpse at image segmentation, which is an example of shape/geometric identification problems. Many chapters of this book are of direct interest to imaging sciences. Section 10 presents some background and perspectives. A fundamental issue is to find tractable and preferably analytical representations of a geometry as a variable that are compatible with the problems at hand. The generic examples suggest two types of representations: the ones where the geometry is parametrized by functions and the ones where a family of functions is parametrized by the geometry. As is always the case, the choice is very much problem dependent. In the first case, the topology of the variable sets is fixed; in the second case the families of sets are much larger and topological changes are included. The book presents the two points of view. Finally, section 11 sketches the material in the second edition of the book.

2

A Simple One-Dimensional Example

A general feature of minimization problems with respect to a shape or a geometry subject to a state equation constraint is that they are generally not convex and that, when they have a solution, it is generally not unique. This is illustrated in ´a [2]: minimize the objective function the following simple example from J. Ce  a def 2 |ya (x) − 1| dx, J(a) = 0

where a ≥ 0 and ya is the solution of the boundary value problem (state equation) d2 ya def (x) = −2 in Ωa = (0, a), dx2

dya (0) = 0, dx

ya (a) = 0.

(2.1)

4

Chapter 1. Introduction: Examples, Background, and Perspectives

Here the one-dimensional geometric domain Ωa = ]0, a[ is the minimizing variable. We recognize the classical structure of a control problem, except that the minimizing variable is no longer under the integral sign but in the limits of the integral sign. One consequence of this difference is that even the simplest problems will usually not be convex or convexifiables. They will require a special analysis. In this example it is easy to check that the solution of the state equation is ya (x) = a2 − x2

and J(a) =

8 5 4 3 a − a + a. 15 3

The graph of J, shown in Figure 1.1, is not the graph of a convex function. Its global minimum in a0 = 0, local maximum in a1 , and local minimum in a2 ,       3 3 1 1 def def a1 = 1 − √ , a2 = 1+ √ , 4 4 3 3 are all different.

a1

a2

Figure 1.1. Graph of J(a). To avoid a trivial solution, a strictly positive lower bound must be put on a. A unique minimizing solution is obtained for a ≥ a1 where the gradient of J is zero. For 0 < a < a2 , the minimum will occur at the preset lower or upper bound on a.

3

Buckling of Columns

The next example illustrates the fact that even simple problems can be nondifferentiable with respect to the geometry. This is generic of all eigenvalue problems when the eigenvalue is not simple. One of the early optimal design problems was formulated by J. L. Lagrange [1] in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by the Danish mathematician and astronomer T. Clausen [1] in 1849. It consists in finding the best profile of a vertical column of fixed volume to prevent buckling.

3. Buckling of Columns

5

It turns out that this problem is in fact a hidden maximization of an eigenvalue. Many incorrect solutions had been published until 1992. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. The difficulty was that the eigenvalue is not simple and hence not differentiable with respect to the geometry. Consider a normalized column of unit height and unit volume (see Figure 1.2). Denote by  the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies  1  1 EI |u |2 dx −  |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For sufficiently small load  the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest  for which this minimum is zero. This is equivalent to finding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (3.1) 1  |2 dx 0=u∈V |u 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

1

x

normalized load 

cross section area A(x)

0

Figure 1.2. Column of height one and cross section area A under the load .

6

Chapter 1. Introduction: Examples, Background, and Perspectives

cross-sectional area A(x),



I(x) = c A(x)

1

A(x) dx = 1.

and 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we finally get

1

A |u |2 dx , 1  |2 dx 0=u∈V A∈A |u 0    1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(3.2) (3.3)

0

4

Eigenvalue Problems

Let D be a bounded open Lipschitzian domain in RN and A ∈ L∞ (D; L(RN , RN )) be a matrix function defined on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.1)

for some coercivity and continuity constants 0 < α ≤ β and ∗A is the transpose of A. Consider the minimization or the maximization of the first eigenvalue  sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω  inf 1 , (4.2) λ (Ω) = A |ϕ|2 dx 0=ϕ∈H0 (Ω) inf λ (Ω) Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D (cf., for instance, sections 2, 7, and 9 of Chapter 8). In the vectorial case, consider the following linear elasticity problem: find U ∈ H01 (Ω)3 such that   Cε(U )·· ε(W ) dx = F · W dx (4.3) ∀W ∈ H01 (Ω)3 , Ω

Ω

for some distributed loading F ∈ L2 (Ω)3 and a constitutive law C which is a bilinear symmetric transformation of  def def

σij τij Sym3 = τ ∈ L(R3 ; R3 ) : ∗ τ = τ , σ·· τ = 1≤i,j≤3

(L(R3 ; R3 ) is the space of all linear transformations of R3 or 3 × 3-matrices) under the following assumption. Assumption 4.1. The constitutive law is a transformation C ∈ Sym3 for which there exists a constant α > 0 such that Cτ ·· τ ≥ α τ ·· τ for all τ ∈ Sym3 .

5. Optimal Triangular Meshing

7

For instance, for the Lam´e constants µ > 0 and λ ≥ 0, the special constitutive law Cτ = 2µ τ + λ tr τ I satisfies Assumption 4.1 with α = 2µ. The associated bilinear form is  def Cε(U )·· ε(W ) dx, aΩ (U, W ) = Ω

where U is a vector function, D(U ) is the Jacobian matrix of U , and def

ε(U ) =

1 (D(U ) + ∗D(U )) 2

is the strain tensor. The first eigenvalue is the minimum of the Rayleigh quotient   aΩ (U, U ) 1 3 (Ω) , U = 0 . : ∀U ∈ H λ(Ω) = inf  0 |U |2 dx Ω A typical problem is to find the sensitivity of the first eigenvalue with respect to the shape of the domain Ω. In 1907, J. Hadamard [1] used displacements along the normal to the boundary Γ of a C ∞ -domain to compute the derivative of the first eigenvalue of the clamped plate. As in the case of the column, this problem is not differentiable with respect to the geometry when the eigenvalue is not simple.

5

Optimal Triangular Meshing

The shape calculus that will be developed in Chapters 9 and 10 for problems governed by partial differential equations (the continuous model ) will be readily applicable to their discrete model as in the finite element discretization of elliptic boundary value problems. However, some care has to be exerted in the choice of the formula for the gradient, since the solution of a finite element discretization problem is usually less smooth than the solution of its continuous counterpart. Most shape objective functionals will have two basic formulas for their shape gradient: a boundary expression and a volume expression. The boundary expression is always nicer and more compact but can be applied only when the solution of the underlying partial differential equation is smooth and in most cases smoother than the finite element solution. This leads to serious computational errors. The right formula to use is the less attractive volume expression that requires only the same smoothness as the finite element solution. Numerous computational experiments confirm that fact (cf., for instance, E. J. Haug and J. S. Arora [1] or E. J. Haug, K. K. Choi, and V. Komkov [1]). With the volume expression, the gradient of the objective function with respect to internal and boundary nodes can be readily obtained by plugging in the right velocity field. A large class of linear elliptic boundary value problems can be expressed as the minimum of a quadratic function over some Hilbert space. For instance, let Ω be a bounded open domain in RN with a smooth boundary Γ. The solution u of the boundary value problem −∆u = f in Ω,

u = 0 on Γ

8

Chapter 1. Introduction: Examples, Background, and Perspectives

is the minimizing element in the Sobolev space H01 (Ω) of the energy functional  def E(v, Ω) = |∇v|2 − 2f v dx, Ω  def |∇u|2 dx. J(Ω) = inf1 E(v, Ω) = E(u, Ω) = − v∈H0 (Ω)

Ω

The elements of this problem are a Hilbert space V , a continuous symmetrical coercive bilinear form on V , and a continuous linear form  on V . With this notation ∃u ∈ V,

E(u) = inf E(v), v∈V

def

E(v) = a(v, v) − 2 (v)

and u is the unique solution of the variational equation ∃u ∈ U, ∀v ∈ V,

a(u, v) = (v).

In the finite element approximation of the solution u, a finite-dimensional subspace Vh of V is used for some small mesh parameter h. The solution of the approximate problem is given by ∃uh ∈ Vh ,

E(uh ) = inf E(vh ), vh ∈Vh

⇒ ∃uh ∈ Uh , ∀vh ∈ Vh ,

a(uh , vh ) = (vh ).

It is easy to show that the error can be expressed as follows: a(u − uh , u − uh ) = u − uh 2V = 2 [E(uh ) − E(u)] . Assume that Ω is a polygonal domain in RN . In the finite element method, the ¯ domain is partitioned into a set τh of small triangles by introducing nodes in Ω def

M = {Mi ∈ Ω : 1 ≤ i ≤ p}

def

def

∂M = {Mi ∈ ∂Ω : p + 1 ≤ i ≤ p + q}

and M = M ∪ ∂M

for some integers p ≥ N + 1 and q ≥ 1 (see Figure 1.3). Therefore the triangularization τh = τh (M ), the solution space Vh = Vh (M ), and the solution uh = uh (M ) are functions of the positions of the nodes of the set M . Assuming that the total 0

0

0 Mi

1 0

0 0

Figure 1.3. Triangulation and basis function associated with node Mi .

5. Optimal Triangular Meshing

9

number of nodes is fixed, consider the following optimal triangularization problem: def

inf j(M ), M

j(M ) = E(uh (τh (M )), Ω) = 

inf

vh ∈Vh (M )

E(vh , Ω),

 |∇(u − uh )|2 dx = 2 E(uh , Ω(τh (M ))) − E(u, Ω) Ω 

= 2 J(Ω(τh (M ))) − J(Ω) ,  def |∇uh |2 dx. J(Ω(τh (M ))) = inf E(vh , Ω(τh (M ))) = E(uh , Ω(τh (M ))) = − u −

uh 2V

=

vh ∈Vh

Ω

The objective is to compute the partial derivative of j(M ) with respect to the th component (Mi )
of the node Mi : ∂j (M ). ∂(Mi )
This partial derivative can be computed by using the velocity method for the special ´sio [3]) velocity field (cf. M. C. Delfour, G. Payre, and J.-P. Zole Vi
(x) = bMi (x) e
, where bMi ∈ Vh is the (piecewise P 1 ) basis function associated with the node Mi : bMi (Mj ) = δij for all i, j. In that method each point X of the plane is moved according to the solution of the vector differential equation dx (t) = V (x(t)), dt

x(0) = X.

def

This yields a transformation X → Tt (X) = x(t; X) : R2 → R2 of the plane, and it is natural to introduce the following notion of semiderivative: def

dJ(Ω; V ) = lim

t0

J(Tt (Ω)) − J(Ω) . t

For t ≥ 0 small, the velocity field must be chosen in such a way that triangles are moved onto triangles and the point Mi is moved in the direction e
: Mi → Mit = Mi + t e
. This is achieved by choosing the following velocity field: Vi
(t, x) = bMit (x) e
, where bMit is the piecewise P 1 basis function associated with node Mit : bMit (Mj ) = δij for all i, j. This yields the family of transformations Tt (x) = x + t bMi (x) e
which moves the node Mi to Mi + t e
and hence ∂j (M ) = dJ(Ω; Vi
). ∂(Mi )


10

Chapter 1. Introduction: Examples, Background, and Perspectives Going back to our original example, introduce the shape functional   def 2 |∇u| dx, E(Ω, v) = |∇v|2 − 2 f v dx. J(Ω) = inf1 E(Ω, v) = − v∈H0 (Ω)

Ω

Ω

In Chapter 9, we shall show that we have the following boundary and volume expressions for the derivative of J(Ω):

 dJ(Ω; V ) =

 2 ∂u dJ(Ω; V ) = − V · n dΓ, Γ ∂n A (0)∇u · ∇u − 2 [div V (0)f + ∇f · V (0)] u dx,

Ω 

A (0) = div V (0) I − ∗DV (0) − DV (0). For a P 1 -approximation  def

¯ : v|K ∈ P 1 (K), ∀K ∈ τh Vh = v ∈ C 0 (Ω) and the trace of the normal derivative on Γ is not defined. Thus, it is necessary to use the volume expression. For the velocity field Vi
DVi
= e
∗∇bMi , div DVi
= e
· ∇bMi , A (0) = e
· ∇bMi I − e
∗∇bMi − ∇bMi ∗ e
. Since ∂j (M ) = dJ(Ω; Vi
), ∂(Mi )
we finally obtain the formula for the derivative of the function j(M ) with respect to node Mi in the direction e
:  ∂j (M ) = [ e
· ∇bMi I − e
∗∇bMi − ∇bMi ∗ e
] ∇; uh · ∇uh ∂(Mi )
Ω − 2 [ e
· ∇bMi f + ∇f · e
bMi ] uh dx. Since the support of bMi consists of the triangles having Mi as a vertex, the gradient with respect to the nodes can be constructed piece by piece by visiting each node.

6

Modeling Free Boundary Problems

The first step towards the solution of a shape optimization is the mathematical modeling of the problem. Physical phenomena are often modeled on relatively smooth or nice geometries. Adding an objective functional to the model will usually push the system towards rougher geometries or even microstructures. For instance, in the optimal design of plates the optimization of the profile of a plate led to highly oscillating profiles that looked like a comb with abrupt variations ranging from zero

6. Modeling Free Boundary Problems

11

to maximum thickness. The phenomenon began to be understood in 1975 with the paper of N. Olhoff [1] for circular plates with the introduction of the mechanical notion of stiffeners. The optimal plate was a virtual plate, a microstructure, that is a homogenized geometry. Another example is the Plateau problem of minimal surfaces that experimentally exhibits surfaces with singularities. In both cases, it is mathematically natural to replace the geometry by a characteristic function, a function that is equal to 1 on the set and 0 outside the set. Instead of optimizing over a restricted family of geometries, the problem is relaxed to the optimization over a set of measurable characteristic functions that contains a much larger family of geometries, including the ones with boundary singularities and/or an arbitrary number of holes.

6.1

Free Interface between Two Materials

´a and K. Malanowski [1] Consider the optimal design problem studied by J. Ce in 1970, where the optimization variable is the distribution of two materials with different physical characteristics within a fixed domain D. It cannot a priori be assumed that the two regions are separated by a smooth interface and that each region is connected. This problem will be covered in more details in section 4 of Chapter 5. Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume for the moment that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth interface ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 1.4. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem   − k1 y = f in Ω1  y = 0 on ∂D

and − k2 y = f in Ω2 , ∂y ∂y + k2 = 0 on Ω1 ∩ Ω2 , and k1 ∂n1 ∂n2

(6.1)

where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Assume that k1 > k2 . The objective is to maximize the equivalent of the compliance  f y dx (6.2) J(Ω1 ) = − D

Ω1

Ω2

Figure 1.4. Fixed domain D and its partition into Ω1 and Ω2 .

12

Chapter 1. Introduction: Examples, Background, and Perspectives

over all domains Ω1 in D subject to the following constraint on the volume of material k1 which occupies the part Ω1 of D: m(Ω1 ) ≤ α,

0 < α < m(D)

(6.3)

for some constant α. If χ denotes the characteristic function of the domain Ω1 , / Ω1 , χ(x) = 1 if x ∈ Ω1 and 0 if x ∈ the compliance J(χ) = J(Ω1 ) can be expressed as the infimum over the Sobolev space H01 (D) of an energy functional defined on the fixed set D: J(χ) = def



min

ϕ∈H01 (D)

E(χ, ϕ),

(k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx.

E(χ, ϕ) =

(6.4) (6.5)

D

J(χ) can be minimized or maximized over some appropriate family of characteristic functions or with respect to their relaxation to functions between 0 and 1 that would correspond to microstructures. As in the eigenvalue problem, the objective function is an infimum, but here the infimum is over a space that does not depend on the function χ that specifies the geometric domain. This will be handled by the special techniques of Chapter 10 for the differentiation of the minimum of a functional.

6.2

Minimal Surfaces

The celebrated Plateau’s problem, named after the Belgian physicist and professor J. A. F. Plateau [1] (1801–1883), who did experimental observations on the geometry of soap films around 1873, also provides a nice example where the geometry is a variable. It consists in finding the surface of least area among those bounded by a given curve. One of the difficulties in studying the minimal surface problem is the description of such surfaces in the usual language of differential geometry. For instance, the set of possible singularities is not known. Measure theoretic methods such as k-currents (k-dim surfaces) were used by E. R. Reifenberg [1, 2, 3, 4] around 1960, H. Federer and W. H. Fleming [1] in 1960 (normals and integral currents), F. J. Almgren, Jr. [1] in 1965 (varifolds), and H. Federer [5] in 1969. In the early 1950s, E. De Giorgi [1, 2, 3] and R. Caccioppoli [1] considered a hypersurface in the N -dimensional Euclidean space RN as the boundary of a set. In order to obtain a boundary measure, they restricted their attention to sets whose characteristic function is of bounded variation. Their key property is an associated natural notion of perimeter that extends the classical surface measure of the boundary of a smooth set to the larger family of Caccioppoli sets named after the celebrated Neapolitan mathematician Renato Caccioppoli.2 2 In 1992 his tormented personality was remembered in a film directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano).

7. Design of a Thermal Diffuser

13

Caccioppoli sets occur in many shape optimization problems (or free boundary problems), where a surface tension is present on the (free) boundary, such as in the free interface water/soil in a dam (C. Baiocchi, V. Comincioli, E. Magenes, and G. A. Pozzi [1]) in 1973 and in the free boundary of a water wave (M. Souli ´sio [1, 2, 3, 4, 5]) in 1988. More details will be given in Chapter 5. and J.-P. Zole

7

Design of a Thermal Diffuser

Shape optimization problems are everywhere in engineering, physics, and medicine. We choose two illustrative examples that were proposed by the Canadian Space Program in the 1980s. The first one is the design of a thermal diffuser to condition the thermal environment of electronic devices in communication satellites; the second one is the design of a thermal radiator that will be described in the next section. There are more and more design and control problems coming from medicine. For instance, the design of endoprotheses such as valves, stents, and coils in blood vessels or left ventricular assistance devices (cardiac pumps) in interventional cardiology helps to improve the health of patients and minimize the consequences and costs of therapeutical interventions by going to mini-invasive procedures.

7.1

Description of the Physical Problem

This problem arises in connection with the use of high-power solid-state devices (HPSSD) in communication satellites (cf. M. C. Delfour, G. Payre, and J.´sio [1]). An HPSSD dissipates a large amount of thermal power (typ. P. Zole > 50 W) over a relatively small mounting surface (typ. 1.25 cm2 ). Yet, its junction temperature is required to be kept moderately low (typ. 110◦ C). The thermal resistance from the junction to the mounting surface is known for any particular HPSSD (typ. 1◦ C/W), so that the mounting surface is required to be kept at a lower temperature than the junction (typ. 60◦ C). In a space application the thermal power must ultimately be dissipated to the environment by the mechanism of radiation. However, to radiate large amounts of thermal power at moderately low temperatures, correspondingly large radiating areas are required. Thus we have the requirement to efficiently spread the high thermal power flux (TPF) at the HPSSD source (typ. 40 W/cm2 ) to a low TPF at the radiator (typ. 0.04 W/cm2 ) so that the source temperature is maintained at an acceptably low level (typ. < 60◦ C) at the mounting surface. The efficient spreading task is best accomplished using heatpipes, but the snag in the scheme is that heatpipes can accept only a limited maximum TPF from a source (typ. max 4 W/cm2 ). Hence we are led to the requirement for a thermal diffuser. This device is inserted between the HPSSD and the heatpipes and reduces the TPF at the source (typ. > 40 W/cm2 ) to a level acceptable to the heatpipes (typ. > 4 W/cm2 ). The heatpipes then sufficiently spread the heat over large space radiators, reducing the TPF from a level at the diffuser (typ. 4 W/cm2 ) to that at the radiator (typ. 0.04 W/cm2 ). This scheme of heat spreading is depicted in Figure 1.5. It is the design of the thermal diffuser which is the problem at hand. We may assume that the HPSSD presents a uniform thermal power flux to the diffuser

14

Chapter 1. Introduction: Examples, Background, and Perspectives schematic drawing not to scale

radiator to space heatpipes

heatpipe saddle

thermal diffuser

high-power solid-state diffuser

Figure 1.5. Heat spreading scheme for high-power solid-state devices. at the HPSSD/diffuser interface. Heatpipes are essentially isothermalizing devices, and we may assume that the diffuser/heatpipe saddle interface is indeed isothermal. Any other surfaces of the diffuser may be treated as adiabatic.

7.2

Statement of the Problem

Assume that the thermal diffuser is a volume Ω symmetrical about the z-axis (cf. Figure 1.6 (A)) whose boundary surface is made up of three regular pieces: the mounting surface Σ1 (a disk perpendicular to the z-axis with center in (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the interface Σ3 between the diffuser and the heatpipe saddle (a disk perpendicular to the z-axis with center in (r, z) = (0, L)). The temperature distribution over this volume Ω is the solution of the stationary heat equation k∆T = 0 (∆T , the Laplacian of T ) with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

T = T3 (constant) on Σ3 ,

(7.1)

where n always denotes the outward unit normal to the boundary surface Σ and ∂T /∂n is the normal derivative to the boundary surface Σ, ∂T = ∇T · n (∇T = the gradient of T ). ∂n

(7.2)

The parameters appearing in (7.1) are k = thermal conductivity (typ. 1.8W/cm×◦ C), qin = uniform inward thermal power flux at the source (positive constant).

7. Design of a Thermal Diffuser

15

The radius R0 of the mounting surface Σ1 is fixed so that the boundary surface Σ1 is already given in the design problem. For practical considerations, we assume that the diffuser is solid without interior hollows or cutouts. The class of shapes for the diffuser is characterized by the design parameter L > 0 and the positive function R(z), 0 < z ≤ L, with R(0) = R0 > 0. They are volumes of revolution Ω about the z-axis generated by the surface A between the z-axis and the function R(z) (cf. Figure 1.6 (B)), that is,  def

Ω = (x, y, z) : 0 < z < L, x2 + y 2 < R(z)2 .

(7.3)

So the shape of Ω is completely specified by the length L > 0 and the function R(z) > 0 on the interval [0, L]. z

z Σ3

L

ζ

L A R(z)

S3

1 S4

D

Σ2 0

0

Σ1

S2 y

0

0

y

R0

(A)

0

0

(B)

1 S1

ρ (C)

Figure 1.6. (A) Volume Ω and its boundary Σ; (B) Surface A generating
Ω; (C) Surface D generating Ω. Assuming that the diffuser is made up of a homogeneous material of uniform density (no hollow) the design objective is to minimize the volume def





J(Ω) =

L

R(z)2 dz

dx = π Ω

(7.4)

0

subject to a uniform constraint on the outward thermal power flux at the interface Σ3 between the diffuser and the heatpipe saddle: sup −k

p∈Σ3

∂T ∂T (p) ≤ qout or k + qout ≥ 0 on Σ3 , ∂z ∂n

(7.5)

where qout is a specified positive constant. It is readily seen that the minimization problem (7.4) subject to the constraint (7.5) (where T is the solution of the heat equation with the boundary conditions (7.1)) is independent of the fixed temperature T3 on the boundary Σ3 . In other words the optimal shape Ω, if it exists, is independent of T3 . As a result, from now on we set T3 equal to 0.

16

7.3

Chapter 1. Introduction: Examples, Background, and Perspectives

Reformulation of the Problem

In a shape optimization problem the formulation is important from both the theoretical and the numerical viewpoints. In particular condition (7.5) is difficult to numerically handle since it involves the pointwise evaluation of the normal derivative on the piece of boundary Σ3 . This problem can be reformulated as the minimization of T on Σ3 , where T is now the solution of a variational inequality. Consider the following minimization problem over the subspace of functions that are positive or zero on Σ3 :  def

(7.6) V + (Ω) = v ∈ H 1 (Ω) : v|Σ3 ≥ 0 ,    1 inf |∇v|2 dx − qin v dΣ + qout v dΣ. (7.7) v∈V + (Ω) Ω 2 Σ1 Σ3 H 1 (Ω) is the usual Sobolev space on the domain Ω, and the inequality on Σ3 has to be interpreted quasi-everywhere in the capacity sense. Leaving aside those technicalities, the minimizing solution of (7.7) is characterized by ∂T ∂T = qin on Σ1 , k = 0 on Σ2 , −k ∆T = 0 in Ω, k ∂n ∂n     ∂T ∂T + qout ≥ 0, T k + qout = 0 on Σ3 . T ≥ 0, k ∂n ∂n

(7.8)

The former constraint (7.5) is verified and replaced by the new constraint T = 0 on Σ3 .

(7.9)

If there exists a nonempty domain Ω of the form (7.3) such that T = 0 on Σ3 , the problem is feasible. In this formulation the pointwise constraint on the normal derivative of the temperature on Σ3 has been replaced by a pointwise constraint on the less demanding trace of the temperature on Σ3 . Yet, we now have to solve a variational inequality instead of a variational equation for the temperature T .

7.4

Scaling of the Problem

In the above formulations the shape parameter L and the shape function R are not independent of each other since the function R is defined on the interval [0, L]. This motivates the following changes of variables and the introduction of the dimensionless temperature y: x y z , y → ξ2 = , z → ζ = , 0 ≤ ζ ≤ 1, x → ξ1 = R0 R0 L R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0 k y(ξ1 , ξ2 , ζ) = T (R0 ξ1 , R0 ξ2 , Lζ), Lqin   def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

7. Design of a Thermal Diffuser

17

˜ now appears as a coefficient in the partial differential equation The parameter L  2  ∂ y ∂2y ∂2y 2 ˜ L + 2 + 2 = 0 in D 2 ∂ξ1 ∂ξ2 ∂ζ with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D: ∂y = 1 on S1 , ∂νA

∂y = 0 on S2 , ∂νA

y = 0 on S3 ,

(7.10)

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface S,   ∂y ∂y ∂y ∂y 2 ˜ + ν3 . = L ν1 + ν2 ∂νA ∂ξ1 ∂ξ2 ∂ζ Finally, the optimal design problem depends only on the ratio q = qout /qin through the constraint ∂y qout + ≥ 0 on S3 . ∂νA qin ˜ > 0 and the function R ˜ > 0 now defined The design variables are the parameter L on the fixed interval [0, 1].

7.5

Design Problem

The fact that this specific design problem can be reduced to finding a parameter and a function gives the false or unfounded impression that it can now be solved by standard mathematical programming and numerical methods. Early work on such problems revealed a different reality, such as oscillating boundaries and convergence towards nonphysical designs. Clearly, the geometry refused to be handled by standard methods without a better understanding of the underlying physics and its inception in the modeling of the geometric variable. At the theoretical level, the existence of solution requires a concept of continuity with respect to the geometry of the solution of either the heat equation with an inequality constraint on the TPF or the variational inequality with an equality constraint on the temperature. The other element is the lower semicontinuity of the objective functional that is not too problematic for the volume functional as long as the chosen topology on the geometry preserves the continuity of the volume functional. For instance, the classical Hausdorff metric topology does not preserve the volume. In the context of fluid mechanics (cf., for instance, O. Pironneau [1]), it means that a drag minimizing sequence of sets with constant volume may converge to a set with twice the volume (cf. Example 4.1 in Chapter 6). A wine making industry exploiting the convergence in the Hausdorff metric topology could yield miraculous profits. Other serious issues are, for instance, the lack of differentiability of the solution of a variational inequality at the continuous level that will inadvertently affect the differentiability or the evolution of a gradient method at the discrete level. We shall

18

Chapter 1. Introduction: Examples, Background, and Perspectives

see that there is not only one topology for shapes but a whole range that selectively preserve some but not all of the geometrical features. Again the right choice is problem dependent, much like the choice of the right Sobolev space in the theory of partial differential equations.

8

Design of a Thermal Radiator

Current trends indicate that future communications satellites and spacecrafts will grow ever larger, consume ever more electrical power, and dissipate larger amounts of thermal energy. Various techniques and devices can be deployed to condition the thermal environment for payload boxes within a spacecraft, but it is desirable to employ those which offer good performance for low cost, low weight, and high reliability. A thermal radiator (or radiating fin) which accepts a given TPF from a payload box and radiates it directly to space can offer good performance and high reliability at low cost. However, without careful design, such a radiator can be unnecessarily bulky and heavy. It is the mass-optimized design of the thermal radiator ´which is the problem at hand (cf. M. C. Delfour, G. Payre, and J.-P. Zole sio [2]). We may assume that the payload box presents a uniform TPF (typ. 0.1 to 1.0 W/cm2 ) into the radiator at the box/radiator interface. The radiating surface is a second surface mirror which consists of a sheet of glass whose inner surface has silver coating. We may assume that the TPF out of the radiator/space interface is governed by the T 4 radiation law, although we must account also for a constant TPF (typ. 0.01 W/cm2 ) into this interface from the sun. Any other surfaces of the radiator may be treated as adiabatic. Two constraints restrict freedom in the design of the thermal radiator: (i) the maximum temperature at the box/radiator interface is not to exceed some constant (typ. 50◦ ); and (ii) no part of the radiator is to be thinner than some constant (typ. 1 mm).

8.1

Statement of the Problem

Assume that the radiator is a volume Ω symmetrical about the z-axis (cf. Figure 1.7) whose boundary surface is made up of three regular pieces: the contact surface Σ1 (a disk perpendicular to the z-axis with center at the point (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the radiating surface Σ3 (a disk perpendicular to the z-axis with center at (r, z) = (0, L)). More precisely 

Σ1 = (x, y, z) : z = 0 and x2 + y 2 ≤ R02 ,

 Σ2 = (x, y, z) : x2 + y 2 = R(z)2 , 0 ≤ z ≤ L , (8.1) 

2 2 2 Σ3 = (x, y, z) : z = L and x + y ≤ R(L) , where the radius R0 > 0 (typ. 10 cm), the length L > 0, and the function R : [0, L] → R,

R(0) = R0 ,

are given (R, the field of real numbers).

R(z) > 0, 0 ≤ z ≤ L,

(8.2)

8. Design of a Thermal Radiator

19

z

z

Σ3

L Σ2 y

0

0

R0 Σ1

r

Figure 1.7. Volume Ω and its cross section. The temperature distribution (in Kelvin degrees) over the volume Ω is the solution of the stationary heat equation ∆T = 0

(the Laplacian of T )

(8.3)

with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

k

∂T = −σεT |T |3 + qs on Σ3 , ∂n

(8.4)

where n denotes the outward normal to the boundary surface Σ, ∂T /∂n is the normal derivative on the boundary surface Σ, and ∂T = ∇T · n (∇T = the gradient of T ). ∂n The parameters appearing in (8.1)–(8.4) are k = thermal conductivity (typ. 1.8 W/cm×◦ C), qin = uniform inward thermal power flux at the source (typ. 0.1 to 1.0 W/cm2 ), σ = Boltzmann’s constant (5.67 × 10−8 W/m2 K4 ), ε = surface emissivity (typ. 0.8), qs = solar inward thermal power flux (0.01 W/cm2 ). The optimal design problem consists in minimizing the volume  L def R(z)2 dz J(Ω) = π

(8.5)

0

over all length L > 0 and shape function R subject to the constraint T (x, y, z) ≤ Tf

(typ. 50◦ C),

∀(x, y, z) ∈ Σ1 .

In this analysis we shall drop the requirement (ii) in the introduction.

(8.6)

20

8.2

Chapter 1. Introduction: Examples, Background, and Perspectives

Scaling of the Problem

As in the case of the diffuser, it is convenient to introduce the following dimensionless coordinates and temperature y (see Figure 1.8): x → ξ1 =

x , R0

y → ξ2 =

y , R0

z → ζ =

z , L

0 ≤ ζ ≤ 1,

R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0  1/3 σεR0 T (R0 ξ1 , R0 ξ2 , Lζ), y(ξ1 , ξ2 , ζ) = k   def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

˜ now appears as a coefficient in the partial differential equation The parameter L  2  ∂ y ∂2y ∂ 2y 2 ˜ L + = 0 in D + ∂ξ12 ∂ξ22 ∂ζ 2 with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D:   1/3 qin ∂y σεR0 ˜ on S1 , R0 =L ∂νA k k ∂y =0 ∂νA

(8.7)

on S2 ,

∂y 3 ˜ ˜ + Ly|y| =L ∂νA



σεR0 k

1/3



qin qs R0 k qin

on S3 ,

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface Σ,   ∂y ˜ 2 ν1 ∂y + ν2 ∂y + ν3 ∂y . =L ∂νA ∂ξ1 ∂ξ2 ∂ζ

ζ

ζ

˜3 Σ

1

˜2 Σ

S3

1 A

r 1 ˜1 Σ

0

S2 r

0

1 S1


and its generating surface A. Figure 1.8. Volume Ω

9. A Glimpse into Segmentation of Images

9

21

A Glimpse into Segmentation of Images

The study of the problem of linguistic or visual perceptions was initiated by several pioneering authors, such as H. Blum [1] in 1967, D. Marr and E. Hildreth [1] in 1980, and D. Marr [1] in 1982. It involves specialists of psychology, artificial intelligence, and experimentalists such as D. H. Hubel and T. N. Wiesel [1] in 1962 and F. W. Campbell and J. G. Robson [1] in 1968. In the first part of this section, we revisit the pioneering work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image by convolution with a sufficiently differentiable normalized function as a function of the scaling parameter. We extend the space-frequency uncertainty principle to N -dimensional images. It is the analogue of the Heisenberg uncertainty principle of quantum mechanics. We revisit the Laplacian filter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. In the second part, we show how shape analysis methods and the shape and tangential calculuses can be applied to objective functionals defined on the whole contour of an image. We anticipate Chapters 9 and 10 on shape derivatives by the velocity method and show how they can be applied to snakes, active geodesic contours, and level sets (cf., for instance, the book of S. Osher and N. Paragios [1]). It shows that the Eulerian shape semiderivative is the basic ingredient behind those representations, including the case of the oriented distance function. In all cases, the evolution equation for the continuous gradient descent method is shown to have the same structure. For more material along the same lines, the reader is referred to M. Dehaes [1] and M. Dehaes and M. C. Delfour [1].

9.1

Automatic Image Processing

The first level of image processing is the detection of the contours or the boundaries of the objects in the image. For an ideal image I : D → R defined in an open two-dimensional frame D with values in an interval of greys continuously ranging from white to black (see Figure 1.9), the edges of an object correspond to the loci of discontinuity of the image I (cf. D. Marr and E. Hildreth [1]), also called “step edges” by D. Marr [1]. As can be seen from Figure 1.9, the loci of discontinuity may only reveal part of an object hidden by another one and a subsequent and different level of processing is required. A more difficult case is the detection of black curves or cracks in a white frame (cf. Figure 1.10), where the function I becomes a measure supported by the curves rather than a function. In practice, the frame D of the image is divided into periodically spaced cells P (square, hexagon, diamond) with a quantized value or pixel from 256 grey levels. For small squares a piecewise linear continuous interpolation or higher degree C 1 -interpolation can be used to remove the discontinuities at the intercell boundaries. In addition, observations and measurements introduce noise or perturbations and the interpolated image I needs to be further smoothed or filtered. When the characteristics of the noise are known, an appropriate filter can do the job (see, for instance, the use of low pass filters in A. Rosenfeld and M. Thurston [1], A. P. Witkin [1], and A. L. Yuille and T. Poggio [1]).

22

Chapter 1. Introduction: Examples, Background, and Perspectives

frame D levels of grey I

open set Ω

Figure 1.9. Image I of objects and their segmentation in the frame D.

frame D

Figure 1.10. Image I containing black curves or cracks in the frame D. Given an ideal grey level image I defined in a fixed bounded open frame D we want to identify the edges or boundaries of the objects contained in the image as shown in Figure 1.9. The edge identification or segmentation problem is a lowlevel processing of a real image which should also involve higher-level processings or tasks (cf. M. Kass, M. Witkin, and D. Terzopoulos [1]). Intuitively the edges coincide with the loci of discontinuities of the image function I. At such points the norm |∇I| of the gradient ∇I is infinite. Roughly speaking the segmentation of an ideal image I consists in finding the loci of discontinuity of the function I. The idea is now to first smooth I by convolution with a sufficiently differentiable function. This operation will be followed and/or combined with the use of an edge detector as the zero crossings of the Laplacian. In the literature the term filter often applies to both the filter and the detector. In this section we shall make the distinction between the two operations.

9.2

Image Smoothing/Filtering by Convolution and Edge Detectors

In this section we revisit the work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image I by convolution with a sufficiently differentiable normalized function ρ as a function of the scaling parameter ε > 0. In the second part of this

9. A Glimpse into Segmentation of Images

23

section we revisit the Laplacian filter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. 9.2.1

Construction of the Convolution of I

Let ρ : RN → R, N ≥ 1, be a sufficiently smooth function such that  ρ(x) dx = 1. ρ≥0 and RN

Associate with the image I, ρ, and a scaling parameter ε > 0 the normalized convolution    x−y 1 def Iε (x) = (I ∗ ρε )(x) = N dy, (9.1) I(y) ρ ε ε RN where for x ∈ RN def

ρε (x) =

1 x ρ εN ε

 and RN

ρε (x) dx = 1.

(9.2)

The function ρε plays the role of a probability density and ρε (x) dx of a probability measure. Under appropriate conditions Iε converges to the original image I as ε → 0. For a sufficiently small ε > 0 the loci of discontinuity of the function I are transformed into loci of strong variation of the gradient of the convolution Iε . When ρ has compact support, the convolution acts locally around each point in a neighborhood whose size is of order ε. A popular choice for ρ is the Gaussian with integral normalized to one in RN defined by  2 1 1 def e− 2 |x| and GN (x) dx = 1 GN (x) = √ ( 2π)N RN and the normalized Gaussian of variance ε GN ε (x) =

1 x 2 1 1 √ e− 2 | ε | N ε ( 2π)N

 and RN

GN ε (x) dx = 1.

(9.3)

As noted in L. Alvarez, P.-L. Lions, and J.-M. Morel [1] in dimension N = 2, the function u(t) = G2√2t ∗ I is the solution of the parabolic equation ∂u (t, x) = ∆u(t, x), ∂t 9.2.2

u(0, x) = I(x).

Space-Frequency Uncertainty Relationship

N It is interesting to compute the Fourier transform F(GN ε ) of Gε to make explicit the N relationship between the mean square deviations of Gε and its Fourier transform N F(GN ε ). For ω ∈ R , define the Fourier transform of a function f by  1 def f (x) e−i ω·x . (9.4) F(f )(ω) = √ ( 2π)N RN

24

Chapter 1. Introduction: Examples, Background, and Perspectives

In applications the integral of the square of f often corresponds to an energy. We shall refer to the L2 -norm of f as the energy norm and the function f 2 (x) as the energy density. The Fourier transform (9.4) of the normalized Gaussian (9.3) is  2 1 1 1 def −i ω·x √ )(ω) = GN dx = √ e− 2 |εω| . (9.5) F(GN ε ε (x) e N N ( 2π) ( 2π) RN D. Marr and E. Hildreth [1] notice that there is an uncertainty relationship between the mean square deviation with respect to the energy density f (x)2 and the mean square deviation with respect to the energy density F(f )(ω)2 of its Fourier transform in dimension 1 using a result√of R. Bracewell [1, pp. 160–161], where he uses the constant 1/(2π) instead of 1/ 2π in the definition of the Fourier transform that yields 1/(4π) instead of 1/2 as a lower bound to the product of the two mean square deviations. This uncertainty relationship generalizes to dimension N . First define the notions of centro¨ıd and variance with respect to the energy density f (x)2 . Given a function f ∈ L1 (RN ) ∩ L2 (RN ) and x ∈ RN , define the centro¨ıd as  2 N x f (x) dx x = R f (x)2 dx RN def

(9.6)

and the variance as  def

∆x2 = x − x2 =

|x|2 f (x)2 dx − |x|2 . 2 dx f (x) RN

RN

Theorem 9.1 (Uncertainty relationship). Given N ≥ 1 and f ∈ W 1,1 (RN ) ∩ W 1,2 (RN ) such that −ix f ∈ L1 (RN ) ∩ L2 (RN ), ∆x ∆ω ≥

N . 2

(9.7)

For f = GN ε 1 x 2 1 1 N ε2 √ , e− 2 | ε | ⇒ ∆x2 = N ε ( 2π)N 2 2 1 1 N fˆ(ω) = F(GN e− 2 |εω| ⇒ ∆ω2 = ε )(ω) = √ 2 ε2 ( 2π)N

f (x) = GN ε (x) =

⇒ ∆x∆ω =

N 2

N for GN ε and F(Gε ).

(9.8) (9.9) (9.10)

So the normalized Gaussian filter is indeed an optimal filter since it achieves the lower bound in all dimensions as stated by D. Marr and E. Hildreth [1] in dimension 1 in 1980. In the context of quantum mechanics, this relationship is the analogue of the Heisenberg uncertainty principle.

9. A Glimpse into Segmentation of Images 9.2.3

25

Laplacian Detector

One way to detect the edges of a regular object is to start from the convolutionsmoothed image Iε . Given a direction v, |v| = 1, and a point x ∈ R2 an edge point will correspond to a local minimum or maximum of the directional derivative def f (t) = ∇Iε (x + tv) · v with respect to t. Denote by tˆ such a point. Then a necessary condition for an extremal is D2 Iε (x + tˆv)v · v = 0. The point x ˆ = x + tˆv is a zero crossing following the terminology of D. Marr and E. Hildreth [1] of the second-order directional derivative in the direction v. Thus we are looking for the pairs (ˆ x, vˆ) verifying the necessary condition D2 Iε (ˆ x)ˆ v · vˆ = 0

(9.11)

and, more precisely, lines or curves C such that ∀x ∈ C, ∃v(x), |v(x)| = 1,

D2 Iε (x)v(x) · v(x) = 0.

(9.12)

This condition is necessary in order that, in each point x ∈ C, there exists a direction v(x) such that ∇Iε (x) · v is extremal. In order to limit the search to points x rather than to pairs (x, v), the Laplacian detector was introduced in D. Marr and E. Hildreth [1] under two assumptions: the linear variation of the intensity along the edges C of the object and the condition of zero crossing of the second-order derivative in the direction normal to C. Assumption 9.1 (Linear variation condition). The intensity Iε in a neighborhood of lines parallel to C is locally linear (affine). Assumption 9.2. The zero-crossing condition is verified in all points of C in the direction of the normal n at the point, that is, D2 Iε n · n = 0 on C.

(9.13)

Under those two assumptions and for a line C, it is easy to show that the points of C verify the necessary condition ∆Iε = 0 on C.

(9.14)

Such conditions can be investigated for edges C that are the boundary Γ of a smooth domain Ω ⊂ RN of class C 2 by using the tangential calculus developed in section 5.2 of Chapter 9 and M. C. Delfour [7, 8] for objects in RN , N ≥ 1, and not just in dimension N = 2. Indeed we want to find points x ∈ Γ such that the function ∇Iε (x)·n(x) is an extremal of the function f (t) = ∇Iε (x+t ∇bΩ (x))·∇bΩ (x) in t = 0. This yields the local necessary condition D2 Iε ∇bΩ · ∇bΩ = D2 Iε n · n = 0 on Γ of Assumption 9.2. As for Assumption 9.1, its analogue is the following.

26

Chapter 1. Introduction: Examples, Background, and Perspectives

Assumption 9.3. The restriction to the curve C of the gradient of the intensity Iε is a constant vector c, that is, ∇Iε = c,

c is a constant vector on C.

(9.15)

Indeed the Laplacian of Iε on C can be decomposed as follows (cf. section 5.2 of Chapter 9): ∆Iε = div Γ ∇Iε + D2 Iε n · n, where div Γ ∇Iε is the tangential divergence of ∇Iε that can be defined as follows: div Γ ∇Iε = div (∇Iε ◦ pC )|C and pC is the projection onto C. Hence, under Assumption 9.3, div Γ (∇Iε ) = 0 on C since div Γ (∇Iε ) = div (∇Iε ◦ pΓ )|Γ = div (c)|Γ = 0, and, in view of (9.13) of Assumption 9.2, D2 Iε n · n = 0 on C. Therefore ∆Iε = div Γ (∇Iε ) + D2 Iε n · n = 0 on C, and we obtain the necessary condition ∆Iε = 0 on C.

9.3

(9.16)

Objective Functions Defined on the Whole Edge

With the pioneering work of M. Kass, M. Witkin, and D. Terzopoulos [1] in 1988 we go from a local necessary condition at a point of the edge to a global necessary condition by introducing objective functionals defined on the entire edge of an object. Here many computations and analytical studies can be simplified by adopting the point of view that a closed curve in the plane is the boundary of a set and using the whole machinery developed for shape and geometric analysis and the tangential and shape calculuses in Chapter 9. 9.3.1

Eulerian Shape Semiderivative

In this section we briefly summarize the main elements of the velocity method. Definition 9.1. Given D, ∅ = D ⊂ RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D is the holdall or the universe. A shape functional is a map J : A → E from an admissible family A in P(D) with values in a topological vector space E (usually R).

9. A Glimpse into Segmentation of Images

27

Given a velocity field V : [0, τ ] × RN → RN (the notation V (t)(x) = V (t, x) will often be used), consider the transformations def

T (t, X) = x(t, X),

t ≥ 0, X ∈ RN ,

(9.17)

where x(t, X) = x(t) is defined as the flow of the differential equation dx (t) = V (t, x(t)), t ≥ 0, dt

x(0, X) = X

(9.18)

(here the notation x → Tt (x) = T (t, x) : RN → RN will be used). The Eulerian shape semiderivative of J in Ω in the direction V is defined as def

dJ(Ω; V ) = lim

t0

J(Ωt ) − J(Ω) t

(9.19)

(when the limit exists in E), where Ωt = Tt (Ω) = {Tt (x) : x ∈ Ω}. Under appropriate assumptions on the family {V (t)}, the transformations {Tt } are homeomorphisms that transport the boundary Γ of Ω onto the boundary Γt of Ωt and the interior Ω onto the interior of Ωt . 9.3.2

From Local to Global Conditions on the Edge

For simplicity, drop the subscript ε of the convolution-smoothed image Iε and assume that the edge of the object is the boundary Γ of an open domain Ω of class C 2 . Following V. Caselles, R. Kimmel, and G. Sapiro [1], it is important to choose objective functionals that are intrinsically defined and do not depend on an arbitrary parametrization of the boundary. For instance, given a frame D = ]0, a[ × ]0, b[ and a smoothed image I : D → R, find an extremum of the objective functional  ∂I def dΓ, (9.20) E(Ω) = Γ ∂n where the integrand is the normal derivative of I. Using the velocity method the Eulerian shape directional semiderivative is given by the expression (cf. Chapter 9)     ∂I ∂ ∂I + V · n dΓ, (9.21) dE(Ω; V ) = H ∂n ∂n ∂n Γ where H = ∆bΩ is the mean curvature and n = ∇bΩ is the outward unit normal. Proceeding in a formal way a necessary condition would be   ∂I ∂ ∂I + = 0 on Γ H ∂n ∂n ∂n ⇒ ∆bΩ ∇I · ∇bΩ + ∇(∇I · ∇bΩ ) · ∇bΩ = 0 ⇒ ∆bΩ ∇I · ∇bΩ + D2 I∇bΩ · ∇bΩ + D2 bΩ ∇I · ∇bΩ = 0 ⇒ D2I n · n + H

∂I = 0 on Γ. ∂n

(9.22)

28

Chapter 1. Introduction: Examples, Background, and Perspectives

This global condition is to be compared with the local condition (9.13). It can also be expressed in terms of the Laplacian and the tangential Laplacian of I as ∆I − ∆Γ I = 0 on Γ,

(9.23)

which can be compared with the local condition (9.16), ∆I = 0. This arises from the decomposition of the Laplacian of I with respect to Γ using the following identity for a smooth vector function U : div U = div Γ U + DU n · n,

(9.24)

where the tangential divergence of U is defined as div Γ U = div (U ◦ pΓ )|Γ and pΓ is the projection onto Γ. Applying this to U = ∇I and recalling the definition of the tangential Laplacian ∆I = div Γ ∇I + D 2 I n · n = ∆Γ I + H

∂I + D2 I n · n, ∂n

where the tangential gradient ∇Γ I and the Laplace–Beltrami operator ∆Γ I are defined as ∇Γ I = ∇(I ◦ pΓ )|Γ

9.4 9.4.1

and

∆Γ I = div Γ (∇Γ I).

Snakes, Geodesic Active Contours, and Level Sets Objective Functions Defined on the Contours

In the literature the objective or energy functional is generally made up of two terms: one (image energy) that depends on the image and one (internal energy) that specifies the smoothness of Γ. A general form of objective functional is  def E(Ω) = g(I) dΓ, g(I) is a function of I. (9.25) Γ

The directional semiderivative with respect to a velocity field V is given by    ∂ Hg(I) + dE(Ω; V ) = g(I) n · V dΓ. (9.26) ∂n Γ This gradient will make the snakes move and will activate the contours. 9.4.2

Snakes and Geodesic Active Contours

If a gradient descent method is used to minimize (9.25) starting from an initial curve C 0 = C, the iterative process is equivalent to following the evolution C t (boundary of the smooth domain Ωt ) of the closed curve C given by the equation ∂C t = − [Ht gI + (∇gI · nt )] nt on C t , ∂t

(9.27)

9. A Glimpse into Segmentation of Images

29

where Ht is the mean curvature, nt is the unit exterior normal, and the right-hand side of the equation is formally the “derivative” of (9.25) given by (9.26). Equation (9.27) is referred to as the geodesic flow. For gI = 1 it is the motion by mean curvature ∂C t = −Ht nt on C t . ∂t 9.4.3

(9.28)

Level Set Method

The idea to represent the contours C t by the zero-level set of a function ϕt (x) = ϕ(t, x) for a function ϕ : [0, τ ] × RN → R by setting

 C t = x ∈ RN : ϕ(t, x) = 0 = ϕ−1 t {0} and to replace (9.27) by an equation for ϕ is due to S. Osher and J. A. Sethian [1] in 1988. This approach seems to have been simultaneously introduced in image ´, T. Coll, and F. Dibos [1] in 1993 under processing by V. Caselles, F. Catte the name “geometric partial differential equations” with, in addition to the mean curvature term, a “transport” term and by R. Malladi, J. A. Sethian, and B. C. Vemuri [1] in 1995 under the name “level set approach” combined with the notion of “extension velocity.” Let (t, x) → ϕ(t, x) : [0, τ ] × RN → R be a smooth function and Ω be a subset N of R of boundary Γ = Ω ∩ Ω such that



 int Ω = x ∈ RN : ϕ(0, x) < 0 and Γ = x ∈ RN : ϕ(0, x) = 0 . (9.29) Let V : [0, τ ]×RN → RN be a sufficiently smooth velocity field so that the transformations {Tt } are diffeomorphisms. Moreover, assume that the images Ωt = Tt (Ω) verify the following properties: for all t ∈ [0, τ ]



 int Ωt = x ∈ RN : ϕ(t, x) < 0 and Γt = x ∈ RN : ϕ(t, x) = 0 . (9.30) Assuming that the function ϕt (x) = ϕ(t, x) is at least of class C 1 and that ∇ϕt = 0 on ϕ−1 t {0}, the total derivative with respect to t of ϕ(t, Tt (x)) for x ∈ Γ yields d ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · Tt (x) = 0. ∂t dt

(9.31)

By substituting the velocity field in (9.18), we get ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · V (t, Tt (x)) = 0, ∀ Tt (x) ∈ Γt ∂t ∂ ϕt + ∇ϕt · V (t) = 0 on Γt , t ∈ [0, τ ]. ⇒ ∂t

(9.32)

This last equation, the level set evolution equation, is verified only on the boundaries or fronts Γt , 0 ≤ t ≤ τ . Can we find a representative ϕ in the equivalence class

 −1 [ϕ]Ω,V = ϕ : ϕ−1 t {0} = Γt and ϕt {< 0} = int Ωt , ∀t ∈ [0, τ ]

30

Chapter 1. Introduction: Examples, Background, and Perspectives

of functions ϕ that verify conditions (9.29) and (9.30) in order to extend (9.32) from Γt to the whole RN or at least almost everywhere in RN ? Another possibility is to consider the larger equivalence class

 [ϕ]Γ,V = ϕ : ϕ−1 t {0} = Γt , ∀t ∈ [0, τ ] . 9.4.4

Velocity Carried by the Normal

In D. Adalsteinsson and J. A. Sethian [1], J. Gomes and O. Faugeras [1], and R. Malladi, J. A. Sethian, and B. C. Vemuri [1], the front moves under the effect of a velocity field carried by the normal with a scalar velocity that depends on the curvatures of the level set or the front. So we are led to consider velocity fields V of the form V (t)|Γt = v(t) nt ,

x ∈ Γt ,

(9.33)

for a scalar velocity (t, x) → v(t)(x) : [0, τ ] × RN → R. By using the expression ∇ϕt /|∇ϕt | of the exterior normal nt as a function of ∇ϕt , (9.32) now becomes ∂ ϕt + v(t) |∇ϕt | = 0 on Γt . ∂t

(9.34)

For instance, consider the example of the minimization of the total length of the curve C of (9.27) for the metric g(I) dC as introduced by V. Caselles, R. Kimmel, and G. Sapiro [1]. By using the computation (9.26) of the shape semiderivative with respect to the velocity field V of the objective functional (9.25), a natural direction of descent is given by 

v(t)nt ,



∂ v(t) = − Ht g(I) + g(I) ∂nt

on Γt .

(9.35)

The normal nt and the mean curvature Ht can be expressed as a function of ∇ϕt :     ∇ϕt ∇ϕt ∇ϕt and Ht = div Γt nt = div Γt = div nt = . (9.36) |∇ϕt | |∇ϕt | |∇ϕt | Γt By substituting in (9.34), we get the following evolution equation:   ∂ ∇ϕt ∂ϕt g(I) ∇ϕt · − Ht g(I) + = 0 on Γt ∂t ∂nt |∇ϕt |

(9.37)

and      ∂ϕt − Ht g(I) + ∂ g(I) |∇ϕt | = 0 ∂t ∂nt   ϕ = ϕ0 0

on Γt , on Γ0 .

(9.38)

9. A Glimpse into Segmentation of Images

31

By substituting expressions (9.36) for nt and Ht in terms of ∇ϕt , we finally get      ∇ϕt   ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt |   ϕ = ϕ0 0

on Γt ,

(9.39)

on Γ0 .

The negative sign arises from the fact that we have chosen the outward rather than the inward normal. The main references to the existence and uniqueness theorems related to (9.39) can be found in Y. G. Chen, Y. Giga, and S. Goto [1]. 9.4.5

Extension of the Level Set Equations

Equation (9.39) on the fronts Γt is not convenient from both the theoretical and the numerical viewpoints. So it would be desirable to be able to extend (9.39) in a small tubular neighborhood of thickness h  def

Uh (Γt ) = x ∈ RN : dΓt (x) < h

(9.40)

of the front Γt for a small h > 0 (dΓt , the distance function to Γt ). In theory, the velocity field associated with Γt is given by     ∇ϕt ∇ϕt ∇ϕt (9.41) V (t) = − div g(I) + ∇g(I) · on Γt . |∇ϕt | |∇ϕt | |∇ϕt | Given the identities (9.36), the expressions of nt and Ht extend to a neighborhood of Γt and possibly to RN if ∇ϕt is sufficiently smooth and ∇ϕt = 0 on Γt . Equation (9.39) can be extended to RN at the price of violating the assumptions (9.29)–(9.30), either by loss of smoothness of ϕt or by allowing its gradient to be zero:      ∇ϕt   ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt |   ϕ = ϕ0 0

in RN ,

(9.42)

in RN .

´, T. Coll, and F. Dibos [1] In fact, the starting point of V. Caselles, F. Catte was the following geometric partial differential equation:        ∂ϕt − div ∇ϕt g(I) + ν g(I) |∇ϕt | = 0 in RN , ∂t |∇ϕt | (9.43)   ϕ = ϕ0 in RN 0

for a constant ν > 0 and the function g(I) = 1/(1+|∇I|2 ). They prove the existence, in dimension 2, of a viscosity solution unique for initial data ϕ0 ∈ C([0, 1] × [0, 1]) ∩ W 1,∞ ([0, 1] × [0, 1]) and g ∈ W 1,∞ (R2 ).

32

9.5 9.5.1

Chapter 1. Introduction: Examples, Background, and Perspectives

Objective Function Defined on the Whole Image Tikhonov Regularization/Smoothing

The convolution is only one approach to smoothing an image I ∈ L2 (D) defined in a frame D. Another way is to use the Tikhonov regularization: given ε > 0, find a function Iε ∈ H 1 (D) that minimizes the objective functional  def 1 ε|∇ϕ|2 + |ϕ − I|2 dx, Eε (ϕ) = 2 D where the value of ε can be suitably adjusted to get the desired degree of smoothness. The minimizing function Iε ∈ H 1 (D) is solution of the following variational equation:  ∀ϕ ∈ H 1 (D),

ε ∇Iε · ∇ϕ + (Iε − I) ϕ dx = 0.

(9.44)

D

The price to pay now is that the computation is made on the whole image. This can be partly compensated by combining in a single operation or level of processing the smoothing of the image with the detection of the edges. To do that D. Mumford and J. Shah [1] in 1985 and [2] in 1989 introduced a new objective functional that received a lot of attention from the mathematical and engineering communities. 9.5.2

Objective Function of Mumford and Shah

Given a grey level ideal image in a fixed bounded open frame D we are looking for an open subset Ω of D such that its boundary reveals the edges of the two-dimensional objects contained in the image of Figure 1.9. Definition 9.2. Let D be a bounded open subset of RN with Lipschitzian boundary. (i) An image in the frame D is specified by a function I ∈ L2 (D). (ii) We say that {Ωj }j∈J is an open partition of D if {Ωj }j∈J is a family of disjoint connected open subsets of D such that mN (∪j∈J Ωj ) = mN (D)

and

mN (∂ ∪j∈J Ωj ) = 0,

where mN is the N -dimensional Lebesgue measure. Denote by P(D) the family of all such open partitions of D. The idea behind the formulation of D. Mumford and J. Shah [1] in 1985 is to do a Tikhonov regularization Iε,j ∈ H 1 (Ωj ) of I on each element Ωj of the partition and then minimize the sum of the local minimizations over Ωj over all open

9. A Glimpse into Segmentation of Images

33

partitions of the image D: find an open partition P = {Ωj }j∈J in P(D) solution of the minimization problem  inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx (9.45) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some fixed constant ε > 0. Observe that without the condition mN (∪j∈J Ωj ) = mN (D) the empty set would be a solution of the problem. The question of existence requires a more specific family of open partitions or a penalization term that controls the “length” of the interfaces in some sense:  inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx + c HN −1 (∂ ∪j∈J Ωj ) (9.46) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some c > 0, where HN −1 is the Hausdorff (N − 1)-dimensional measure. Equivalently, we are looking for Ω = ∪j∈J Ωj open in D such that χΩ = χD a.e. in D (χΩ and χD , characteristic functions) solution of the following minimization problem:  inf inf1 ε|∇ϕ|2 + |ϕ − I|2 dx + c HN −1 (∂Ω). (9.47) Ω open ⊂D ϕ∈H (Ω) χΩ =χD a.e.

Ω

In general, the Hausdorff measure is not a lower semicontinuous functional. So either the (N − 1)-Hausdorff measure HN −1 is relaxed to a lower semicontinuous notion of perimeter or the minimization problem is reformulated with respect to a more suitable family of open domains and/or a space of functions larger than H 1 (Ω). Another way of looking at the problem would be to minimize the number J of connected subsets of the open partition, but this seems more difficult to formalize. 9.5.3

Relaxation of the (N − 1)-Hausdorff Measure

The choice of a relaxation of the (N − 1)-Hausdorff measure HN −1 is critical. Here the finite perimeter of Caccioppoli (cf. section 6 in Chapter 5) reduces to the perimeter of D since the characteristic function of Ω is almost everywhere equal to the characteristic function of D. However, the relaxation of HN −1 (∂Ω) to the ´sio [8] (N − 1)-dimensional upper Minkowski content by D. Bucur and J.-P. Zole is much more interesting and in view of its associated compactness theorem (cf. section 12 in Chapter 7) it yields a unique solution to the problem. 9.5.4

Relaxation to BV-, H s -, and SBV-Functions

The other avenue to explore is to reformulate the minimization problem with respect to a space of functions larger than H 1 (Ω). One possibility is to use functions with possible jumps, such as functions of bounded variations. In dimension 1, the BV-functions can be decomposed into an

34

Chapter 1. Introduction: Examples, Background, and Perspectives

absolutely continuous function in W 1,1 (0, 1) plus a jump part at a countable number of points of discontinuity. Such functions have their analogue in dimension N . With this in mind one could consider the penalized objective function  def JBV (ϕ) = |ϕ − I|2 dx + εϕ2BV(D) , (9.48) D

which is similar to the Tikhonov regularization when rewritten in the form   def JH 1 (ϕ) = |ϕ − I|2 dx + εϕ2H 1 (D) = (1 + ε)|ϕ − I|2 dx + ε|∇ϕ|2 dx, D

D

where we square the penalization term to make it differentiable. Unfortunately, the square of the BV-norm is not differentiable. Going back to Figure 1.9, assume that the triangle T has an intensity of 1, the circle C has an intensity of 2/3, and the remaining part of the square S has an intensity of 1/3. The remainder of the image is set equal to 0. Then the image functional is precisely 2 1 I(x) = χT (x) + χC + χS\(T ∪C) . 3 3 We know from Theorem 6.9 in section 6.3 of Chapter 5 that the characteristic function of a Lipschitzian set is a BV-function. Therefore, if we minimize the objective functional (9.48) over all ϕ ∈ BV(D), we get exactly ϕˆ = I. If we insist on formulating the minimization problem over a Hilbert space as for the Tikhonov regularization, the norm of the penalization term can be chosen in the space H s (D) for some s ∈ (0, 1]:  def JH s (ϕ) = |ϕ − I|2 dx + ε ϕ2H s (D) . D

For 0 < s < 1/2, the norm is given by the expression def





ϕH s (D) =

dx D

dy D

|ϕ(y) − ϕ(x)|2 |y − x|N +2s

and the relaxation is very close to the one in BV(D) since BV(D)∩L∞ (D) ⊂ H s (D), 0 < s < 1/2 (cf. Theorem 6.9 (ii) in Chapter 5). However, as seducing as those relaxations can be, it is not clear that the objectives of the original formulation are preserved. Take the BV-formulation. It is implicitly assumed that there are clean jumps across the interfaces. This is true in dimension 1, but not in dimension strictly greater than 1 as explained in L. Ambrosio [1], where he points out that the distributional gradient of a BV-function which is a vector measure has three parts: an absolutely continuous part, a jump part, and a nasty Cantor part. The space BV(D) contains pathological functions of Cantor–Vitali type that are continuous with an approximate differential 0 almost everywhere, and this class of functions is dense in L2 (D), making the infimum of

9. A Glimpse into Segmentation of Images

35

JBV over BV(D) equal to zero without giving information about the segmentation. To get around this difficulty, he introduces the smaller space SBV(D) of functions whose distributional gradient does not have a Cantor part. He considers the mathematical framework for the minimization of the objective functional  def |ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (K) J(ϕ, K) = D\K

with respect to all closed sets K ⊂ Ω and ϕ ∈ H 1 (Ω\K), where I ∈ L∞ (D). This problem makes sense and has a solution in SBV(D) when K is replaced by the set Sϕ of points where the function ϕ ∈ SBV(D) has a jump, that is, def

 |ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (Sϕ ),

J(ϕ) =

D

that turns out to be well-defined on SBV(D). 9.5.5

Cracked Sets and Density Perimeter

The alternate avenue to explore is to reformulate the minimization problem with respect to a more suitable family of open domains (cf. section 15 of Chapter 7). An additional reason to do that would be to remove the term on the length of the interface. When a penalization on the length of the segmentation is included, long slender objects are not “seen” by the numerical algorithms since they have too large a perimeter. In order to retain a segmentation of piecewise H 1 -functions ´sio [38] introduced the without a perimeter term, M. C. Delfour and J.-P. Zole family of cracked sets (see Figure 1.11) that yields a compactness theorem in the W 1,p -topology associated with the oriented distance function (cf. Chapter 7).

Figure 1.11. Example of a two-dimensional strongly cracked set. The originality of this approach is that it does not require a penalization term on the length of the segmentation and that, within the set of solutions, there exists one with minimum density perimeter as defined by D. Bucur and ´sio [8]. J.-P. Zole

36

Chapter 1. Introduction: Examples, Background, and Perspectives

Theorem 9.2. Let D be a bounded open subset of RN and α > 0 and h > 0 be real numbers.3 Consider the families   Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,   def , F(D, h, α) = Ω ⊂ D : dΓ (x + td)  ≥ α such that inf (9.49) 0
Cbh,α (D) = {bΩ : Ω ∈ F(D, h, α)} ,   Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,   def , Fs (D, h, α) = Ω ⊂ D : dΓ (x + td)  ≥ α such that inf |t| 0<|t|
(9.50)

def

(Cbh,α )s (D) = {bΩ : Ω ∈ Fs (D, h, α)} . Then Cbh,α (D) and (Cbh,α )s (D) are compact in W 1,p (D), 1 ≤ p < ∞, where dΓ is the distance function to the boundary of Ω. In view of the above compactness, it can be shown that there exists a solution to the following minimization problem:  inf inf1 ε |∇ϕ|2 + |ϕ − f |2 dx. (9.51) Ω∈F (D,h,α) ϕ∈H (Ω) Ω open ⊂D, mN (Ω)=mN (D)

Ω

Cracked sets form a very rich family of sets with a huge potential that is not yet fully exploited in the image segmentation problem. Indeed, they can be used not only to partition the frame of an image but also to detect isolated cracks and points provided an objective functional sharper than the one of Mumford and Shah is used. For instance, in view of the connection between image segmentation and fracture theory hinted at in J. Blat and J.-M. Morel [1], the theory may have potential applications in problems related to the detection of fractures or cracks or fracture branching and segmentation in geomaterials (cf. K. B. Broberg [1]), but this is way beyond the scope of this book. Some initial considerations about the numerical approximation of cracked sets can be found in M. C. Delfour and ´sio [41]. J.-P. Zole

10 10.1

Shapes and Geometries: Background and Perspectives Parametrize Geometries by Functions or Functions by Geometries?

In the buckling of the column and in the design of the thermal diffuser and the thermal radiator the geometry was a volume of revolution generated by rotation around the z-axis of the hypograph between the axis and the graph of the function 3 In view of the fact that the distance function d is Lipschitzian with constant 1, we necessarily Γ have 0 < α ≤ 1.

10. Shapes and Geometries: Background and Perspectives

37

defined in a variable interval [0, L]. They are examples of sets parametrized by functions or scalars. A well-documented weakness of this approach is that numerical computations of optimal shapes often yield boundary oscillations. This is typical of the parametrization of the boundary by a function. It provides a good control of the displacement in the direction normal to the graph but little control in the tangential direction. Extending this approach to sets locally defined by several graphs becomes tricky. For instance, it is not possible to represent a sphere or a torus from a biLipschitzian mapping from some domain in the plane unless we replace the plane by a surface with facets as illustrated in Figure 1.12 for a ball (cf. M. C. Delfour [8] for the construction of a surface with facets from a C 1,1 -surface). Moreover, the surface with facets of Figure 1.12 is closely associated with the ball and would not be appropriate for a torus.

Figure 1.12. Example of a surface with facets associated with a ball. Another family of function-parametrized sets is the family of images of a fixed set by a family of homeomorphisms or diffeomorphisms. This can be advantageous in problems where it is desirable to work with a fixed mesh of the original domain, thus avoiding remeshing the domain at each step of the optimization process. Function-parametrized sets are also used in the identification of objects (cf., for instance, the school of R. Azencott, A. Trouv´e, and L. Younes). This approach can be traced back to R. Courant and D. Hilbert [1] in 1953 and the construction of Courant metrics and complete metric spaces of images of a fixed closed or crack-free open set by A. M. Micheletti [1] in 1972. This approach is less interesting in optimization problems where the topology of the set is part of the unknown features of the set. Indeed, the images of a fixed set by an homeomorphism cannot change the topology of the fixed set. For instance, it cannot create holes that are not present in the fixed set. The example of the distribution of two materials in a fixed domain is generic of problems where the topology of the set is part of the unknowns. The variable chosen to represent the set is the characteristic function χA , a function parametrized by the set A. As such, arbitrary geometries that are only measurable can be used as independent variables. The topology (for instance, the number of holes) is not fixed a priori and extremely complex design can be obtained. Moreover, the underlying

38

Chapter 1. Introduction: Examples, Background, and Perspectives

Lp -spaces where the characteristic functions naturally live induce a metric ρ([A1 ], [A2 ]) = χA2 − χA2 Lp (D) on the family of equivalence classes of Lebesgue measurable subsets of a bounded measurable holdall D. This type of topology has been used in the proof of the Maximum Principle by Pontryagin and later by Ekeland. Other examples of set-parametrized functions include the uniformly Lipschitzian distance function dA to a set A. The family of such functions for subsets of a holdall D, D bounded open, can be considered as a family of functions in C 0 (D) or W 1,p (D). For instance, the metric ρ([A1 ], [A2 ]) = dA2 − dA2 C 0 (D) ([A] is the equivalence class of all subsets of D with the same closure) coincides ´iu [1] in his thesis with the ´ecart mutuel between two sets introduced by D. Pompe presented in Paris in March 1905 that was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914. Unfortunately that metric does not preserve the volume functional, but the norm can be changed to obtain the new metric ρ([A1 ], [A2 ]) = dA2 − dA2 W 1,p (D) that preserves the volume functional. Another example of a set-parametrized function is the support function def

σA (x) = sup a · x a∈A

of convex analysis. Here the equivalence classes are  def

[A] = B ⊂ D : co B = co A , that is, sets with the same closed convex hull. The associated metric is ρ([A1 ], [A2 ]) = σA2 − σA2 C 0 (B) , ´sio [17]). where B is the unit open ball in RN (cf. M. C. Delfour and J.-P. Zole The general pattern that emerges from the last three examples is the following one. Choose a fixed holdall D and let A denote a variable set in D. Select a family of set parametrized functions fA and let  def

[A] = B ⊂ RN : fA = fB be the associate equivalence class. Finally, assume that the family of functions {fA } is contained in a Banach space B(D). Choose as a metric ρ([A1 ], [A2 ]) = fA2 − fA2 B(D) . This approach will be detailed in Chapters 5, 6, and 7 for the characteristic function χA , the distance function dA , and the oriented distance function bA , but many other set-parametrized functions can be used according to the requirements of the problem at hand.

10. Shapes and Geometries: Background and Perspectives

10.2

39

Shape Analysis in Mechanics and Mathematics

The terminology shape analysis has been introduced independently in at least two different contexts: continuum mechanics and mathematical theory of partial differ´a [1]). ential equations (cf., for instance, E. J. Haug and J. Ce In continuum mechanics, shape analysis encompasses contributions to structural mechanics of elastic bodies such as beams, plates, shells, arches, and trusses. In such problems, the objective is to optimize the compliance, e.g., the work of the applied loadings, by choosing the design parameters of the structure. Many of the early contributions are in dimension 2 and complete analytical solutions are often provided. Yet, it is not always easy to distinguish between a shape optimization problem, such as the shape of a two-dimensional plate, and a distributed parameter problem, such as the optimal thickness of that plate. The thickness that is often considered as a shape parameter is really a distributed parameter over the two-dimensional domain, which also specifies the plate. When the thickness goes to zero in some parts of the plate, holes are created and induce changes in the topology and the shape of the associated two-dimensional domain. This naturally leads to topological optimization, which deals with the connectivity of a domain, the number of holes, the fractal dimension of the boundary, and ultimately the appearance of a microstructure. This was exemplified by K.-T. Cheng and N. Olhoff [2]’s celebrated optimization of the compliance of the circular plate with respect to its thickness under prescribed loading and a constraint on the volume of material. Such questions have received a lot of attention in specific cases and have been analyzed by homogenization methods or Γ-convergence (cf., for instance, F. Murat and L. Tartar [1], the conference proceedings edited by M. Bendsøe and C. A. Mota Soares [1], the book by M. Bendsøe [1], and the book edited by A. Cherkaev and R. Kohn [1] that contains a selection of translations of key papers written in French or Russian). However, many fundamental questions still remain open. For instance, how does such an analysis affect the validity of the underlying mechanical or physical models? For convenience we shall refer to this viewpoint as the compliance analysis that generally involves the extremum of the minimum of an energy or work functional with respect to some design parameters. In the mathematical theory of partial differential equations, the analysis dealt with the sensitivity of the solution of boundary value problems with respect to the shape of the geometric domain on which the partial differential equation is defined. This was done for different applications, including free boundary problems, noncylindric problems,4 and shape identification problems. This shape sensitivity analysis was simultaneously developed for the solution of the partial differential equation and for shape functionals depending on their solution. In that context, the compliance analysis becomes a special case that, quite remarkably, does not require the shape sensitivity analysis of the solution of the associated partial differential equation (cf. section 2 of Chapter 10). This important 4 A cylindric problem is a partial differential equation where the geometric domain is fixed and independent of the time variable. A noncylindric problem is a partial differential equation where the geometric domain changes with time.

40

Chapter 1. Introduction: Examples, Background, and Perspectives

simplification arises from the fact that the compliance is the minimum of an energy or work functional. An historical example that also benefits from this property is the shape derivative of the first eigenvalue of the plate studied at the beginning of the 20th century by J. Hadamard [1]. As for the compliance, the shape sensitivity analysis of the first eigenfunction is not required even when the first eigenvalue is repeated. This follows from the fact that the first eigenvalue can be expressed as a minimum through Rayleigh’s quotient or G. Auchmuty [1]’s dual variational principle. Shape sensitivity analysis deals with a larger class of shape functionals (e.g., minimal drag, noise reduction) and partial differential equations (e.g., the wave equation, viscous or non-Newtonian fluids), where variational energy functionals are usually not available and for which the compliance analysis is no longer applicable. Yet, the shape sensitivity analysis of the solution of the partial differential equation can again be avoided by incorporating the partial differential equation into a Hamiltonian or Lagrangian formulation. As in control theory, this yields a partial differential equation for the adjoint state that is coupled with the initial partial differential equation or state equation. A precise mathematical justification of this approach can be given when the Lagrangian has saddle points and the shape derivative can be obtained from theorems on the differentiability of saddle points with respect to a parameter even when the saddle point solution is not unique (cf. section 5 in Chapter 10). The shape sensitivity analysis through a family of diffeomorphisms which preserve the smoothness of the images of a fixed domain is primarily a local analysis. It is used to establish continuity, to define derivatives, or to optimize in a narrow class of domains with fixed regularities and topologies: it cannot create holes or singularities that were not present in the initial domain. Consider the problem of finding the best location and shape of a hole of given volume in a homogeneous elastic plate to optimize the compliance or some other criterion under a given loading. The expectation is that the presence of the hole would improve the compliance over a homogeneous plate without holes. This problem has its analogue in control theory, for instance the optimal placement of sensors and actuators for the control and stabilization of large flexible space structures or flexible arms of robots. Another important example is the localization of sensors and actuators to achieve noise reduction in structures. The optimal placement is usually an integral part of the control synthesis. ´a, A. J. Gioan, and J. Michel [1] proposed to In the early 1970s J. Ce introduce relaxed problems in which the optimal domain (here an optimal hole, the optimal location of the support of the optimal control) was systematically replaced by a density function ranging between zero and one and that would hopefully be a characteristic function (a bang bang control) in the optimal regime. Furthermore, in order to work on a fixed domain D without holes, they replaced the holes by a very weak elastic material. This changed the original topological optimization problem into the identification of distributed coefficients over the fixed domain D. Equivalently, this identification problem reduces to finding the optimal distribution of two materials for a transmission equation in D under a volume constraint on one of the two materials. Under appropriate conditions, the solution of this problem is a characteristic function, even when the space of distributed parameters is relaxed

10. Shapes and Geometries: Background and Perspectives

41

to the closed convex hull of the set of characteristic functions (cf. Chapter 5). This general technique can also be used to study the continuity of the solution of the homogeneous Dirichlet boundary value problems with respect to the domain Ω, by introducing a transmission problem over a fixed domain D and letting the distributed coefficient go to infinity in the complement of Ω with respect to D. Of course, the coefficient over the complement of Ω could be replaced by a Lagrange multiplier, thus providing a formulation over the fixed domain D. This is one of the many ways to make the domain fictitious and avoid dealing directly with the ´riaux [1]). geometry (cf. R. Glowinski, T.-W. Pan, and J. Pe

10.3

Characteristic Functions: Surface Measure and Geometric Measure Theory

The main advantage of the relaxation to a characteristic function in the formulation of problems involving a domain integral and/or a volume constraint on Lebesgue measurable sets is that the unknown set is completely described by a single function instead of a family of local diffeomorphisms. Yet, such measurable sets can be quite unstructured. For example, in problems involving a surface tension on the free boundary of a fluid or on the interface between two fluids, the domain must have a locally finite boundary measure. Another example is when the objective functional is a function of the normal derivative of the state variable along the boundary (e.g., a flow or a thermal power flux through the boundary). In both cases the use of characteristic functions is limited by the fact that they are not differentiable on the boundary of the set and cannot be readily used to describe very smooth domains. Fortunately, there is enough latitude to make sense of a locally finite boundary measure for sets for which the characteristic function is of bounded variation; that is, its gradient is a vector of bounded measures. Such sets are known as Caccioppoli or finite perimeter sets. They were a key ingredient in the contribution of E. De Giorgi to the theory of minimal surfaces in the 1950s, since the norm of that vector measure (which is the total variation of the characteristic function) turns out to be a relaxation of the boundary measure or “perimeter” of the set. To get a compactness result for a family of Caccioppoli sets it is sufficient to put a uniform bound on their perimeters. This field of activities is known as geometric measure theory, and its tools have been used very successfully in the theory of free and moving boundary problems. Even though it came rather late to shape analysis, this material is both important and fundamental.

10.4

Distance Functions: Smoothness, Normal, and Curvatures

Another set-parametrized function that plays a role similar to the characteristic function is the distance function between closed subsets of a fixed holdall5 D of ´iu [1] in RN . The “´ecart mutuel ” between two sets was introduced by D. Pompe 5 This set D will play several roles in this book. It is the universe in which the variable subsets live. It will often be referred to as the underlying holdall. In other circumstances it will have a purely technical role, much like the control volume in fluid mechanics.

42

Chapter 1. Introduction: Examples, Background, and Perspectives

his thesis presented in Paris in March 1905. This is the first example of a metric between two sets in the literature. It was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914. The Pomp´eiu–Hausdorff metric between two sets corresponds to the uniform norm of the difference of their respective distance functions. For most applications it is not a very interesting topology since the volume functional is not continuous in that topology. The continuity of the volume functional can be recovered by replacing the uniform norm by the W 1,p -norm since the characteristic function of the closure of the set can be expressed in terms of the gradient of the distance function. In 1951, H. Federer [1] introduced the family of sets of positive reach and gave a first access to curvatures from the distance function in much the same spirit as the perimeter from the characteristic function. They are sets for which the projection onto the set is unique in a neighborhood of the set. Since the projection can be expressed in terms of the gradient of the square of the distance function, this is equivalent to requiring that the square of the distance function be C 1,1 in a neighborhood of the set. Since the gradient of the distance function has a jump discontinuity at the boundary of the set, he managed to recover from the distance function the curvature measures of the boundary and made sense of the classical Steiner formula for sets with positive reach. The jump discontinuity of the characteristic function across the boundary can be bypassed by going to the oriented distance function. For smooth domains this function is quite remarkable since it inherits the same degree of smoothness in a neighborhood of the boundary as the boundary itself. Then the gradient, the Hessian matrix, and the higher-order derivatives in the neighborhood of the boundary can be used to characterize and compute normals, curvatures, and their derivatives along the boundary. This correspondence remains true for domains of class C 1,1 . The gradient of the oriented distance function coincides with the outward unit normal on the boundary. This implicit orientation is at the origin of the terminology oriented distance function that was introduced by M. C. Delfour ´sio [17] in 1994 to distinguish it from the algebraic distance function and J.-P. Zole to a submanifold that can be defined only in terms of some discriminating criterion that distinguishes between what is above and what is below the submanifold. This is not always possible, while the oriented distance function to a set is always welldefined. The restriction of the Hessian matrix of the oriented distance function to the boundary coincides with the second fundamental form of differential geometry. Its eigenvalues are zero and the principal curvatures of the boundary. A nice relaxation of the curvatures is obtained by considering sets for which the elements of the Hessian matrix of the oriented distance function are bounded measures. They are called sets of locally bounded curvature (cf. M. C. Delfour ´sio [17, 32]). To get a compactness result, it is sufficient to put a and J.-P. Zole uniform bound on the total variation of the gradient (the elements of the Hessian matrix are bounded measures). The oriented distance function also provides a framework to compare the smoothness of sets ranging from arbitrary sets with a nonempty boundary to sets of class C ∞ via the smoothness of the oriented distance function in a neighborhood of the boundary of the set. As a result Sobolev spaces can be used to introduce the

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43

notion of a Sobolev domain, which becomes intertwined with the classical notion of C k -domains.

10.5

Shape Optimization: Compliance Analysis and Sensitivity Analysis

As in the vector space case, optimization and control problems with respect to geometry are of various degrees of difficulty. When the objective functional does not depend on the solution of a state equation or variational inequality defined on the variable domain, it is sufficient to invoke compactness and continuity arguments. In special cases such as the optimization of the compliance or of the first eigenvalue, the problem can be transformed into the optimization of an objective functional that is itself the minimum of some appropriate functional defined on a fixed function space. There a direct study of the dependence of the solution of the state equation with respect to the underlying domain can be bypassed. In the general case, a state equation constraint has to be handled very carefully from both the mathematical and the application viewpoints. When the analysis can be restricted to families of Lipschitzian or convex domains, it is usually possible to give a meaning and prove the continuity of the solution of the state equation with respect to the underlying varying domain. When families of arbitrary bounded open domains are considered, new phenomena can occur. As the domains converge in some sense, the corresponding solutions may only converge in a weak sense to the solution of a different type of state equation over the limit domain: boundary conditions may no longer be satisfied, strange terms6 may occur on the right-hand side of the equation, etc. In such cases, it is often a matter of modeling of the physical or technological phenomenon. Is it natural to accept a generalized solution or a relaxed formulation of the state equation or should the family of domains be sufficiently restricted to preserve the form of the original state equation and maintain the continuity of the solution with respect to the domain? The most suitable relaxation is not necessarily the most general mathematical relaxation. It must always be compatible with the physical or technological problem at hand. Too general a relaxation of the problem or too restrictive conditions on the varying domains can yield completely unsatisfactory solutions however nice the underlying mathematics are. A good balance of mathematical, physical, and engineering intuitions is essential. The study of the shape continuity of the solution of the partial differential equation is also of independent mathematical interest. In the literature, this issue has been addressed in various ways. Some authors simply introduce a stability assumption that essentially says that the limiting domain is such that continuity occurs. Others introduce a set of more technical assumptions that correspond to a minimal set of conditions to make the crucial steps of the proof of the continuity work. On the constructive side the really challenging issue is to characterize the families of domains for which the continuity holds. 6 Cf.,

for instance, D. Cioranescu and F. Murat [1, 2].

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Chapter 1. Introduction: Examples, Background, and Perspectives

Many authors have constructed compact families for that purpose. For instance the Courant metric topology was used by A. M. Micheletti [1] in 1972 for C k -domains, the uniform cone condition by D. Chenais [1] in 1973 for uniformly Lipschitzian domains, and other metric topologies by F. Murat and J. Simon [1] in 1976 for Lipschitzian domains. More general capacity conditions were introduced after 1994 by D. Bucur and ´sio [5] in order to obtain compact subfamilies of domains with respect to J.-P. Zole the complementary Hausdorff topology and control the curvature of the boundaries ˇ ´ k [2] of the domains. In dimension 2 they recover the nice result of V. Sver a in 1993, which involves a uniform bound on the number of connected components of the complement of the sets. Intuitively the capacity conditions are such that, locally, the complement of the domains in the chosen family has “enough capacity” to preserve the homogeneous Dirichlet boundary condition in the limit. Yet, the capacity conditions might not be easy to use in a specific example. So D. Bucur ´sio [1, 5] introduced a simpler geometric constraint, called the flat and J.-P. Zole cone condition, under which the continuity and compactness results still hold. This generalizes the uniform cone property to a much larger class of open domains. All this will be further generalized in section 9 of Chapter 8.

10.6

Shape Derivatives

For functionals defined on a family of domains, it is important to distinguish between a function with values in a topological vector space over a fixed domain and a function such as the solution of a partial differential equation that lives in a Sobolev space defined on the varying domain. In the latter case special techniques have to be used to transport the solution onto a fixed domain or to first embed the varying domains into a fixed holdall D, extend the solutions to D, and enlarge the Sobolev space to a large enough space of functions defined over the fixed holdall D. In both cases the function is defined over a family of domains or sets belonging to some shape space which is generally nonlinear and nonconvex. Thus, defining derivatives on those spaces is more related to defining derivatives on differentiable manifolds than in vector spaces. It is perhaps for groups of diffeomorphisms that the most complete theory of shape derivatives is available. This material will be covered in detail in Chapters 2, 3, and 9, but it is useful to briefly introduce the main ideas and definitions here. Given a Banach space Θ of mappings from a fixed open holdall D ⊂ RN into RN , first consider the group of diffeomorphisms  def

F(Θ) = F : D → D : F − I ∈ Θ and F −1 − I ∈ Θ . Then consider the images of a fixed domain Ω0 in D by F(Θ): def

X (Ω0 ) = {F (Ω0 ) : ∀F ∈ F(Θ)} . They can be identified with the quotient group F(Θ)/G(Ω0 ),

def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

10. Shapes and Geometries: Background and Perspectives

45

of diffeomorphisms of D. For the specific choices of D, Θ, and Ω0 that are of interest, this quotient group can be endowed with the so-called Courant metric to make it a complete metric space. This metric space is neither linear nor convex. For unconstrained domains (D = RN ) and “sufficiently small” elements θ ∈ Θ, transformations of the form F = I + θ belong to F(Θ) and hence perturbations of the identity (i.e., of Ω0 ) can be chosen in the vector space Θ. This makes it possible to define directional derivatives and speak of Gateaux and Fr´echet differentiability with respect to Θ as in the classical case of functions defined on vector spaces. However, this approach does not extend to submanifolds D of RN or to domains that are constrained in one way or another (e.g., constant volume, perimeter, etc.). In the unconstrained case, we get an infinite-dimensional differentiable manifold structure on the quotient group F(Θ)/G(Ω0 ) and the tangent space to F(Θ) is the whole linear space Θ. As a result ideas and techniques from differential geometry are readily applicable. For instance, the continuity in I (i.e., at Ω0 ) can be characterized by the continuity along one-dimensional flows of velocity fields in F(Θ) through the point I (that is, I(Ω0 ) = Ω0 ). This suggests defining a notion of directional derivative along one-dimensional flows associated with “velocity fields” V which keep the flows inside D. When D is a smooth submanifold of RN such velocities are tangent to D and the set of all such velocities has a vector space structure. This key property generalizes to other types of holdalls. So it is possible to define directional derivatives and speak of shape gradient and shape Hessian with respect to the associated vector space of velocities. This second approach has been known in the literature as the velocity method. Another very nice property of that method is that perturbations of the identity, as previously defined in the unconstrained case, can be recovered by a special choice of velocity field, thus creating a certain unity in the methodology. The concept of topological derivative was introduced by J. Soko
lowski and ˙ A. Zochowski [1] for problems where the knowledge of the optimal topology of the domain is important. It gives some sensitivity of a shape functional to the presence of a small hole at a point of the domain as the size of that hole goes to zero. For a domain integral, that is, the integral of a locally Lebesgue integrable function, this derivative is the negative of the classical set-derivative which is equal to the function at almost every point. This is a direct consequence of the Lebesgue differentiation theorem. So, at least in its simplest form, this approach aims at extending the classical concept of set-differentiation as the inverse of integration over sets. This type of derivative becomes more intricate as we look at the sensitivity of the solution of a boundary value problem or at shape functionals which are functions of that solution. For functions that are the integrals of an integrable function with respect to a Radon measure, the Lebesgue differentiation extends in the form of the Lebesgue–Besicovitch differentiation theorem, which says that the set-derivative of that integral is again equal to the function almost everywhere with respect to the Radon measure. So one could try to determine the class of shape functionals that can be expressed as an integral with respect to some Radon measure. Since this material requires extensive technical results to fully appreciate its impact, we did not include a chapter on this topic in the book, but the approach is an important

46

Chapter 1. Introduction: Examples, Background, and Perspectives

addition to the global arsenal! Fortunately, the theoretical and numerical work is well documented in the literature and books by J. Sokolowski and his coauthors should be available in the near future.

10.7

Shape Calculus and Tangential Differential Calculus

The velocity method has been used in various applications and contexts, and a very complete shape calculus is now available (for instance, the reader is referred to the ´sio [9]). In the computations of derivatives, book of J. Soko
lowski and J.-P. Zole and especially of second-order derivatives, an intensive use is made of the tangential differential calculus. In order to avoid parametrizations and local bases, tangential derivatives are defined through extensions of functions from the boundary to some small Euclidean neighborhood. Their importance should not be underestimated from both the theoretical and the computational points of view. The use of an intrinsic tangential gradient, divergence, or Laplacian can considerably simplify the computation and the final form of the expressions making more apparent their fine structure. Too many computations using local coordinates, Christoffel symbols, or intricate parametrizations are often difficult to decipher or to use effectively. This book provides the latest and most important developments of that calculus. They arises from the systematic use of the oriented distance function in the theory of thin and asymptotic shells.7 In that context, it has been realized that extending functions defined on the boundary Γ of a domain Ω by composition with the projection onto Γ results in sweeping simplifications in the tangential calculus. This is due to the fact that the projection can be expressed in terms of the gradient of the oriented distance function. Computing derivatives on Γ becomes as easy as computing derivatives in the Euclidean space. Curvature terms, when they occur, appear in the right place and in the right form through the Hessian matrix of the oriented distance function that coincides with the second fundamental form of Γ. Chapter 9 will provide a self-contained introduction to these techniques and show how the combined strengths of the shape calculus and the tangential differential calculus considerably simplify computations and expand our capability to tackle highly complex and challenging problems.

10.8

Shape Analysis in This Book

Problems in which the design, control, or optimization variable is no longer a vector of parameters or functions, but is the shape of a geometric domain, a set, or even a “fuzzy entity,” cover a much broader range of applications than those for which the compliance or the shape sensitivity analysis have been used. Yet their analysis makes use of common mathematical techniques: partial differential equations, functional analysis, geometry, modern optimization and control theories, finite element analysis, large scale constrained numerical optimization, etc. In this book the terminology shape will be used for domains ranging from unstructured sets to C ∞ -domains. Relaxations of their geometric characteristics such as the volume, perimeter, connectivity, curvatures, and their derivatives will 7 Cf.

´sio [19, 20, 25] and M. C. Delfour [3, 4, 6, 7]. M. C. Delfour and J.-P. Zole

11. Shapes and Geometries: Second Edition

47

be considered. Shape spaces (often metric and complete) corresponding to different levels of smoothness or degrees of relaxation of the geometry will be systematically constructed. For smooth domains we will emphasize the use of groups of diffeomorphisms endowed with the Courant metric. For more general domains we will use a generic construction based on the use of set-parametrized functions. The characteristic function will be associated with metric spaces of equivalence classes of Lebesgue measurable sets. The distance function will be associated not only with the uniform Hausdorff metric topology in the space of continuous functions but also with W 1,p -topologies for which the characteristic function and hence the volume function are continuous. The oriented distance function will be used in the same way to generate new metric topologies. In each case, the metric is constructed from the norm of one of the set-parametrized functions in an appropriate function space. The construction is generic and applies to other choices of set-parametrized functions and function spaces. For instance, the support function of convex analysis can be used to generate a complete metric topology on equivalence classes of sets ´sio [17]). with the same closed convex hull (cf. M. C. Delfour and J.-P. Zole The nice property of the characteristic and distance functions over classical local diffeomorphisms is that the set is globally described in terms of the analytical properties of a single function. For instance, the gradient of the characteristic function yields a relaxed definition of the perimeter, and the Hessian of the distance function yields the boundary measure and curvature terms. But the characteristic function and the gradient of the distance function are both discontinuous at the boundary of the set. This seriously limits their use in the description of smooth domains. In contrast, the oriented distance function can describe a broad spectrum of sets ranging from arbitrary sets with nonempty boundary to open C ∞ -domains according to its degree of smoothness in a neighborhood of the boundary of the set. It readily combines the advantages of local diffeomorphisms and the characteristic and distance functions that are readily obtained from it. This provides, as in functional analysis, a common framework for the classification and comparison of domains according to their relative degree or lack of smoothness. As in geometric measure theory, compact families of sets will be introduced based on the degree of differentiability. One interesting family is made up of the sets with locally bounded curvature, which provide a sufficient degree of relaxation for most applications and for which nice compactness theorems are available (cf. sections 5 and 11 in Chapter 7). This family includes Federer’s sets of positive reach and hence closed convex and semiconvex sets.

11

Shapes and Geometries: Second Edition

Except for the last two chapters, this second edition is almost a new book. Chapters 2, 3, and 4 give the classical description of sets and domains from the point of view of differential geometry. Special attention is paid to domains that verify some segment properties and have a local epigraph representation and to domains that are the image of a fixed set by a family of diffeomorphisms of the Euclidean space.

48

Chapter 1. Introduction: Examples, Background, and Perspectives

Chapters 5, 6, and 7 give a function analytic description of sets and domains via the set-parametrized characteristic functions, distance functions, and oriented distance functions. It emphasizes the fact that we are now dealing with equivalence classes of sets that may or may not have an invariant open or closed set representative in the class. In particular, they include Lebesgue measurable sets and Federer’s sets of positive reach. Many of the classical properties of sets can be recovered from the regularity or function analytic properties of those functions. We concentrate on the basic properties, the construction of spaces of domains, metrics, and topologies, and the characterization of compact families. Chapter 8 deals with problem formulations and shape continuity and optimization for some generic examples. Chapters 9 and 10 are devoted to a modern version of the shape calculus, an introduction to the tangential differential calculus, and the shape derivatives under a state equation constraint.

11.1

Geometries Parametrized by Functions

Chapter 2 gives several classical characterizations and properties of sets or domains from the point of view of differential geometry: sets locally described in a neighborhood of each point of their boundary by an homeomorphism or a diffeomorphism olderian diffeomorphisms), by the epigraph of a C 0 function (e.g., Lip(e.g., C k or H¨ schitzian or H¨ olderian domains), or by a geometric property (e.g., segment, cone, cusp), or sets globally described by the level sets of a C 1 -function. The sections on sets that are locally the epigraph of a C 0 function and sets having one of the segment properties have been completely reorganized, rewritten, and enriched with new results and older ones that are difficult to find in the literature other than in the form of folk theorems 8 or theorems without satisfying proofs. Special attention is given to the uniform fat segment property that was introduced in the first edition of the book under the name uniform cusp property. Several equivalent properties are given for domains that satisfy a uniform segment property and the stronger uniform fat segment property that can be expressed in terms of a dominating function and its modulus of continuity. All this is specialized to domains verifying a uniform cone or cusp property. Chapter 3 adopts another point of view by considering families of sets that are the images of a fixed subset of RN by some family of transformations of RN . The structure and the topology of the images can then be specified via the natural algebraic and topological structures of transformations or equivalence classes of transformations for which the full power of function analytic methods is available. In 1972, A. M. Micheletti [1] introduced the so-called Courant metric and gave what seems to be the first construction of a complete metric topology on the images 8 As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that are found by investigation and may circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. Knowledge of folklore is the coin of the realm of academic mathematics, showing the relative insight of investigators (http://en.wikipedia.org/ wiki/Mathematical folklore).

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of a fixed set. It is the nonlinear and nonconvex character of such shape spaces that will make the differential calculus and the analysis of shape optimization problems more challenging than their counterparts in topological vector spaces. The construction of A. M. Micheletti [1] is generic and readily extends to many families of transformations of RN . The new version of Chapter 3 considerably expands the material and ideas of the first edition by extracting the fundamental assumptions behind the generic framework of A. M. Micheletti that leads to the Courant metrics on the quotient space of families of transformations by subgroup of isometries such as identities, rotations, translations, or flips. Constructions are given for a large spectrum of transformations of the Euclidean space and for arbitrary closed subgroups. New complete metrics on the whole spaces of homeomorphisms and C k -diffeomorphisms are also introduced. They extend classical results for transformations of compact manifolds to general unbounded closed sets and open sets that are crack-free. This material is central in classical mechanics and physics and in modern applications such as imaging and detection. The former Chapter 7 on transformations versus flows of velocities has been moved right after the Courant metrics as Chapter 4 and considerably expanded. It now specializes the results of Chapter 3 to spaces of transformations that are generated by the flow of a velocity field over a generic time interval. The main motivation is to introduce a notion of semiderivatives as well as a tractable criterion for continuity with respect to Courant metrics. The velocity point of view was also adopted by R. Azencott and A. Trouv´e starting in 1994 to construct complete metrics and geodesic paths in spaces of diffeomorphisms generated by a velocity field with applications to imaging. Chapter 4 also gives general equivalences between transformations and flows of velocity fields for unconstrained and constrained families of domains and further sharpens the results to the specific families of transformations associated with the Courant metrics studied in Chapter 2. Several examples of transformations and velocities associated with widely used families of domains are given: C ∞ -domains, C k -domains, Cartesian graphs, polar coordinates and star-shaped domains, and level sets. The chapter clarifies the long-standing issue of the equivalence of the continuity of shape functionals with respect to the Courant metrics and along the flow of velocity fields. This chapter also prepares the ground for and motivates the definition of shape semiderivatives that will be given in Chapter 9. As in the case of the continuity the equivalent characterizations via transformations and flows of velocities are very much in the background and at the origin of the many seemingly different definitions which can be found in the literature. Preliminary considerations are first given to the definition of a shape functional and to two candidates for the definition of a directional shape semiderivative. They respectively correspond to perturbations of the identity associated with any one of the metric spaces constructed in Chapter 2 and to the velocity method associated with the flow of a generally nonautonomous vector field. The first one seems to be limited to domains in RN , while the second one naturally extends to domains living in a fixed smooth submanifold of RN . Moreover, the shape directional derivative obtained by perturbations of the identity

50

Chapter 1. Introduction: Examples, Background, and Perspectives

can be recovered by a special choice of velocity field. Most definitions of shape derivatives which can be found in the literature can be brought down to one of the two approaches.

11.2

Functions Parametrized by Geometries

Almost all compactness theorems for families of sets are specified by function analytic conditions on special set-parametrized families of functions. In this book we single out the characteristic function, the distance function, and the oriented distance function. In the last case, a complete and convivial tangential differential calculus on the boundary of smooth sets will be obtained without local bases or Christoffel symbols and will be exploited in the computation of shape derivatives later in Chapters 9 and 10. Chapter 5 relaxes the family of classical domains to the equivalence classes of Lebesgue measurable sets. Using the characteristic function associated with a set, complete metric groups of equivalence classes of characteristic functions are constructed. As for Courant metrics on groups, they are nonlinear and nonconvex. On one hand, this type of relaxation is desirable in optimization problems where the topology of the optimal set is not a priori specified; on the other hand, it necessitates the relaxation of the theory of partial differential equations on a smooth open domain to measurable sets that have no smoothness and may not even be open. Furthermore some optimization problems yield optimal solutions where the characteristic function is naturally relaxed to a function between zero and one. Such solutions can be interpreted as microstructures, fuzzy sets, probability measures, etc. As a first illustration of the use of characteristic functions in optimization, ´a and K. Malanowski [1] for the the solution of the original problem of J. Ce optimization of the compliance with respect to the distribution of two materials is given with complete details. A second example deals with the buckling of columns, which is one of the very early optimal design problems formulated by Lagrange in 1770. This is followed by the construction of the nice representative of an equivalence class of measurable functions: the measure theoretic representative. The last section is devoted to the Caccioppoli or finite perimeter sets, which have been introduced to solve the Plateau problem of minimal surfaces (J. A. F. Plateau [1]). Even if, by nature, a characteristic function is discontinuous at the boundary of the associated set, the characteristic function of Caccioppoli sets has some smoothness: it belongs to W ,p (D), 1 ≤  < 1/p, p ≥ 1. This is sufficient to obtain compact families of sets by putting a uniform bound on the perimeter. One such family is the set of (locally) Lipschitzian (epigraph) domains contained in a fixed bounded holdall and satisfying a uniform cone property. This property puts a uniform bound on the perimeter of the sets. The use of the theory of finite perimeter sets is illustrated by an application to a free boundary problem in fluid mechanics: the modeling of the Bernoulli wave where the surface tension of the water enters via the perimeter of the free boundary. The chapter concludes with an approximation of the Dirichlet problem by transmission problems over a fixed larger space in order to study the continuity of its solution with respect to the underlying moving domains.

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Chapter 6 moves on to the classical Pomp´eiu–Hausdorff metric topology which is associated with the space of equivalence classes of distance functions of sets with the same closure. As in Chapter 2, the construction of the metric on equivalence classes of sets with the same closure is generic. By going to the distance function of the complement of the set in the uniform topology of the continuous functions, we get the complementary Hausdorff topology. These uniform topologies are often too coarse for applications to physical or technological systems. But, since the distance function is uniformly Lipschitzian, it can also be embedded into W 1,p Sobolev spaces, and finer metric topologies can be generated. They offer definite advantages over the uniform Hausdorff topology in the sense that they preserve the continuity of the volume of sets since the characteristic function is continuous with respect to W 1,p -topologies. Yet, we lose the compactness of the family of subsets of a fixed bounded holdall of RN . Compact families are recovered by imposing some smoothness on the Hessian matrix as in the case of the characteristic functions of Chapter 5. Sets for which the gradient of the distance function is a vector of functions of bounded variation are said to be of bounded curvature since their Hessian matrix is intimately connected with the curvatures of the boundary. This class of sets is sufficiently large for applications and at the same time sufficiently structured to obtain interesting theoretical results. For instance, such sets turn out to be Caccioppoli sets. Furthermore, the squared distance function is directly related to the projection of a point onto the set and is used to characterize Federer’s sets of positive reach and construct compact families. Closed convex sets that are completely characterized by the convexity of their distance function are also of locally bounded curvature. To complete the list of families of sets that are associated with the distance function we introduce Federer’s sets of positive reach and a first compactness theorem. The chapter is complemented with a general compactness theorem for families of sets of global or local bounded curvature in a tubular neighborhood of their boundary. The use of the distance function of Chapter 6 to characterize the smoothness of sets is limited by the fact that its gradient presents a jump discontinuity at the boundary. This is similar to the jump discontinuity of the characteristic function. To get around this difficulty we introduce the oriented distance function that is obtained by subtracting from the distance function to a set the distance function to its complement, providing a level set description of the set. One remarkable property of this function is the fact that a set is of class C 1,1 (resp., C k , k ≥ 2) if and only if its oriented distance function is locally C 1,1 (resp., C k ) in a neighborhood of its boundary. It also provides an orientation of the boundary since its gradient coincides with the unit outward normal. This is why we use the terminology oriented distance function. As in Chapter 6, Hausdorff and W 1,p -topologies and sets of global or locally bounded curvature can be introduced. We now have a continuous classification of sets ranging from sets with a nonempty boundary to C ∞ -sets, much as in the theory of functions. Closed convex sets are characterized by the convexity of their oriented distance function. Convex sets and semiconvex sets are of locally bounded curvature. This property extends to Federer’s sets of positive reach. Several compactness theorems in the W 1,p -topology of the oriented distance function are presented in this chapter for families of subsets of a bounded open

52

Chapter 1. Introduction: Examples, Background, and Perspectives

holdall with either bounded curvature in tubular neighborhoods of their boundary, ´sio [8], or or with a bound on the density perimeter of D. Bucur and J.-P. Zole with a uniform fat segment property or equivalently the uniform boundedness and equicontinuity of all the local graphs of the sets in the family. The theorem on the compactness under the uniform fat segment property is specialized to the family of subsets of a bounded holdall satisfying a uniform cone or cusp property. It is more general than the one we obtained for Lipschitzian domains in Chapter 5, where the perimeter of each set was finite and uniformly bounded for all subsets of the holdall. H¨ olderian domains do not necessarily have a locally finite boundary measure. Yet, the fact that the boundary of a H¨ olderian domain may have cusp does not mean that all H¨ olderian domains do not have a locally finite boundary measure. In order to include applications where the perimeter is bounded, the general compactness theorem is specialized to families of subsets of a holdall that verify the uniform fat segment property with a uniform bound on either the De Giorgi perimeter or the density perimeter. A last section 15 deals with the family of cracked sets. They have ´sio [38] in the context of the image been used in M. C. Delfour and J.-P. Zole segmentation problem of D. Mumford and J. Shah [2] that will be detailed in this section. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks, and sets that can be made up of components of different codimensions. The Hausdorff (N − 1) measure of their boundary is not necessarily finite. Yet, compact families (in the W 1,p -topology) of such sets can be constructed.

11.3

Shape Continuity and Optimization

Chapter 8 deals with problem formulations, continuity, or semicontinuity of shape functionals or of the solutions of boundary value problems with respect to their underlying domain of definition for selected generic examples. Combined with compact families of sets studied in the previous chapters, they are all essential to getting the existence of optimal shapes. This is illustrated to some extent in Chapter 5 for the modeling of the transmission problem with the help of the characteristic function that occurs both in the model and in the specification of the metric on the equivalence classes of measurable sets. This chapter first reviews the continuity of the transmission problem using characteristic functions. It characterizes the upper semicontinuity of the first eigenvalue of the generalized Laplacian with respect to the domain using the complementary Hausdorff topology. Then it studies the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of definition since they require different constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial differential equations. In problems where the objective functional depends explicitly on the domain and the solution of an elliptic equation defined on the same domain, the strong continuity of the solution with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains, such as the uniform fat segment property.

11. Shapes and Geometries: Second Edition

53

The second part extends some of the results to a larger family of domains satisfying capacity conditions which turn out to be important to obtain the continuity of solutions of partial differential equations with homogeneous Dirichlet boundary conditions with respect to their underlying domain of definition. One special case of a capacity condition is the flat cone condition that generalizes the condition of ˇ ´ k [2] involving a bound on the number of connected components of the V. Sver a complement of the sets.

11.4

Derivatives, Shape and Tangential Differential Calculuses, and Derivatives under State Constraints

Chapter 9 is devoted to the essential shape calculus and the no less essential tangential or boundary calculuses since traces, normals, or tangential gradients will naturally occur in the final expressions. After a self-contained review of differentiation in topological vector spaces that emphasizes Gateaux and Hadamard differentials, it introduces basic definitions of first- and second-order Eulerian shape semiderivatives and derivatives by the velocity method. General structure theorems are given for Eulerian semiderivatives of a shape functional. They arise from the fact that shape functionals are usually defined over equivalence classes of sets, and hence only the normal part of the velocity along the boundary really affects the shape functional. Bridges are provided with the method of perturbations of the identity. A section is devoted to a modern version of the shape calculus. It gives the general formulae for the shape derivative of domain and boundary integrals. From these formulae several examples are worked out, including the semiderivative of the boundary integral of the square of the normal derivative. For the computation of a broader range of shape derivatives ´sio [9]. In the reader is referred to the book by J. Soko
lowski and J.-P. Zole most cases, both domain and boundary expressions are available for derivatives. The boundary expression usually contains more information on the structure of the derivative than its domain counterpart. Finally, to effectively deal with the differential calculus in boundary integrals, we provide the latest version of the tangential calculus on C 2 -submanifolds of codimension 1 which has been developed in the con´sio [28, 33] and text of the theory of shells (cf. M. C. Delfour and J.-P. Zole M. C. Delfour [3, 7]). This calculus has been significantly simplified by using the projection associated with the oriented distance function studied in Chapter 7. This powerful tool combined with the shape calculus makes it possible to obtain clean explicit expressions of second-order shape derivatives of domain integrals along with a better understanding of their fine structure. Chapter 10, the final chapter, completes the shape calculus by introducing the basic theoretical results and computational tools for the shape derivative of functionals that depend on a state variable that is usually the solution of a partial differential equation or inequality defined over the varying underlying domain. The first section concentrates on shape functionals that are of compliance type; that is, they are the minimum of an energy functional associated with the state equation or inequality. Such functionals are very nice in the sense that they do not generate an adjoint state equation, and their derivative can be obtained by theorems on the

54

Chapter 1. Introduction: Examples, Background, and Perspectives

differentiability of a minimum with respect to a parameter even when the minimizers are not unique. A detailed generic example is provided to illustrate how to use the function space parametrization to transport the functions in Sobolev spaces over variable domains to a Sobolev space over the fixed larger holdall. These techniques extend to more complex situations. For instance, Sobolev spaces of vector functions with zero divergence can be transported by the so-called Piola transformation. Domain and boundary expressions are provided. The main theorem is applied to the example of the buckling of columns. An explicit expression of the semiderivative of Euler’s buckling load with respect to the cross-sectional area is obtained from the main theorem, and a necessary and sufficient analytical condition is given to characterize the maximum Euler’s buckling load with respect to a family of crosssectional areas. The theory is further illustrated by providing the semiderivative of the first eigenvalue of several boundary value problems over a bounded open domain: Laplace equation, bi-Laplace equation, linear elasticity. In general, the first eigenvalue is not simple over an arbitrary bounded open domain and the eigenvalue is not differentiable; yet the main theorem provides explicit domain and boundary expressions of the semiderivatives. For general shape functionals, a Lagrangian formulation is used to incorporate the state equation and to avoid the study of the derivative of the state equation with respect to the domain. The computation of the shape derivative of a stateconstrained functional reduces to the computation of the derivative of a saddle point with respect to a parameter even when the saddle point solution is not unique. It yields an expression that depends on the associated adjoint state equation, much like in control theory, but here the domains play the role of the controls. The technique is illustrated on both the homogeneous Dirichlet and the Neumann boundary value problems by function space parametrization. An alternative to this method is the function space embedding combined with the use of Lagrange multipliers. It consists in extending solutions of the boundary value problems over the variable domains to a larger fixed holdall, rather than transporting them. This approach offers many technical advantages over the other one. The computations are easier and they apply to larger classes of problems. This is illustrated on the nonhomogeneous Dirichlet boundary value problem. Again domain and boundary expressions for the shape gradient are obtained. Yet the relative advantages of one method over the other are very much problem and objective dependent. Finally, it is important to acknowledge that the above techniques seem quite robust and are systematically used for nonlinear state equations and in contexts where optimization or saddle point formulations are not available.

Chapter 2

Classical Descriptions of Geometries and Their Properties 1

Introduction

This chapter is devoted to the classical descriptions of nonempty subsets of the finite-dimensional Euclidean space that are characterized by the smoothness or properties of their boundary. A first approach is to assume that we can associate with each point of the boundary a diffeomorphism from a neighborhood of that point that locally flattens the boundary. Another way is to assume that the set is the union of the positive level sets of a continuous function and that the zero level is its boundary. A third way is to assume that in each point of the boundary the set is locally the epigraph of a function. The smoothness of the set is then characterized by the smoothness of the corresponding diffeomorphism, level function, or graph function. Those definitions are equivalent for sufficiently smooth sets. Domains that are locally the epigraph of a continuous function in the third category also belong to the first category, but the converse is generally not true. The basic definitions and constructions for sets locally described by an isomorphism or a diffeomorphism are given in section 3, for sets locally described by the level sets of a function in section 4, and for sets locally described by the epigraph of a function in section 5. Section 6 deals with sets characterized by geometric segment properties. A set satisfying the basic segment property has the property that its boundary ∂Ω be locally a C 0 epigraph (cf. section 6.2). The stronger uniform segment property is further strengthened by introducing the uniform fat segment property and the concept of a dominating function that will be necessary to construct compact families of sets later in Chapter 7. Complete equivalences are provided between the three C 0 epigraph properties and the three geometric segment properties in sections 6.2 and 6.3. For sets with a compact boundary, the six properties are shown to be equivalent in section 6.2. Section 6.4 specializes the fat segment property to the geometric uniform cusp and cone properties that respectively characterize H¨ olderian and Lipschitzian domains. The uniform cone property has provided one of the early examples of a compact family of Lipschitzian domains (cf. section 6.4 in Chapter 5) in 1973. 55

56

2 2.1

Chapter 2. Classical Descriptions of Geometries and Their Properties

Notation and Definitions Basic Notation

N is the set of integers {1, 2, . . . } and R the field of real numbers. The interior and the closure of a subset A in RN will be denoted, respectively, by int A and A. The relative complement of A in B will be written as def

/ A}. B A (or B\A) = {x ∈ B : x ∈ When B = RN , we also write A or Ac . The boundary ∂A of A is defined as A ∩A. Since the superscript t will often appear in the book, the transpose of a vector v and a matrix A will be denoted, respectively, by ∗v and ∗A. The inverse of A will be denoted by A−1 . The inner product and norm in RN will be written as def

x·y =

N

|x| =

xi y i ,

√ x · x.

i=1

For a linear transformation A : RN → RK , the norm of A is defined as def

|A| =

max |Ax|RK ,

|x|RN ≤1

and if Aij and Bij are the matrix representations of A and B with respect to some bases {a1 , . . . , aN } and {b1 , . . . , bK } of RN and RK , respectively, the double inner product is defined as def

A · ·B =

K N

Aij Bij

(2.1)

i=1 j=1

and the associated norm

2.2

√

A · · A is equivalent to the norm |A| of A.

Abelian Group Structures on Subsets of a Fixed Holdall D

Given a nonempty set D, consider the power set of D def

P(D) = {A : A ⊂ D} . The power set is closed under the operation of union, intersection, and complement, but those operations do not have an inverse with respect to the neutral element ∅: B ∪ A = A ∪ B,

A ∪ ∅ = A,

A ∪ B = ∅ ⇐⇒ A = B = ∅. Similarly, with the neutral element RN A ∩ B = B ∩ A,

A ∩ RN = A,

A ∩ B = RN ⇐⇒ A = B = RN .

2. Notation and Definitions 2.2.1

57

First Abelian Group Structure on (P(D), ∆)

Denote by
the symmetric difference of two sets in P(D):  

 def A
B = A ∩ B ∪ B ∩ A = [A ∪ B] ∩ A ∪ B = [A ∪ B] ∩  [A ∩ B] . By definition,
is commutative and associative: A
B = B
A,

(A
B)
C = A
(B
C).

∅ is a neutral element: A
∅ = A. An inverse must verify

  A
B = ∅ ⇐⇒ A ∩ B ∪ B ∩ A = ∅ ⇐⇒ B ∩ A = ∅ = A ∩ B ⇐⇒ A = B. Therefore A
A = ∅, every element A of P(D) is its own inverse, A−1 = A, and (P(D),
) is an Abelian group. This yields a kind of triangle inequality: (A
B)
(B
C) = A
C. The symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C. (But note that for the diameter of the symmetric difference the triangle inequality does not hold.) Because every element in this group is its own inverse, (P(D),
) is in fact a vector space over the field Z2 with two elements. Finally, the intersection distributes over the symmetric difference: A ∩ (B
C) = (A ∩ B)
(A ∩ C). Hence the power set P(D) becomes a ring with symmetric difference as addition and intersection as multiplication. It is the prototypical example of a Boolean ring. For operations involving the complement A
B = A
B, (A
B) = [A ∩ B] ∪ [A ∩ B] = B
A = A
B ⇒ A
RN = A,

A
A = RN .

For operations involving the closure   ∂A =  A
A = A
A = A
A ⇒ ∂A
A = A

and ∂A
A = A.

58 2.2.2

Chapter 2. Classical Descriptions of Geometries and Their Properties Second Abelian Group Structure on (P(D), ∇)

If for some reason ∅ is not acceptable as the multiplicative neutral element, RN can play that role with the following new multiplication: def

AB = 



    A ∩ B ∪ B ∩ A =  [A ∪ B] ∩  [A ∩ B] =  [A ∪ B] ∪ [A ∩ B] .

By definition AB = BA,

A RN = A.

A multiplicative inverse must verify

  A ∩ B ∪ B ∩ A = ∅ AB = RN ⇔ ⇔ B ∩ A = ∅ = A ∩ B ⇔ A = B. Therefore A−1 = A and (P(D), ) are an Abelian group. When D = RN , some identities are interesting: AB = AB,

 AB =  [A ∩ B] ∪ B ∩ A = (AB) = AB, ⇒ A∅ = A, AA = ∅. 

2.3

Connected Space, Path-Connected Space, and Geodesic Distance

Definition 2.1. (i) A connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets. (ii) A topological space X is said to be path-connected (or pathwise connected or 0-connected ) if for any two points x and y in X there exists a continuous function f from the unit interval [0, 1] to X with f (0) = x and f (1) = y. (This function is called a path in X from x to y.) (iii) Given a path-connected subset S of RN , the geodesic distance distS (x, y) between two points x and y of S is the infimum of the lengths of the paths in S joining x and y. The definitions of a connected space and a path-connected space are compatible with the convention that the empty set ∅ be both connected and pathconnected. Theorem 2.1. (i) The closure of a connected set is connected. But the closure of a path-connected set need not be path-connected. (ii) Every path-connected space is connected. But a connected space need not be path-connected.

2. Notation and Definitions

59

(iii) Y is path-connected if and only if it is connected, and each y ∈ Y has a pathconnected neighborhood. In particular, open subsets of RN are connected if and only if they are path-connected. Proof. Cf. J. Dugundji [1]: (i) Thm. 1.4 and Ex. 4, p. 108. (ii) Thm. 5.3, p. 115. The topologist’s sine curve {(x, y) : y = sin 1/x, x > 0} ∪ {(0, y) : |y| ≤ 1} in R2 is an example of a connected space that is not path-connected. (iii) Thm. 5.5, p. 116, Cor. 5.6, p. 116. Remark 2.1. Subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R. Definition 2.2. Given a subset A of RN , def

#(A) =

number of connected components of A, 0,

if A = ∅, if A = ∅.

We say that the set A is hole-free if A is connected or A = RN . If we adopt the convention that the empty set ∅ is both connected and pathconnected, then a set A is connected if and only if its complement A is hole-free. Theorem 2.2. Let A and Ω be subsets of RN . (i) A set A = ∅ is connected if and only if #(A) = 1; a set A is hole-free if and only if #(A) ≤ 1. (ii) An open set Ω = ∅ is path-connected if and only if #(Ω) = 1; an open set Ω is hole-free if and only if #(Ω) ≤ 1. Remark 2.2. Observe that for a nonempty open subset Ω of an open holdall D in RN , #(Ω ∩ D) can be strictly greater than #(Ω). As an example, let D be the open square of the side equal to 2 centered in (0, 0) and Ω be the open ball of center (0, 0) and radius 1: #(Ω ∩ D) = 4 > 1 = #(Ω).

2.4

Bouligand’s Contingent Cone, Dual Cone, and Normal Cone

Definition 2.3. Let ∅ = Ω ⊂ RN and let x ∈ Ω. The dual cone associated with Ω is defined as def

Ω∗ = x∗ ∈ RN : ∀x ∈ Ω,

Ω∗ is a closed convex cone in 0.

 x∗ · x ≥ 0 .

(2.2)

60

Chapter 2. Classical Descriptions of Geometries and Their Properties

Definition 2.4. Let ∅ = Ω ⊂ RN and let x ∈ Ω. (i) The vector h ∈ RN is an admissible direction for Ω in x if there exists a sequence {εn > 0}, εn  0 as n → ∞, such that ∀n, ∃xn ∈ Ω and

lim

n→∞

xn − x = h. εn

(2.3)

Tx Ω will denote the set of all admissible directions for Ω in x and it will be referred to as Bouligand’s contingent cone 1 to Ω in x. (ii) The dual cone (−Tx Ω)∗ associated with Ω and a point x ∈ Ω will be referred to as the normal cone to Ω in x ∈ Ω. In general, Tx Ω is a closed cone in 0. If Ω is convex, then Tx Ω = R(Ω − x) is a closed convex cone in 0. When x ∈ int Ω, TΩ (x) = RN .

2.5 2.5.1

Sobolev Spaces Definitions

For a detailed analysis of Sobolev spaces the reader is referred to R. A. Adams [1], ˇas [1], and Vo-Khac Khoan [1]. Let Ω be a bounded open subset of RN . J. Nec D(Ω) is the space of infinitely continuously differentiable functions with compact support in Ω endowed with Schwartz’s topology. Its topological dual D(Ω)∗ is called the space of distributions. Denote by NN the set of all N -tuples α = (α1 , . . . , αn ) ∈ NN . An element of NN will be called a multi-index. For each α ∈ NN , define the order |α| of α and the partial derivative ∂ α as follows: |α| =

N

αi ,

∂α =

i=1

1 ∂xα 1

∂ |α| . N . . . ∂xα N

(2.4)

The Sobolev space W m,p (Ω), 1 ≤ p ≤ ∞, m ∈ N, is the space of all distributions T ∈ Lp (Ω)∗ with distributional partial derivatives Dα T ∈ Lp (Ω)∗ for all α, |α| ≤ m. For p = 2 we shall use the notation H m (Ω) = W m,2 (Ω). By density of D(Ω) in Lp (Ω) for 1 ≤ p < ∞, we shall often identify a distribution T ∈ Lp (Ω)∗ with a function f ∈ Lp (Ω), p−1 + q −1 = 1:  ∀ϕ ∈ D(Ω), T, ϕ = f ϕ dx. Ω

Endowed with the norm  1/p def ∂ α vpLp (Ω) dx , vm,p,Ω =  |α|≤m

def

vm,∞,Ω = max ∂ α vL∞ (Ω) dx (2.5) |α|≤m

1 The contingent cone T (x) was introduced in the 1930s by G. Bouligand [1]. It naturally Ω occurs in the viability theory of differential equations (cf. M. Nagumo [1] or J.-P. Aubin and A. Cellina [1, p. 174 and p. 180]).

2. Notation and Definitions

61

the space W m,p (Ω) is a Banach space. We shall also use the seminorm  |v|m,p,Ω = 

 |α|=m

Ω

1/p |∂ α v|pLp (Ω) dx

,

def

|v|m,∞,Ω = ∂ m vL∞ (Ω) dx.

(2.6)

When p = 2 we drop the index p and write vm,Ω and |v|m,Ω . By the Meyers and Serrin theorem W m,p (Ω) coincides with the completion of {ϕ ∈ C m (Ω) : ϕm,p < ∞} with respect to the norm ϕm,p for 1 ≤ p < ∞.2 The reader is referred to R. A. Adams [1] for details and extension of this definition to the case where m is not an integer. When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by W s,p (Ω)m or W s,p (Ω, Rm ). 2.5.2

The Space W0m,p (Ω)

Define the Sobolev space W m,p

W0m,p (Ω) = D(Ω)

,

(2.7)

where the closure is with respect to the norm  · m,p,Ω of (2.5). When Ω is bounded and 1 ≤ p < ∞, there exists a constant C(diam Ω, m, p) such that ∀ v ∈ W0m,p (Ω),

vm,p,Ω ≤ C(diam Ω, m, p) |v|m,p,Ω

(2.8)

and the seminorm | · |m,p,Ω is a norm on the space W0m,p (Ω) equivalent to the norm  · m,p,Ω (cf. W. P. Ziemer [1, Thm. 4.4.1, p. 188]). We have the chain of continuous embeddings W0m,p (Ω) → W m,p (Ω) → Lp (Ω). For p = 2, m ≥ 1, and a bounded open Ω, the injection of H0m (Ω) into H0m−1 (Ω) is compact (cf. R. A. Adams [1, Rellich–Kondrachov compactness Theorem 6.2, p. 144] for other embedding theorems). Now assume that the set Ω is Lipschitzian in the sense of Definition 5.2 (ii) in section 5.2. There exists a constant C(Ω) such as ∀ v ∈ C ∞ (Ω),

vL2 (Γ) ≤ C(Ω) v1,Ω .

(2.9)

But, by Theorem 6.3 in section 6.1 C ∞ (Ω) = H 1 (Ω), where the closure is with respect to the norm · 1,Ω . Hence there exists a continuous linear map v ∈ H 1 (Ω) → v ∈ L2 (Γ) that will be referred to as the trace operator . From this we get the following characterization:

 H01 (Ω) = v ∈ H 1 (Ω) : v = 0 on Γ . (2.10) 2 C m (Ω)

is defined in section 2.6.1.

62

Chapter 2. Classical Descriptions of Geometries and Their Properties

Since the exterior unit normal ν = (ν1 , . . . , νN ) exists almost everywhere along Γ, the normal derivative operator ∂ ∂ = νi ∂ν ∂x i i=1 N

(2.11)

is also defined almost everywhere along Γ and the following characterization also holds:   ∂v 2 2 = 0 on Γ . (2.12) H0 (Ω) = v ∈ H (Ω) : v = 0 and ∂ν Given two functions u, v in H 1 (Ω), we get the following Green formula:    ∂v ∂u u dx = − v dx + uv νi dΓ ∂xi Ω Ω ∂xi Γ

(2.13)

is true for 1 ≤ i ≤ N . From that formula, other Green formulae can be obtained. For instance, by replacing u by ∂u/∂xi and summing from 1 to N , we get  ∇u · ∇v dx = Ω

   N ∂u ∂v ∂u v dΓ dx = − ∆u v dx + ∂x ∂x i i Ω i=1 Ω Γ ∂ν

(2.14)

for all u ∈ H 2 (Ω) and v ∈ H 1 (Ω). 2.5.3

Embedding of H01 (Ω) into H01 (D)

Let Ω and D be two open subsets of RN such that Ω ⊂ D. Denote by e0 (ϕ) the extension by zero of an element ϕ of D(Ω) to D and consider the linear injection ϕ → e0 (ϕ) : D(Ω) → D(D). By definition, for k ≥ 1, ϕH k (Ω) = e0 (ϕ)H k (D) and e0 extends by continuity and density to a linear isometric map def

Hk

e0 : H0k (Ω) = D(Ω)

def

Hk

→ H0k (D) = D(D)

.

Denote by H0k (Ω; D) the image of H0k (Ω) by e0 . Theorem 2.3. Let Ω ⊂ D be two open subsets of RN such that D is bounded. The linear subspace H0k (Ω; D) of H0k (D) is closed and isometrically isomorphic to H0k (Ω) with the following properties: for all ψ ∈ H0k (Ω; D) ψ|Ω ∈ H0k (Ω) and ∀α, |α| ≤ k,

∂ α ψ = 0 a.e. in D\Ω.

Any convergent sequence in H0k (Ω)-weak converges in H0k−1 (Ω)-strong.

(2.15)

2. Notation and Definitions

63

Proof. (i) It is sufficient to prove that H0k (Ω; D) is closed. The other properties are easy to check. Pick a sequence {ϕn } in H0k (Ω) such that {e0 (ϕn )} is Cauchy in H0k (D) and denote by Φ its limit in H0k (D). Since e0 is an isometry, then {ϕn } is also Cauchy in H0k (Ω). Denote by ϕ its limit in H0k (Ω). Then Φ − e0 (ϕ)H k (D) ≤ Φ − e0 (ϕn )H k (D) + e0 (ϕn − ϕ)H k (D) = Φ − e0 (ϕn )H k (D) + ϕn − ϕH k (Ω) → 0 and there exists ϕ ∈ H0k (Ω) such that Φ = e0 (ϕ) in H0k (D). Hence Φ ∈ H0k (Ω; D). (ii) Since Ω is bounded, there exists a sufficiently large open ball B such that Ω ⊂ B. By the embedding of H0k (Ω) into H0k (B), if a sequence ϕn converges to ϕ in H0k (Ω)-weak, then eΩ (ϕn ) converges to eΩ (ϕ) in H0k (B)-weak. Since the ball is sufficiently smooth, by Rellich’s theorem, the sequence converges in H0k−1 (B)strong, and in view of the linear isometric isomorphism, ϕn converges to ϕ in H0k−1 (Ω)-strong. 2.5.4

Projection Operator

Given u ∈ H01 (D), let y = y(Ω) ∈ H01 (Ω; D) be the solution of the variational problem   1 ∀ϕ ∈ H0 (Ω; D), ∇y · ∇ϕ dx = ∇u · ∇ϕ dx. D

D

The projection operator PΩ of H01 (D) onto H01 (Ω) or H01 (Ω; D) is the mapping def

u → PΩ u = y(Ω) : H01 (D) → H01 (Ω; D) that is linear and continuous since ∇PΩ uL2 (D) ≤ ∇uL2 (D) . A fundamental issue will be the continuity of the projection PΩ with respect to Ω.

2.6 2.6.1

Spaces of Continuous and Differentiable Functions Continuous and C k Functions

Let Ω be an open subset of RN . Denote by C(Ω) or C 0 (Ω) the space of continuous functions from Ω to R. For a multi-index α ∈ NN , let |α| be the order of α and ∂ α be the partial derivative as defined in (2.4) of section 2.5.1. For an integer k ≥ 1,  def

C k (Ω) = f ∈ C k−1 (Ω) : ∂ α f ∈ C(Ω), ∀α, |α| = k . By convention ∂ 0 f will be the function f in order to make sense of the case α = 0. When |α| = 1, we also use the standard notation ∂i f or ∂f /∂xi . Dk (Ω) or Cck (Ω) (resp., D(Ω) or Cc∞ (Ω)) will denote the space of all k-times (resp., infinitely)

64

Chapter 2. Classical Descriptions of Geometries and Their Properties

continuously differentiable functions with compact support contained in the open set Ω. Denote by B0 (Ω) the space of bounded continuous functions from Ω to R, and, for an integer k ≥ 1, the space  def

B k (Ω) = f ∈ Bk−1 (Ω) : ∂ α f ∈ B 0 (Ω), ∀α, |α| = k , that is, the space of all functions in B0 (Ω) whose derivatives of order less than or equal to k are continuous and bounded in Ω. Endowed with the norm def

f C k (Ω) =

max

sup |∂ α f (x)|,

(2.16)

0≤|α|≤k x∈Ω

B k (Ω) is a Banach space. When D is open but not necessarily bounded, the space C 0 (D) of continuous functions on D is endowed with the Fr´echet topology of uniform convergence on compact subsets K of D, which is defined by the family of seminorms ∀K compact ⊂ D,

def

qK (f ) = max |f (x)|.

(2.17)

x∈K

It is metrizable since the topology induced by the family of seminorms {qK } is equivalent to the one generated by the subfamily {qKk }k≥1 , where the compact sets {Kk }k≥1 are chosen as follows:   1 def (2.18) Kk = x ∈ D : d
D (x) ≥ and |x| ≤ k , k ≥ 1 k ´ th [1, Ex. 3, p. 116]). Thus the Fr´echet topology on (cf., for instance, J. Horva C 0 (D) is equivalent to the topology defined by the metric def

δ(f, g) =

∞ 1 qKk (f − g) . 2k 1 + qKk (f − g)

(2.19)

k=1

0 When C 0 (D) is endowed with that topology, it will be denoted by Cloc (D). 3 If a function f is bounded and uniformly continuous on Ω, it possesses a unique, continuous extension to the closure Ω of Ω. Denote by C k (Ω) the space of functions f in C k (Ω) for which ∂ α f is bounded and uniformly continuous on Ω for all α, 0 ≤ |α| ≤ k. A function f in C k (Ω) is said to vanish at the boundary of Ω if for every α, 0 ≤ |α| ≤ k, and ε > 0 there exists a compact subset K of Ω such that, for all x ∈ Ω ∩ K, |∂ α f (x)| ≤ ε. Denote by C0k (Ω) the space of all such functions. Clearly C0k (Ω) ⊂ C k (Ω) ⊂ Bk (Ω) ⊂ C k (Ω). Endowed with the norm (2.16), C0k (Ω), C k (Ω), and Bk (Ω) are Banach spaces. Finally % % % def def def C k (Ω), C0∞ (Ω) = C0k (Ω), and B(Ω) = Bk (Ω). C ∞ (Ω) = k≥0

k≥0

k≥0

function f : Ω → R is uniformly continuous if for each ε > 0 there exists δ > 0 such that for all x and y in Ω such that |x − y| < δ we have |f (x) − f (y)| < ε. 3A

2. Notation and Definitions

65

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C0k (Ω)m or C0k (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), B k (Ω)m or Bk (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), etc. We quote the following classical compactness theorem. Theorem 2.4 (Ascoli–Arzel`a theorem). Let Ω be a bounded open subset of RN . A subset K of C(Ω) is precompact in C(Ω) provided the following two conditions hold: (i) there exists a constant M such that for all f ∈ K and x ∈ Ω, |f (x)| ≤ M ; (ii) for every ε > 0 there exists δ > 0 such that if f ∈ K, x, y in Ω, and |x−y| < δ, then |f (x) − f (y)| < ε. 2.6.2

H¨ older (C 0,
) and Lipschitz (C 0,1 ) Continuous Functions

Given λ, 0 < λ ≤ 1, a function f is (0, λ)-H¨ older continuous in Ω if ∃c > 0, ∀x, y ∈ Ω,

|f (y) − f (x)| ≤ c |x − y|λ .

When λ = 1, we also say that f is Lipschitz or Lipschitz continuous. Similarly for k ≥ 1, f is (k, λ)-H¨ older continuous in Ω if |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ .

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω,

Denote by C k,λ (Ω) the space of all (k.λ)-H¨older continuous functions on Ω. Define for k ≥ 0 the subspaces4 & ∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω def (2.20) C k,λ (Ω) = f ∈ C k (Ω) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y|λ of C k (Ω). By definition for each α, 0 ≤ |α| ≤ k, ∂ α f has a unique, bounded, continuous extension to Ω. Endowed with the norm      |∂ α f (y) − ∂ α f (x)|  def f C k,λ (Ω) = max f C k (Ω) , max sup (2.21)   |x − y|λ 0≤|α|≤k x,y∈Ω   x=y

C k,λ (Ω) is a Banach space. Finally denote by C0k,λ (Ω) the space C k,λ (Ω) ∩ C0k (Ω). 2.6.3

Embedding Theorem

In general C k+1 (Ω) ⊂ C k,1 (Ω), but the inclusion is true for a large class of domains, including convex domains.5 We first quote the following embedding theorem. 4 The notation C k,λ (Ω) should not be confused with the notation C k,λ (Ω) for (k, λ)-H¨ older continuous functions in Ω without the uniform boundedness assumption in Ω. In particular, C k,λ (RN ) is contained in but not equal to C k,λ (RN ). 5 For instance the convexity can be relaxed to the more general condition: there exists M > 0 such that for all x and y in Ω, there exists a path γx,y in Ω (that is, a C 1 injective map from the

interval [0, 1] into Ω with γ(0) = x, γ(1) = y, and

1 0

γ
(t) dt ≤ M |x − y|).

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 2.5 (R. A. Adams [1]). Let k ≥ 0 be an integer and 0 < ν < λ ≤ 1 be real numbers. Then the following embeddings exist: C k+1 (Ω) → C k (Ω),

(2.22)

(Ω) → C (Ω),

(2.23)

C C

k,λ

k,λ

k

(Ω) → C

k,ν

(Ω).

(2.24)

If Ω is bounded, then the embeddings (2.23) and (2.24) are compact. If Ω is convex, we have the further embeddings C k+1 (Ω) → C k,1 (Ω), C

k+1

(Ω) → C

k,ν

(Ω).

(2.25) (2.26)

If Ω is convex and bounded, then embeddings (2.22) and (2.26) are compact. As a consequence of the second part of the theorem, the definition of C k,λ (Ω) simplifies when Ω is convex: def

C k,λ (Ω) = f ∈ Bk (Ω) : ∀α, |α| = k, ∃c > 0, ∀x, y ∈ Ω,  |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ ,

(2.27)

and its norm is equivalent to the norm def

|∂ α f (y) − ∂ α f (x)| . |x − y|λ x,y∈Ω

f C k,λ (Ω) = f C k (Ω) + max sup |α|=k

(2.28)

x=y

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C k,λ (Ω)m or C k,λ (Ω, Rm ). 2.6.4

Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance

By Rademacher’s theorem (cf., for instance, L. C. Evans and R. F. Gariepy [1]) k+1,∞ C k,1 (Ω) ⊂ Wloc (Ω)

and, when Ω is convex, C k,1 (Ω) = W k+1,∞ (Ω). This last convexity assumption can be relaxed as follows. From Rademacher’s ´zis [1, p. 154]) theorem and the inequality (cf. H. Bre ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω, we have the following theorem.

|u(x) − u(y)| ≤ ∇uL∞ (Ω) distΩ (x, y)

3. Sets Locally Described by an Homeomorphism or a Diffeomorphism

67

Theorem 2.6. Let Ω be a bounded, path-connected, open subset of RN such that ∃c, ∀x, y ∈ Ω,

distΩ (x, y) ≤ c |x − y|.

(2.29)

Then we have W 1,∞ (Ω) = C 0,1 (Ω) algebraically and topologically, and there exists a constant c such that ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω,

|u(x) − u(y)| ≤ c ∇uL∞ (Ω) |x − y|.

Hence for all integers k ≥ 0, W k+1,∞ (Ω) = C k,1 (Ω) algebraically and topologically. This is true for Lipschitzian domains (D. Gilbarg and N. S. Trudinger [1]). Corollary 1. Let Ω be a bounded, open, path-connected, and locally Lipschitzian6 subset of RN . Then property (2.29) and Theorem 2.6 are verified. Proof. By Theorem 5.8 later in section 5.4 of this chapter.

3

Sets Locally Described by an Homeomorphism or a Diffeomorphism

Open domains in the Euclidean space RN are classically described by introducing at each point of their boundary a local diffeomorphism (that is, defined in a neighborhood of the point) that locally flattens the boundary. The smoothness of the boundary is determined by the smoothness of the local diffeomorphism. After introducing the main definition in section 3.1, we briefly recall the definition of an (embedded) submanifold of codimension greater than or equal to 1 in section 3.2 in order to make sense of the associated boundary integral and, more generally, the integral over smooth submanifolds of arbitrary dimension in RN . The definition of the integral can be generalized by introducing the Hausdorff measures that extend the integration theory from smooth submanifolds to arbitrary Hausdorff measurable subsets of RN , thus making the writing of the integral completely independent of the choice of the local diffeomorphisms. Section 3.3 completes the section by defining the fundamental forms and curvatures for a smooth submanifold of RN of codimension 1.

3.1

Sets of Classes C k and C k,


Let {e1 , . . . , eN } be the standard unit orthonormal basis in RN . We use the notation ζ = (ζ  , ζN ) for a point ζ = (ζ1 , . . . , ζN ) in RN , where ζ  = (ζ1 , . . . , ζN −1 ). Denote by B the open unit ball in RN and define the sets def

B0 = {ζ ∈ B : ζN = 0} , def

B+ = {ζ ∈ B : ζN > 0} ,

def

B− = {ζ ∈ B : ζN < 0} .

(3.1) (3.2)

The main elements entering Definition 3.1 are illustrated in Figure 2.1 for N = 2. 6 Cf. Definition 5.2 of section 5 and Theorem 5.3, which says that a locally Lipschitzian domain is equi-Lipschitzian when ∂Ω is bounded.

68

Chapter 2. Classical Descriptions of Geometries and Their Properties ζN gx B+ ζ

set Ω B0 ∂Ω

x U (x)

hx = gx−1

hx (x) = (0, 0) B−

ball B

Figure 2.1. Diffeomorphism gx from U (x) to B. Definition 3.1. Let Ω be a subset of RN such that ∂Ω = ∅, 0 ≤ k be an integer or +∞, and 0 ≤  ≤ 1 be a real number. (i)

- Ω is said to be locally of class C k at x ∈ ∂Ω if there exist (a) a neighborhood U (x) of x, and (b) a bijective map gx : U (x) → B with the following properties:     def gx ∈ C k U (x); B , hx = gx−1 ∈ C k B; U (x) , int Ω ∩ U (x) = hx (B+ ), def Γx = ∂Ω ∩ U (x) = hx (B0 ), B0 = gx (Γx ).

(3.3) (3.4) (3.5)

- Given 0 <  < 1, Ω is said to be locally (k, )-H¨ olderian at x ∈∂Ω if ∈ C k,
U (x), B with conditions (a) and (b) are satisfied with a map g x   −1 k,
B, U (x) . inverse hx = gx ∈ C - Ω is said to be locally k-Lipschitzian at x ∈ ∂Ω  if conditions (a) and−1(b) k,1 are satisfied with a map g U (x), B with inverse hx = gx ∈ ∈ C x   C k,1 B, U (x) . (ii)

- Ω is said to be locally of class C k if, for each x ∈ ∂Ω, Ω is locally of class C k at x. - Given 0 <  < 1, Ω is said to be locally (k, )-H¨ olderian if, for each x ∈ ∂Ω, Ω is locally (k, )-H¨olderian at x. - Ω is said to be locally k-Lipschitzian if, for each x ∈ ∂Ω, Ω is locally k-Lipschitzian at x.

Remark 3.1. By definition int Ω = ∅ and int Ω = ∅. The above definitions are usually given for an open set Ω called a domain. This terminology naturally arises in partial differential equations, where this open set is indeed the domain in which the solution

3. Sets Locally Described by an Homeomorphism or a Diffeomorphism

69

of the partial differential equation is defined.7 At this point it is not necessary to assume that the set Ω is open. The family of diffeomorphisms {gx : x ∈ ∂Ω} characterizes the equivalence class of sets [Ω] = {A ⊂ RN : int A = int Ω and ∂A = ∂Ω} with the same interior and boundary. The sets int Ω and ∂Ω are invariants for the equivalence class [Ω] of sets of class C k,
. The notation Γ for ∂Ω and the standard terminology domain for the unique (open) set int Ω associated with the class [Ω] will be used. In what follows, a C k,
-mapping gx with C k,
-inverse will be called a C k,
-diffeomorphism. Consider the solution y = y(Ω) of the Dirichlet problem on the bounded open domain Ω: −∆y = f in Ω,

y = g on ∂Ω.

(3.6)

When Ω is of class C ∞ , the solution y(Ω) can be sought in any Sobolev space H m (Ω), m ≥ 1, by choosing sufficiently smooth data f and g and appropriate compatibility conditions. However, when Ω is only of class C k , 1 ≤ k < ∞, m cannot be made arbitrary large by choosing smoother data f and g and appropriate compatibility conditions. The reader is referred to R. Dautray and J.-L. Lions [1, Chap. VII, sect. 3, pp. 1271–1304] for classical smoothness results of solutions to elliptic problems in domains of class C k . It is important to understand that we shall consider shape problems involving the solution of a boundary value problem over domains with minimal smoothness. Since we know that, at least for elliptic problems, the smoothness of the solution depends not only on the smoothness of the data but also on the smoothness of the domain. This issue will be of paramount importance to make sure that the shape problems are well posed. Remark 3.2. We shall see in section 5 that domains that are locally the epigraph of a C k,
, k ≥ 1 (resp., Lipschitzian), function are of class C k,
(resp., C 0,1 ), but domains which are of class C 0,1 are not necessarily locally the epigraph of a Lipschitzian function (cf. Examples 5.1 and 5.2). Nevertheless globally C 0,1 -mappings with a C 0,1 inverse are important since they transport Lp -functions onto Lp -functions and W 1,p -functions ˇas [1, Lems. 3.1 and 3.2, pp. 65–66]). onto W 1,p -functions (cf. J. Nec For sets of class C 1 , the unit exterior normal to the boundary Γ = ∂Ω can be characterized through the Jacobian matrices of gx and hx . By definition of B0 , {e1 , . . . , eN −1 } ⊂ B0 and the tangent space Ty Γ, Γ = ∂Ω, at y to Γx is the vector space spanned by the N − 1 vectors {Dhx (ζ  , 0)ei : 1 ≤ i ≤ N − 1} ,

(ζ  , 0) = gx (y) ∈ B0 ,

(3.7)

7 Classically for k ≥ 1 a bounded open domain Ω in RN is said to be of class C k if its boundary is a C k -submanifold of RN of codimension 1 and Ω is located on one side of its boundary Γ = ∂Ω (cf. S. Agmon [1, Def. 9.2, p. 178]).

70

Chapter 2. Classical Descriptions of Geometries and Their Properties

where Dhx (ζ) is the Jacobian matrix of hx at the point ζ: def

(Dhx )
m = ∂m (hx )
. So from (3.7) a normal vector field to Γx at y ∈ Γx is given by mx (y) = − ∗(Dhx )−1 (ζ  , 0) eN = − ∗Dgx (y) eN ,

hx (ζ  , 0) = y,

(3.8)

since −mx (y) · Dhx (ζ  , 0) ei = eN · ei = δiN ,

1 ≤ i ≤ N.

Thus the outward unit normal field n(y) at y ∈ Γx is given by (Dhx )−1 (ζ  , 0) eN , | ∗(Dhx )−1 (ζ  , 0) eN | ∗ (Dhx )−1 (h−1 x (y)) eN . n(y) = − ∗ −1 | (Dhx ) (h−1 x (y)) eN |

∀hx (ζ  , 0) = y ∈ Γx , ∀y ∈ Γx ,

n(y) = −

∗

(3.9)

It can be verified that n is uniquely defined on Γ by checking that for y ∈ Γx ∩ Γx , n is uniquely defined by (3.9).

3.2 3.2.1

Boundary Integral, Canonical Density, and Hausdorff Measures Boundary Integral for Sets of Class C 1

The family of neighborhoods U (x) associated with all the points x of Γ is an open cover of Γ. If Γ = ∂Ω is assumed to be compact, then there exists a finite open subcover: that is, there exists a finite sequence of points {xj : 1 ≤ j ≤ m} of Γ such that Γ ⊂ U1 ∪ · · · ∪ Um , where Uj = U (xj ). For simplicity, index all the previous symbols by j instead of xj . The boundary integration on Γ is obtained by using a partition of unity {rj : 1 ≤ j ≤ m} for the family of open neighborhoods {Uj : 1 ≤ j ≤ m} of Γ:   r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1,   j m (3.10)  rj (x) = 1 in a neighborhood U of Γ,   j=1

such that U ⊂ ∪m j=1 Uj , where D(Uj ) is the set of all infinitely continuously differentiable functions with compact support in Uj . If f ∈ C(Γ), then (f rj ) ◦ hj ∈ C(B0 ),

1 ≤ j ≤ m.

Define the boundary integral of f rj on Γj as   def f rj dΓ = (f rj ) ◦ hj (ζ  , 0) ωj (ζ  ) dζ  , Γj

B0

Γj = U (xj ) ∩ Γ,

(3.11)

(3.12)

3. Sets Locally Described by an Homeomorphism or a Diffeomorphism

71

where ωj = ωxj and ωx is the density term ωx (ζ  ) = |mx (hx (ζ  , 0))| | det Dhx (ζ  , 0)|, ∗

mx (y) = − (Dhx )

−1

(h−1 x (y)) eN

= − Dgx (y) eN .

From this define the boundary integral of f on Γ as  m  def f dΓ = f rj dΓ. Γ

j=1

(3.13)

∗

(3.14)

Γj

The expression on the right-hand side of (3.12) results from the parametrization of the boundary Γj by B0 through the diffeomorphism hj . 3.2.2

Integral on Submanifolds

In differential geometry there is a general procedure to define the canonical density for a d-dimensional submanifold V in RN parametrized by a C k -mapping, k ≥ 1 (cf., for instance, M. Berger and B. Gostiaux [1, Def. 2.1.1, p. 48 (56) and Prop. 6.62, p. 214 (239)]). Definition 3.2. Let ∅ = S ⊂ RN , let k ≥ 1 and 1 ≤ d ≤ N be integers, and let 0 ≤  ≤ 1 be a real number. (i) Given x ∈ S, S is said to be locally a C k - (resp., C k,
-) submanifold of dimension d at x in RN if there exists an open subset U (x) of RN containing k k,
x and  a C - (resp., C -) diffeomorphism gx from U (x) onto its open image gx U (x) , such that     gx U (x) ∩ S = gx U (x) ∩ Rd ,   where Rd = (x1 , . . . , xd , 0, . . . , 0) ∈ RN : ∀(x1 , . . . , xd ) ∈ Rd . (ii) S is said to be a C k - (resp., C k,
-) submanifold of dimension d in RN if, for each x ∈ S, it is locally a C k - (resp., C k,
-) submanifold of dimension d at x in RN . For k ≥ 1 the canonical density ωx on the submanifold S at a point y ∈ U (x) ∩ S is given by ' ωx = | det B|, where the d × d matrix B is given by (B)ij = (Dhx ei ) · (Dhx ej ), 1 ≤ i, j ≤ d,

hx = gx−1 on gx (U (x)).

In the special case d = N − 1, it is easy to verify that  ∗ C C ∗C c ∗ , B = ∗C C, Dhx Dhx = ∗ ∗ cC cc

72

Chapter 2. Classical Descriptions of Geometries and Their Properties

where C is the (N × (N − 1))-matrix and c is the N -vector defined by Cij = {Dhx }ij ,

1 ≤ i ≤ N, 1 ≤ j ≤ N − 1,

c = Dhx eN .

Denote by M (A) the matrix of cofactors associated with a matrix A: M (A)ij is equal to the determinant of the matrix obtained after deleting the ith row and the jth column times (−1)i+j . Then M (A) = (det A) ∗ A−1 , M ( ∗ A) = ∗ M (A), and for two invertible matrices A1 and A2 , M (A1 A2 ) = M (A1 )M (A2 ). As a result det B = M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN , where M ( ∗Dhx Dhx )N N is the N N -cofactor of the matrix ∗Dhx Dhx . Then M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN = eN · M ( ∗Dhx ) M (Dhx )eN = |M (Dhx )eN |2 . In view of the previous considerations and (3.13) √ det B = |M (Dhx )eN | = | det Dhx | | ∗(Dhx )−1 eN | = ωx . 3.2.3

Hausdorff Measures

Definition 3.2 gives the classical construction of a d-dimensional surface measure on the boundary of a C 1 -domain. In 1918, F. Hausdorff [1] introduced a ddimensional measure in RN which gives the same surface measure for smooth submanifolds but is defined on all measurable subsets of RN . When d = N , it is equal to the Lebesgue measure. To complete the discussion we quote the definition from F. Morgan [1, p. 8] or L. C. Evans and R. F. Gariepy [1, p. 60]. Definition 3.3. For any subset S of RN , define the diameter of S as def

diam (S) = sup{|x − y| : x, y ∈ S}. Let α(d) denote the Lebesgue measure of the unit ball in Rd . The d-dimensional Hausdorff measure Hd (A) of a subset A of RN is defined by the following process. For δ small, cover A efficiently by countably many sets Sj with diam (Sj ) ≤ δ, add up all the terms d

α(d) (diam (Sj )/2) , and take the limit as δ → 0: def

Hd (A) = lim

δ0

inf

A⊂∪Sj diam (Sj )≤δ

j

 α(d)

d diam (Sj ) , 2

where the infimum is taken over all countable covers {Sj } of A whose members have diameter at most δ.

3. Sets Locally Described by an Homeomorphism or a Diffeomorphism

73

For 0 ≤ d < ∞, Hd is a Borel regular measure, but not a Radon measure for d < N since it is not necessarily finite on each compact subset of RN . The Hausdorff dimension of a set A ⊂ RN is defined as def

Hdim (A) = inf{0 ≤ s < ∞ : Hs (A) = 0}.

(3.15)

By definition Hdim (A) ≤ N and ∀k > Hdim (A),

Hk (A) = 0.

If a submanifold S of dimension d, 1 ≤ d < N , of Definitions 3.2 and 3.3 is characterized by a single C 1 -diffeomorphism g, that is, h = g −1 ,

g(S) = Rd and S = h(Rd ), then for any Lebesgue-measurable set E ⊂ Rd  ω dx = Hd (h(E)). E

This is a generalization to submanifolds of codimension greater than 1 of formula (3.12). The definition of the Hausdorff measure and the Hausdorff dimension extend from integers d to reals s, 0 ≤ s ≤ ∞, by modifying Definition 3.3 as follows. Definition 3.4. For any real s, 0 ≤ s ≤ ∞, the s-dimensional Hausdorff measure Hs (A) of a subset A of RN is defined by the following process. For δ small, cover A efficiently by countably many sets Sj with diam (Sj ) ≤ δ, and add all the terms α(s) (diam (Sj )/2)d , where def

α(s) =

π s/2 , Γ(s/2 + 1)

Take the limit as δ → 0: def

Hs (A) = lim

δ0

inf

def



∞

Γ(t) =

A⊂∪Sj diam (Sj )≤δ

e−x xt−1 .

(3.16)

0

j

 α(s)

d diam (Sj ) , 2

where the infimum is taken over all countable covers {Sj } of A whose members have diameter at most δ. The Hausdorff dimension is defined by the same formula (3.15).

3.3

Fundamental Forms and Principal Curvatures

Consider a set Ω locally of class C 2 in RN . Its boundary Γ = ∂Ω is an (N − 1)dimensional submanifold of RN of class C 2 . At each point x ∈ Γ there is a C 2 diffeomorphism gx from a neighborhood U (x) of x onto B. Denoting its inverse by

74

Chapter 2. Classical Descriptions of Geometries and Their Properties

hx = gx−1 , the covariant basis at a point y ∈ U (x) ∩ Γ is defined as def

aα (y) =

∂hx  (ζ , 0), ∂ζα

α = 1, . . . , N − 1,

hx (ζ  , 0) = y,

and aN is chosen as the inward unit normal : def

aN (y) =

∗

−1 (Dhx (h−1 eN x (y))) . −1 | ∗(Dhx (hx (y)))−1 eN |

The standard convention that the Greek indices range from 1 to N − 1 and the Roman indices from 1 to N will be followed together with Einstein’s rule of summation over repeated indices. The associated contravariant basis {ai } = {ai (y)} is defined from the covariant one {ai } = {ai (y)} as ai · aj = δij , where δij is the Kronecker index function. The first, second, and third fundamental forms a, b, and c are defined as def

aαβ = aα · aβ ,

def

bαβ = −aα · aN,β ,

def

cαβ = bλα bλβ ,

where aN,β =

∂aN , ∂ζβ

bλα = aλ · aµ bµα .

The above definitions extend to sets of class C 1,1 for which hx ∈ C 1,1 (B) and hence hx ∈ C 1,1 (B0 ). So by Rademacher’s theorem in dimension N − 1 (cf., for instance, L. C. Evans and R. F. Gariepy [1]), hx ∈ W 2,∞ (B0 ) and the definitions of the second and third fundamental forms still make sense HN −1 almost everywhere on Γ. The eigenvalues of bαβ are the (N − 1) principal curvatures κi , 1 ≤ i ≤ N − 1, of the submanifold Γ. The mean curvature H and the Gauss curvature K are defined as N −1 N −1 ( 1 def def H = κα and K = κα . N − 1 α=1 α=1 The choice of the inner normal for aN is necessary to make the principal curvatures of the sphere (boundary of the ball) positive. The factor 1/(N − 1) is used to make the mean curvature of the unit sphere equal to 1 in all dimensions. The reader should keep in mind the long-standing differences in usage between geometry and partial differential equations, where the outer unit normal is used in the integration-byparts formulae for the Euclidean space RN . For integration by parts on submanifolds of RN the sum of the principal curvatures will naturally occur rather than the mean curvature. It will be convenient to redefine H as the sum of the principal curvatures and introduce the notation H for the classical mean curvature: N −1 1 κα , H = κα . H = N − 1 α=1 α=1 def

N −1

def

(3.17)

4. Sets Globally Described by the Level Sets of a Function

4

75

Sets Globally Described by the Level Sets of a Function

From the definition of a set of class C k,
, k ≥ 1 and 0 ≤  ≤ 1, the set Ω can also be locally described by the level sets of the C k,
-function def

fx (y) = gx (y) · eN ,

(4.1)

since by definition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. Thus the exterior normal to Ω is given by n(y) = −

∗ ∗ ∇fx (y) Dgx (y)eN (Dhx (gx (y)))−1 eN =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

From this local construction of the functions {fx : x ∈ ∂Ω}, a global function on RN can be constructed to characterize Ω. Theorem 4.1. Given k ≥ 1, 0 ≤  ≤ 1, and a set Ω in RN of class C k,
with compact boundary, there exists a Lipschitz continuous f : RN → R such that int Ω = {y ∈ RN : f (y) > 0}

and

∂Ω = {y ∈ RN : f (y) = 0}

(4.2)

and a neighborhood W of ∂Ω such that f ∈ C k,
(W ) and ∇f = 0 on W and n = −

∇f , |∇f |

(4.3)

where n is the outward unit normal to Ω on ∂Ω. Proof. Construction of the function f . Fix x ∈ ∂Ω and consider the function fx defined by (4.1). By continuity of ∇fx , let V (x) ⊂ U (x) be a neighborhood of x such that 1 ∀y ∈ V (x), |∇fx (y) − ∇fx (x)| ≤ |∇fx (x)|. 2 Furthermore, let W (x) be a bounded open neighborhood of x such that W (x) ⊂ V (x). For ∂Ω compact there exists a finite subcover {Wj = W (xj ) : 1 ≤ j ≤ m} of ∂Ω for a finite sequence {xj : 1 ≤ j ≤ m} ⊂ ∂Ω. Index all the previous symbols by j instead of xj and define W = ∪m j=1 Wj . Let {rj : 1 ≤ j ≤ m} be a partition of unity for {Vj = V (xj )} such that   r ∈ D(Vj ), 0 ≤ rj (x) ≤ 1, 1 ≤ j ≤ m,   j m (4.4)  rj (x) = 1 in W ,   j=1

76

Chapter 2. Classical Descriptions of Geometries and Their Properties

where, by definition, W is an open neighborhood of ∂Ω such that W ⊂ V = ∪m j=1 Vj and D(Vj ) is the set of all infinitely continuously differentiable functions with compact support in Vj . Define the function f : RN → R: def

f (y) = dW ∪Ωc − dW ∪Ω +

m j=1

rj (y)

fj (y) , |∇fj (xj )|

where dA (x) = inf{|y − x| : y ∈ A} is the distance function from a point x to a nonempty set A of RN . The function dA is Lipschitz continuous with Lipschitz constant equal to 1. Since for all j, fj ∈ C k,
(Vj ), rj ∈ D(Vj ), and W j ⊂ Vj , rj fj ∈ C k,
(Vj ) with compact support in Vj . Therefore since k ≥ 1, rj fj is Lipschitz continuous in RN . So, by definition, f is Lipschitz continuous on RN as the finite sum of Lipschitz continuous functions on RN . Properties (4.2). Introduce the index set def

J(y) = {j : 1 ≤ j ≤ m, rj (y) > 0} for y ∈ V . For all y ∈ W , J(y) = ∅, dW ∪Ωc = 0 = dW ∪Ω , and f (y) =



rj (y)

j∈J(y)

fj (y) . |∇fj (xj )|

(4.5)

For y ∈ ∂Ω = W ∩ ∂Ω, fj (y) = 0 for all j ∈ J(y) and hence f (y) = 0; for y ∈ int Ω ∩ W , fj (y) > 0 and rj (y) > 0 for all j ∈ J(y) and hence f (y) > 0; for y ∈ int Ωc ∩ W , fj (y) < 0 and)rj (y) > 0 for all j ∈ J(y) and hence f (y) < 0. For y ∈ Ω\W , fj ≥ 0, rj ≥ 0, m rj fj ≥ 0, dW ∪Ωc > 0, dW ∪Ω = 0, and f > 0; )mj=1 c for y ∈ Ω \W , fj ≤ 0, rj ≥ 0, j=1 rj fj ≤ 0, dW ∪Ωc = 0, dW ∪Ω > 0, and f < 0. So we have proven that ∂Ω ⊂ f −1 (0), int Ω ⊂ {f > 0}, and int Ωc ⊂ {f < 0}. Hence f has the properties (4.2). Properties (4.3). Recall that on W the function f is given by expression (4.5). It belongs to C k,
(W ), and, a fortiori, to C 1 (W ), as the finite sum of C k,
-functions on W since k ≥ 1. The gradient of f in W is given by ∇f (y) =



∇rj (y)

j∈J(y)

fj (y) ∇fj (y) + rj (y) . |∇fj (xj )| |∇fj (xj )|

It remains to show that it is nonzero on ∂Ω. On ∂Ω ∩ Vj , fj = 0 and ∀j ∈ J(y),

∇fj (y) = −n(y). |∇fj (y)|

Therefore in W , ∇f can be rewritten in the form |∇fj (y)| rj (y) ∇f (y) = −n(y) |∇fj (xj )| j∈J(y) and   |∇fj (y)| rj (y) 1 − ∇f (y) + n(y) = n(y) . |∇fj (xj )| j∈J(y)

4. Sets Globally Described by the Level Sets of a Function Finally, by construction of W ,

|∇f (y)|≥ |n(y)| − |n(y)| ≥1−

j∈J(y)



rj (y)

j∈J(y)

77

|∇fj (y)| rj (y) 1 − |∇fj (xj )|

1 1 = 2 2

and ∇f = 0 in W . This proves properties (4.3). This theorem has a converse. Theorem 4.2. Associate with a continuous function f : RN → R the set def

Ω = {y ∈ RN : f (y) > 0}.

(4.6)

f −1 (0) = {y ∈ RN : f (y) = 0} = ∅

(4.7)

Assume that def

and that there exists a neighborhood V of f −1 (0) such that f ∈ C k,
(V ) for some k ≥ 1 and 0 ≤  ≤ 1 and that ∇f = 0 in f −1 (0). Then Ω is a set of class C k,
, int Ω = Ω

and

∂Ω = f −1 (0).

(4.8)

Proof. By continuity of f , Ω is open, int Ω = Ω, Ω is closed, Ω ⊂ {y ∈ RN : f (y) ≥ 0},

Ω = Ω = {y ∈ RN : f (y) ≤ 0},

and ∂Ω ⊂ f −1 (0). Conversely for each x ∈ ∂Ω, define the function def

g(t) = f (x + t∇f (x)). There exists δ > 0 such that for all |t| < δ, x + t∇f (x) ∈ V and the function g is C 1 in (−δ, δ). Hence since g(0) = 0 and g  (t) = ∇f (x + t∇f (x)) · ∇f (x) 

t

∇f (x + s∇f (x)) · ∇f (x) ds.

f (x + t∇f (x)) = 0

By continuity of ∇f in V and the fact that ∇f (x) = 0, there exists δ  , 0 < δ  < δ, such that ∀s, 0 ≤ |s| ≤ δ  ,

∇f (x + s∇f (x)) · ∇f (x) ≥

1 |∇f (x)|2 > 0. 2

Hence for all t, 0 < t ≤ δ  , f (x + t∇f (x)) > 0. So any point in f −1 (0) can be approximated by a sequence {xn = x + tn ∇f (x) : n ≥ 1, 0 < tn ≤ δ  } in Ω, tn → 0, and f −1 (0) ⊂ Ω. Similarly, using a sequence of negative tn ’s, f −1 (0) ⊂ Ω and hence f −1 (0) ⊂ ∂Ω. This proves (4.8).

78

Chapter 2. Classical Descriptions of Geometries and Their Properties

Fix x ∈ ∂Ω = f −1 (0). Since ∇f (x) = 0, define the unit vector eN (x) = ∇f (x)/|∇f (x)|. Associate with eN (x) unit vectors e1 (x), . . . , eN −1 (x), which form an orthonormal basis in RN with eN (x). Define the map gx : V → RN as   f (y) def N −1 gx (y) = {(y − x) · eα (x)}α=1 , ⇒ gx ∈ C k,
(V ; RN ). |∇f (x)| The transpose of the Jacobian matrix of gx is given by ∗

Dgx (y) = (e1 (x), . . . , eN −1 (x), ∇f (y)/|∇f (x)|)

and Dgx (x) = I, the identity matrix in the {ei (x)} reference system. By the inverse mapping theorem gx has a C k,
inverse hx in some neighborhood U (x) of x in V . Therefore, Ω ∩ U (x) = {y ∈ U (x) : f (x) > 0} = int Ω ∩ U (x), {y ∈ U (x) : f (x) = 0} = ∂Ω ∩ U (x), and the set Ω is of class C k,
. We complete this section with the important theorem of Sard. ´ [3, Vol. III, sect. 16.23, p. 167]). Let X and Y Theorem 4.3 (J. Dieudonne be two differential manifolds, f : X → Y be a C ∞ -mapping, and E be the set of critical points of f . Then f (E) is negligible in Y , and Y − f (E) is dense in Y . Combining this theorem with Theorem 4.1, this means that, for almost all t in the range of a C ∞ -function f : RN → R, the set

 y ∈ RN : f (x) > t is of class C ∞ in the sense of Definition 3.1. The following theorem extends and completes Sard’s theorem. Theorem 4.4 (H. Federer [5, Thm. 3.4.3, p. 316]). If m > ν ≥ 0 and k ≥ 1 are integers, A is an open subset of Rm , B ⊂ A, Y is a normed vector space, and f : A → Y is a map of class k,

dim Im Df (x) ≤ ν for x ∈ B,

then Hν+(m−ν)/k (f (B)) = 0.

5

Sets Locally Described by the Epigraph of a Function

After diffeomorphisms and level sets, the local epigraph description provides a third point of view and another way to characterize the smoothness of a set or a domain. For instance, Lipschitzian domains, which are characterized by the property that their boundary is locally the epigraph of a Lipschitzian function, play a central role in the theory of Sobolev spaces and partial differential equations. They can equivalently be characterized by the geometric uniform cone property. We shall see that sets that are locally the epigraph of C k,
-functions are equivalent to sets that are locally of class C k,
for k ≥ 1.

5. Sets Locally Described by the Epigraph of a Function

5.1

79

Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions

Let eN in RN be a unit reference vector and introduce the following notation: H = {eN }⊥ , def

∀ζ ∈ RN ,

ζ  = PH (ζ), def

def

ζN = eN · ζ

(5.1)

for the associated reference hyperplane H orthogonal to eN , the orthogonal projection PH onto H, and the normal component ζN of ζ. The vector ζ is equal to ζ  + ζN eN and will often be denoted by (ζ  , ζN ). In practice the vector eN is chosen as eN = (0, . . . , 0, 1), but the actual form of the unit vector eN is not important. A direction in RN is specified by a unit vector d ∈ RN , |d| = 1. Alternatively, it can be specified by AeN for some matrix A in the orthogonal subgroup of N × N matrices O(N) = {A : ∗A A = A ∗A = I} , def

(5.2)

where ∗A is the transposed matrix of A. Conversely, for any unit vector d in RN , |d| = 1, there exists8 A ∈ O(N) such that d = AeN . For all x ∈ RN and A ∈ O(N), it is easy to check that |Ax| = |x| = |A−1 x| for all x ∈ RN . The main elements of Definition 5.1 are illustrated in Figure 2.2 for N = 2. ζN graph of ax

set Ω Ax eN ∂Ω

ζ x

0 Vx ⊂ H = {eN }⊥ U(x)

Figure 2.2. Local epigraph representation (N = 2). Definition 5.1. Let eN be a unit vector in RN , H be the hyperplane {eN }⊥ , and Ω be a subset of RN with nonempty boundary ∂Ω. (i) Ω is said to be locally a C 0 epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); the orthonormal basis with respect to d by adding N −1 unit vectors d1 , d2 , . . . , dN −1 and construct the matrix whose columns are the vectors A = [d1 d2 . . . dN −1 d] for which AeN = d. 8 Complete

80

Chapter 2. Classical Descriptions of Geometries and Their Properties (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.3)

(d) and a function ax ∈ C 0 (Vx ) such that ax (0) = 0 and & ζ  ∈ Vx , U(x) ∩ ∂Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : ζN = ax (ζ  ) & ζ  ∈ Vx  U(x) ∩ int Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : . ζN > ax (ζ  ) 

(5.4)

(5.5)

(ii) Ω is said to be a C 0 epigraph if it is locally a C 0 epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

(iii) Ω is said to be an equi-C 0 epigraph if it is a C 0 epigraph and the family of functions {ax : x ∈ ∂Ω} is uniformly bounded and equicontinuous: ∃c > 0 such that ∀x ∈ ∂Ω, ∀ξ  ∈ V, ∀ε > 0, ∃δ > 0 such that ∀x ∈ ∂Ω, ∀y, ∀z ∈ V such that |z − y| < δ,

|ax (ξ  )| ≤ c, (5.6) |ax (z) − ax (y)| < ε.

Remark 5.1. Conditions (5.3), (5.4), and (5.5) are equivalent to the following three conditions: & ζ  ∈ Vx  U(x) ∩ ∂Ω ⊂ x + Ax (ζ + ζN eN ) : , (5.7) ζN = ax (ζ  ) & ζ  ∈ Vx  , (5.8) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN > ax (ζ  ) &  ζ ∈ V x U(x) ∩ int Ω ⊂ x + Ax (ζ  + ζN eN ) : . (5.9) ζN < ax (ζ  ) In particular, Ω is locally a C 0 epigraph if and only if Ω is locally a C 0 epigraph. As a result, conditions (5.3), (5.4), and (5.5) are also equivalent to conditions (5.3), (5.4), and the following condition: & ζ  ∈ Vx  (5.10) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN < ax (ζ  ) in place of condition (5.5).

5. Sets Locally Described by the Epigraph of a Function

81

Remark 5.2. Note that from (5.3) PH (A−1 x (U(x) − x)) ⊂ Vx and that this yields the condition PH (U ) ⊂ V in part (iii) of the definition. Remark 5.3. It is always possible to redefine U(x) − x or Vx to be open balls or open hypercubes centered in 0. For instance, in the case of balls, for Vx = BH (0, ρ) ⊂ Vx , choose the associated neighborhood U  (x) = U(x) ∩ {y ∈ RN : PH (A−1 x (y − x)) ∈ BH (0, ρ)}. Similarly, for U  (x) = B(x, r) ⊂ U(x) choose the associated neighborhood Vx = −1 PH (A−1 x (B(x, r)−x) = BH (0, r) ⊂ Vx , since, from (5.3), Vx ⊃ PH (Ax (U(x)−x)) ⊃ −1 PH (Ax B(x, r) − x)) = BH (0, r). In both cases properties (5.3) to (5.5) are verified for the new neighborhoods. Remark 5.4. Sets Ω ⊂ RN that are locally C 0 epigraphs have a boundary ∂Ω with zero N -dimensional Lebesgue measure. The three cases considered in Definition 5.1 differ only when the boundary ∂Ω is unbounded. But we first introduce some notation. Notation 5.1. (i) Under condition (5.3), a point y in U(x) ⊂ RN is represented by (ζ  , ζN ) ∈ Vx ×R, where ζ  = PH (A−1 x (y − x)) ∈ Vx ⊂ H, def

ζN = A−1 x (y − x) · eN (x). def

We can also identify H with RN −1 = {(z  , 0) ∈ RN : z  ∈ RN −1 }. (ii) The graph, epigraph, and hypograph of ax : Vx → R will be denoted as follows: def

A0x = {x + Ax (ζ  + ζN eN ) : ∀ζ  ∈ Vx , ζN = ax (ζ  )} , def

   A+ x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN > ax (ζ )} , def

   A− x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN < ax (ζ )} . def

The equi-C 0 epigraph case is important since we shall see in Theorem 5.2 that the three cases of Definition 5.1 are equivalent when the boundary ∂Ω is compact. We now show that, for an arbitrary boundary, the equicontinuity of the graph functions {ax : x ∈ ∂Ω} can be expressed in terms of the continuity in 0 of a dominating function h. For this purpose, define the space of dominating functions as follows: def

H = {h : [0, ∞[ → [0, ∞[ : h(0) = 0 and h is continuous in 0} .

(5.11)

By continuity in 0, h is locally bounded in 0: ∀c > 0, ∃ρ > 0 such that ∀θ ∈ [0, ρ],

|h(θ)| ≤ c.

The only part of the function h that will really matter is in a bounded neighborhood of zero. Its extension outside to [0, ∞[ can be relatively arbitrary.

82

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.1. Let Ω be a C 0 -epigraph. The following conditions are equivalent: (i) Ω is an equi-C 0 -epigraph. (ii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ  , ξ  ∈ BH (0, ρ) such that |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |). (5.12)

(iii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ  ∈ BH (0, ρ),

ax (ξ  ) ≤ h(|ξ  |).

(5.13)

Inequalities (5.12) and (5.13) for h ∈ H are also verified with lim sup h that also belongs to H. Proof. Since the ax ’s are uniformly continuous, inequalities (5.12) and (5.13) with h ∈ H imply the same inequalities with lim sup h that also belongs to H by taking the pointwise lim sup of both sides of the inequalities. (ii) ⇒ (i). For h ∈ H, there exists ρ1 > 0 such that 0 < ρ1 ≤ ρ/2 and |h(θ)| ≤ 1 for all θ ∈ [0, ρ1 ]. By construction BH (0, ρ1 ) ⊂ V and from inequality (5.12) ∀ζ  , ξ  ∈ BH (0, ρ1 ), |ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |), ∀ ξ  ∈ BH (0, ρ1 ), |ax (ξ  )| ≤ h(|ξ  |) ≤ 1 and the family {ax : BH (0, ρ1 ) → R : x ∈ ∂Ω} is equicontinuous and uniformly bounded. (i) ⇒ (ii). By Definition 5.1 (iii), there exist a neighborhood V of 0 in H and a neighborhood U of 0 in RN such that for all x ∈ ∂Ω Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U,

and the family {ax ∈ C 0 (V ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous. Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Definition 5.1 

⇒ BH (0, 3r) ⊂ V. U(x) ⊂ y ∈ RN : PH A−1 x (y − x) ∈ V Define the modulus of continuity ∀θ ∈ [0, 3r],

h(θ) = sup sup

sup

ζ  ∈H x∈∂Ω ξ ∈V |ζ  |=1 ξ  +θζ  ∈V

|ax (θζ  + ξ  ) − ax (ξ  )|.

(5.14)

The function h(θ), as the sup of the family θ → |ax (θζ  + ξ  ) − ax (ξ  )| of continuous functions with respect to (ζ  , ξ  ), is lower semicontinuous with respect to θ and bounded in [0, 3r] since the ax ’s are uniformly bounded. By construction h(0) = 0 and by equicontinuity of the ax , for all ε > 0, there exists δ > 0 such that for all θ ∈ [0, 3r], |θ| < δ, ∀x ∈ ∂Ω, ∀ζ  ∈ H, |ζ  | = 1,

|ax (θζ  + ξ  ) − ax (ξ  )| < ε

⇒ |h(θ)| < ε,

5. Sets Locally Described by the Epigraph of a Function

83

where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the new neighborhoods V  = BH (0, ρ) and U  = {ξ ∈ B(0, ρ) : PH ξ ∈ V  } = B(0, ρ) and extend h by the constant h(ρ) to [ρ, ∞[ . Therefore h ∈ H and, by definition of h, for all x ∈ ∂Ω and for all ξ  and ζ  in BH (0, ρ) such that |ξ  − ζ  | < ρ             ξ −ζ     |ax (ξ ) − ax (ζ )| = ax ζ + |ξ − ζ |  − a (ζ ) x ≤ h(|ξ − ζ |) |ξ − ζ  | ⇒ ∀x ∈ ∂Ω, ∀ξ  , ζ  ∈ BH (0, ρ), |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |).

(ii) ⇒ (iii) Choose ζ  = 0 in inequality (5.12): ∀ξ  ∈ BH (0, ρ) such that |ξ  | < ρ,

ax (ξ  ) ≤ |ax (ξ  )| ≤ h(|ξ  |).

(iii) ⇒ (ii). In view of Remark 5.3, choose the new neighborhoods V  = BH (0, ρ) and U  = U ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and the restriction of the functions {ax : V → R : ∀x ∈ ∂Ω} to V  . Choose r > 0 such that B(0, 4r) ⊂ U  (hence 4r ≤ ρ) and consider the region O = {ζ  + ζN eN : ζ  ∈ BH (0, 2r) and h(|ζ  |) < ζN < r} . def

For any x ∈ ∂Ω and ζ ∈ O, consider the point yζ = x + Ax (ζ  + ζN eN ). It is readily ' 2 seen that |yζ − x| = |ζ| < (2r) + r 2 < 3r and that yζ ∈ U(x). In addition     y ζ ∈ A+ x . From (5.13) ax (ζ ) ≤ h(|ζ |) for all ζ ∈ V = BH (0, ρ) and hence ax (ζ ) ≤  + h(|ζ |) < ζN for all ζ ∈ O (2r < ρ). As a result yζ ∈ U(x) ∩ Ax = U(x) ∩ int Ω ⊂ int Ω and ∀x ∈ ∂Ω,

x + Ax O ⊂ int Ω.

(5.15)

Given ζ  and ξ  in BH (0, 2r), consider the two points yζ  = x + Ax (ζ  + ax (ζ  )eN ) ∈ A0x = U(x) ∩ ∂Ω, def

yξ = x + Ax (ξ  + ax (ξ  )eN ) ∈ A0x = U(x) ∩ ∂Ω. def

Since yξ and yζ  belong to U(x) ∩ ∂Ω, yξ does not belong to yζ  + Ax O ⊂ int Ω or / Ax O. Hence equivalently yξ − yζ  ∈ ξ  − ζ  + (ax (ξ  ) − ax (ζ  ))eN = A−1 / O. x (yξ  − yζ  ) ∈ For |ξ  − ζ  | < 2r, this is equivalent to saying that one of the following inequalities is verified: h(|ξ  − ζ  |) ≥ Ax eN · (yξ − yζ  ) = ax (ξ  ) − ax (ζ  ) or r ≤ Ax eN · (yξ − yζ  ) = ax (ξ  ) − ax (ζ  ) by definition of ax and the fact that ξ  − ζ  ∈ BH (0, 2r) ⊂ BH (0, ρ) = V  . But the second inequality contradicts the continuity of ax . Hence the first inequality holds and, by interchanging the roles of ξ  and ζ  , we get inequality (5.12) with a new ρ equal to 2r.

84

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.2. When the boundary ∂Ω is compact, the three properties of Definition 5.1 are equivalent: locally C 0 epigraph

⇐⇒

C 0 epigraph

⇐⇒

equi-C 0 epigraph.

Under such conditions, there exist neighborhoods V of 0 in H and U of 0 in RN such that PH U ⊂ V and the neighborhoods Vx of 0 in H and the neighborhoods U(x) of x in RN can be chosen of the form Vx = V

and

∃Ax ∈ O(N) such that U(x) = x + Ax U.

(5.16)

Moreover, the family of functions {ax : V → R : x ∈ ∂Ω} can be chosen uniformly bounded and equicontinuous with respect to x ∈ ∂Ω. In addition, there exist ρ > 0 such that BH (0, ρ) ⊂ V and a function h ∈ H such that ∀x ∈ ∂Ω, ∀ξ  , ζ  ∈ BH (0, ρ), |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |). (5.17)

Proof. It is sufficient to show that a locally C 0 epigraph is an equi-C 0 epigraph. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, m there exist a finite sequence {xi }m i=1 of points of ∂Ω and a finite subcover {Bi }i=1 , Bi = B(xi , Ri ), Ri = rxi . (i) We first claim that ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi . We proceed by contradiction. If this is not true, then for each n ≥ 1,   ∃xn ∈ ∂Ω, ∀i, 1 ≤ i ≤ m, B xn , 1/n ⊂ Bi ,   / Bi . ⇒ ∀i, 1 ≤ i ≤ m, ∃yin ∈ B xn , 1/n such that yin ∈ Since ∂Ω is compact, there exist a subsequence of {xn } and x ∈ ∂Ω such that xnk → x as k goes to infinity. Hence for each i, 1 ≤ i ≤ m, |yink − x| ≤ |yink − xnk | + |xnk − x| ≤

1 + |xnk − x| → 0 nk

and yink → x as k goes to infinity. For each i, the set Bi is closed. Thus x ∈ Bi and m * / Bi . yink ∈ Bi → x ∈ Bi ⇒ ∃ x ∈ ∂Ω such that x ∈ i=1

But this contradicts the fact that {Bi }m i=1 is an open cover of ∂Ω. This property is obviously also satisfied for all neighborhoods N of 0 such that ∅ = N ⊂ B(0, R). (ii) Construction of the neighborhoods Vx and U  (x) and the maps Ax and ax . From part (i) for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi = B(xi , Ri ) ⊂ U(xi ) ⇒

PH (A−1 xi (B(x, R)

− xi )) ⊂ PH (A−1 xi (U(xi ) − xi )) ⊂ Vxi

⇒ PH (A−1 xi (B(x − xi , R))) ⊂ Vxi .

5. Sets Locally Described by the Epigraph of a Function

85

Since x ∈ ∂Ω ∩ U(xi ), it is of the form x = xi + Axi (ξi + axi (ξi )eN ),

ξi = PH (A−1 xi (x − xi ))

−1 −1 ⇒ PH (A−1 xi (B(x − xi , R))) = PH (Axi (x − xi )) + PH (Axi (B(0, R)))

= ξi + BH (0, R)

⇒ ξi + BH (0, R) ⊂ Vxi .

(5.18)

Choose the neighborhoods def

def

U = B(0, R) and V = BH (0, R). Associate with x ∈ ∂Ω the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ  → ax (ζ  ) = axi (ζ  + ξi ) − axi (ξi ) : V → R . def

It is readily seen from identity (5.18) that ax is well-defined, ax (0) = 0, and ax ∈ C 0 (V ). Since the family {axi : 1 ≤ i ≤ m} is finite, ax C 0 (V ) ≤ 2 axi C 0 (Vx

i

)

≤ 2 max axi C 0 (Vx ) < ∞ 1≤i≤m

i

and the family {ax ; x ∈ ∂Ω} is uniformly bounded. Since each axi is uniformly continuous, for all ε > 0 there exists δi > 0 such that ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δi ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

And since the family {axi : 1 ≤ i ≤ m} is finite pick δ = min1≤i≤m δi to get the following: for all ε > 0, there exists δ > 0 such that for all i ∈ {1, . . . , m}, ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

Therefore, for all ε > 0, there exists δ > 0 such that ∀x ∈ ∂Ω, ∀ζ1 , ζ2 ∈ V such that |ζ1 − ζ2 | < δ,

|ax (ζ1 ) − ax (ζ2 )| < ε

(5.19)

and the family {ax ; x ∈ ∂Ω} is uniformly equicontinuous. By construction, properties (a) and (b) of Definition 5.1 are verified for U  (x) and Ax . By construction, properties (a) and (b) are verified for U  (x) and Ax . Always by construction in x ∈ ∂Ω U  (x) = x + Ax B(0, R) = x + B(0, R) = B(x, R), PH ((Ax )−1 (U  (x) − x)) = PH (B(0, R)) = BH (0, R) = V = Vx

 ⇒ U  (x) ⊂ y ∈ RN : PH ((Ax )−1 (y − x)) ∈ Vx

86

Chapter 2. Classical Descriptions of Geometries and Their Properties

and condition (c) of Definition 5.1 is verified. For condition (d) of Definition 5.1 in x ∈ ∂Ω, let xi be the point associated with x such that B(x, R) ⊂ U(xi ). Since U  (x) = B(x, R), U  (x) ∩ ∂Ω ⊂ U(xi ) ∩ ∂Ω = {xi + Axi (ζ  + axi (ζ  )eN ) : ∀ζ  ∈ Vxi } ∩ ∂Ω and for all y ∈ U  (x) ∩ ∂Ω there exists ξ  ∈ Vxi such that y = xi + Axi (ξ  + axi (ξ  )eN ) = x + Axi (ξ  + axi (ξ  )eN ) − (x − xi ). But x also has a representation with respect to xi x = xi + Axi (ξi + axi (ξi )eN ), 

⇒ y = x + Axi (ξ −

ξi

ξi ∈ Vxi



+ (axi (ξ ) − axi (ξi ))eN )

that can be rewritten as y = x + Axi (ξ  − ξi + (axi ((ξ  − ξi ) + ξi ) − axi (ξi ))eN )   ⇒ ζ  = PH (A−1 xi (y − x)) = ξ − ξi ∈ V def

  −1 (U  (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax ) there exists ζ  ∈ V such that

y = x + Axi (ζ  + (axi (ζ  + ξi ) − axi (ξi ))eN ) = x + Axi (ζ  + ax (ζ  )eN ) and property (5.4) is verified. The constructions and the proof of property (5.5) are similar: U  (x) ∩ int Ω ⊂ U(xi ) ∩ int Ω   = U(xi ) ∩ xi + Axi (ζ  + ζN eN ) : ∀ζ  ∈ Vxi , ∀ζN > axi (ζ  ) and for all y ∈ U  (x)∩int Ω there exists (ξ  , ξN ) ∈ Vxi ×R such that ξN > axi (ξ  ) and y = xi + Axi (ξ  + ξN eN ) = x + Axi (ξ  + ξN eN ) − (x − xi ) = x + Axi (ξ  − ξi + (ξN − axi (ξi ))eN )   ⇒ ζ  = PH (A−1 xi (y − x)) = ξ − ξi ∈ V, def

 ζN = (A−1 xi (y − x)) · eN = ξN − axi (ξi ) def

  −1 (U  (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax )  there exists (ζ , ζN ) ∈ V × R such that

y = x + Axi (ζ  + ax (ζ  )eN ), ζN = ξN − axi (ξi ) = ξN − axi (ξ  ) + axi (ξ  ) − axi (ξi ) = ξN − axi (ξ  ) + axi (ζ  + ξi ) − axi (ξi ) = ξN − axi (ξ  ) + ax (ζ  ) > ax (ζ  )

since ξN > axi (ξ  ) and property (5.5) in (d) is verified. (iii) The other properties and (5.17) follow from Theorem 5.1.

5. Sets Locally Described by the Epigraph of a Function

5.2

87

Local C k,
-Epigraphs and H¨ olderian/Lipschitzian Sets

We extend the three definitions of C 0 epigraphs to C k,
epigraphs. C 0,1 is the important family of Lipschitzian sets and C 0,
, 0 <  < 1, of H¨ olderian sets. Definition 5.2. Let Ω be a subset of RN such that ∂Ω = ∅. Let k ≥ 0, 0 ≤  ≤ 1. (i) Ω is said to be locally a C k,
epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.20)

(d) and a function ax ∈ C k,
(Vx ) such that ax (0) = 0 and U(x) ∩ ∂Ω = U(x) ∩

x + Ax (ζ + ζN eN ) :

U(x) ∩ int Ω = U(x) ∩



&

ζ  ∈ Vx



,

ζN = ax (ζ  )

x + Ax (ζ + ζN eN ) :

(5.21)

&

ζ  ∈ Vx ζN > ax (ζ  )

,

(5.22)

where ζ  = (ζ1 , . . . , ζN −1 ) ∈ RN −1 . We shall say that Ω is locally Lipschitzian in the C 0,1 case and locally H¨ olderian of index  in the C 0,
, 0 <  < 1, case. (ii) Ω is said to be a C k,
epigraph if it is locally a C k,
epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x)−x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

def

Vx = V

def

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

olderian of index  We shall say that Ω is Lipschitzian in the C 0,1 case and H¨ in the C 0,
, 0 <  < 1, case. (iii) Given (k, ) = (0, 0), Ω is said to be an equi-C k.
epigraph if it is a C k,
epigraph and ∃c > 0, ∀x ∈ ∂Ω,

ax C k, (V ) ≤ c,

(5.23)

where f C k, (V ) is the norm on the space C k,
(V ) as defined in (2.21): def

f C k, (V ) = f C k (V ) + max

sup

0≤|α|≤k x,y∈V x=y

def

C k,0 (V ) = C k (V )

and

|∂ α f (y) − ∂ α f (x)| , |y − x|
def

if 0 <  ≤ 1,

f C k,0 (V ) = f C k (V ) .

88

Chapter 2. Classical Descriptions of Geometries and Their Properties Ω is said to be an equi-C 0,0 epigraph if it is an equi-C 0 epigraph in the sense of Definition 5.1 (iii). We shall say that Ω is equi-Lipschitzian in the C 0,1 case and equi-H¨ olderian of index  in the C 0,
, 0 <  < 1, case.

Remark 5.5. In Definition 5.1 (iii) for (k, ) = (0, 0) it follows from condition (5.23) that the functions {ax : x ∈ ∂Ω} are uniformly equicontinuous on V . In the case (k, ) = (0, 0) this is no longer true. So we have to include it by using Definition 5.1 (iii). Theorem 5.2 readily extends to the C k,
case. Theorem 5.3. When the boundary ∂Ω is compact, the three types of sets of Definition 5.2 are equi-C k,
epigraphs in the sense of Definitions 5.2 (iii) and 5.1 (iii). Proof. The proof is the same as the one of Theorem 5.2 for the C 0 case. The equiC k,
property is obtained by choosing in the proof of Theorem 5.2 the common constant as c = max1≤i≤m cxi , where cxi is the constant associated with axi on Vxi , axi C k (Vxi ) ≤ cxi . Sets Ω that are locally a C k,
epigraph are Lebesgue measurable in RN , their “volume” is locally finite, and their boundary has zero “volume.” Theorem 5.4. Let k ≥ 0, let 0 ≤  ≤ 1, and let Ω be locally a C k,
epigraph. (i) The complement Ω is also locally a C k,
epigraph, and int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Moreover, for all x ∈ ∂Ω   m U(x) ∩ ∂Ω = 0

  ⇒ m ∂Ω = 0,

(5.24)

(5.25)

where m is the N -dimensional Lebesgue measure. (ii) If Ω = ∅ is open, then Ω = int Ω. Proof. (i) The fact that the complement of Ω is locally a C k,
epigraph follows from Remark 5.1. To complete the proof, it is sufficient to establish the first two properties of the first line of (5.24). The second line is the first line applied to Ω. Since U(x) is a neighborhood of x, there exists r > 0 such that B(x, 3r) ⊂ U(x). Since ax ∈ C 0 (Vx ), ax (0) = 0, and Vx is a neighborhood of 0, there exists ρ, 0 < ρ < r, such that BH (0, ρ) ⊂ Vx and for all ζ  , |ζ  | < ρ, |ax (ζ  )| ≤ r. Given ζ  ∈ BH (0, ρ), the point yζ  = x + A(ζ  + ax (ζ  )eN ) ∈ ∂Ω ∩ U(x). Given a real α, |α| < r, consider the point zα = x + A(ζ  + (ax (ζ  ) + α)eN ). By construction |zα − x| = |A(ζ  + (ax (ζ  ) + α)eN )| = |ζ  + (ax (ζ  ) + α)eN | < 3r and zα ∈ U(x). Hence the segment (yζ  , yζ  + rAx eN ) ⊂ A+ ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅. Also the segment (yζ  , yζ  − rAx eN ) ⊂ A− ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅.

5. Sets Locally Described by the Epigraph of a Function

89

Finally, associate with yζ  ∈ ∂Ω and the sequence αn = r/n the points zαn ∈ int Ω. As n goes to infinity, zn → yζ  and ∂Ω ⊂ int Ω and hence Ω = int Ω. From the following Lemma 5.1 we get the third identity in the first line of (5.24). (ii) From part (i) ∂(int Ω) = ∂Ω and by Lemma 5.1 (ii) int Ω = Ω. Since Ω is open, Ω = int Ω = Ω and int Ω = Ω = Ω. Lemma 5.1. ⇐⇒

(i) ∂(int Ω) = ∂Ω (ii) ∂(int Ω) = ∂Ω

int Ω = Ω.

⇐⇒

int Ω = Ω.

Proof. It is sufficient to prove (i). (⇒) By definition of the closure, Ω = int Ω ∪ ∂Ω = int Ω ∪ ∂(int Ω) = int Ω. Conversely (⇐) By definition ∂Ω = Ω ∩ Ω, Ω ∩ Ω = int Ω ∩ Ω = int Ω ∩ (int Ω) = int Ω ∩ (int Ω) = ∂(int Ω).

5.3

Local C k,
-Epigraphs and Sets of Class C k,


Sets or domains which are locally the epigraph of a C k,
-function, k ≥ 0, 0 ≤  ≤ 1 (resp., locally Lipschitzian), are sets of class C k,
(resp., C 0,1 ). However, we shall see from Examples 5.1 and 5.2 that a domain of class C 0,1 is generally not locally the epigraph of a Lipschitzian function. Theorem 5.5. Let , 0 ≤  ≤ 1, be a real number. (i) If Ω is locally a C k,
epigraph for k ≥ 0, then it is locally of class C k,
. (ii) If Ω is locally of class C k,
for k ≥ 1, then it is locally a C k,
epigraph. Proof. (i) For each point x ∈ ∂Ω, let U(x), Vx , and ax be the associated neighborhoods and the C k,
-function. Define the C k,
-mappings for ξ = (ξ  , ξN ) ∈ Vx × R hx (ξ) = x + Ax (ξ  + [ξN + ax (ξ  )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

It is easy to verify that hx is the inverse of gx , hx (gx (y)) = y in U(x) and gx (hx (ξ)) = ξ in gx (U(x)), −1 and that since for y ∈ U(x) ∩ ∂Ω, eN · A−1 x (y − x) = ax (PH Ax (y − x)

gx (U(x) ∩ ∂Ω) ∈ H = {(ξ  , ξN ) ∈ RN : ξN = 0}, −1 and that since for y ∈ U(x) ∩ int Ω, eN · A−1 x (y − x) > ax (PH Ax (y − x)

gx (U(x) ∩ int Ω) ⊂ {(ξ  , ξN ) ∈ RN : ξN > 0}. The set Ω is of class C k,
.

(5.26) (5.27)

90

Chapter 2. Classical Descriptions of Geometries and Their Properties

(ii) In the discussion preceding Theorem 4.1, we showed that a set of class C k,
is locally the level set of a C k,
-function. From the definition of a set of class C k,
, k ≥ 1 and 0 ≤  ≤ 1, the set Ω can be locally described by the level sets of the C k,
-function def

fx (y) = gx (y) · eN , since by definition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. The exterior normal to Ω is given by n(y) = −

∗ ∗ Dgx (y)eN (Dhx (gx (y)))−1 eN ∇fx (y) =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

Since fx is C 1 and ∇fx (x) = 0, choose a smaller neighborhood of x, still denoted by U (x), such that ∀y ∈ U (x),

∇fx (y) · ∇fx (x) > 0.

To construct the graph function around the point x ∈ Γ = ∂Ω, choose Ax ∈ O(N) such that Ax eN = −n(x) and H = {eN }⊥ . Consider the C k,
-function, k ≥ 1, λ(ζ  , ζN ) = fx (x + Ax (ζ  + ζN eN )) . def

By construction, λ(0, 0) = 0 and 0 = ∇λ(0, 0) · (0, ξN ) = ∇fx (x) · (0, ξN ) = ξN |∇fx (x)|

⇒ ξN = 0.

Therefore, by the implicit function theorem, there exist a neighborhood Vx ⊂ B0 of (0, 0) and a function ax ∈ C k.
(Vx ) such that λ(0, 0) = 0 and ∀ζ  ∈ Vx ,

λ(ζ  , ax (ζ  )) = fx (x + Ax (ζ  + ax (ζ  )eN ) = 0.

By construction, x + Ax (ζ  + ax (ζ  )eN ) ∈ Γ ∩ U (x). Choose U(x) = U (x) ∩ {x + Ax (ζ  + ζN eN ) : ζ  ∈ Vx and ζN ∈ R}. def

By construction, Vx and U(x) satisfy condition (5.3). Moreover, from (3.4)–(3.5) int Ω ∩ U (x) = hx (B+ ) and ∂Ω ∩ U (x) = hx (B0 ) int Ω ∩ U (x) = {y ∈ U (x) : f (y) > 0}, x ⇒ ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0},

5. Sets Locally Described by the Epigraph of a Function

91

and since U(x) ⊂ U (x) we get conditions (5.4) and (5.5): int Ω ∩ U(x) = {y ∈ U(x) : fx (y) > 0} = A+ ∩ U(x), ∂Ω ∩ U(x) = {y ∈ U(x) : fx (y) = 0} = A0 ∩ U(x). Recall that, by construction of U(x), for all (ζ  , ζN ) ∈ A−1 x (U(x) − {x}), ζN > ax (ζ  ) ⇐⇒ λ(ζ  , ζN ) = fx (x + Ax (ζ  + ζN eN )) > 0, since ∇fx (x) ∂λ  (ζ , ζN ) = ∇fx (x + Ax (ζ  + ζN en )) · Ax eN = ∇fx (x + Ax ζ) · >0 ∂ζN |∇fx (x)| −1 in the neighborhood A−1 x (U(x) − {x}) ⊂ Ax (U (x) − {x}) of (0, 0).

Example 5.1 (R. Adams, N. Aronszajn, and K. T. Smith [1]). Consider the open convex (Lipschitzian) set Ω0 = {ρ eiθ : 0 < ρ < 1, 0 < θ < π/2} and its image Ω = T (Ω0 ) by the C 0,1 -diffeomorphism (see Figure 2.3) T (ρ eiθ ) = ρ ei(θ−log ρ) ,

T −1 (ρ eiθ ) = ρ ei(θ+log ρ) .

It is readily seen that, as ρ goes to zero, the image of the two pieces of the boundary of Ω0 corresponding to θ = 0 and θ = π/2 begin to spiral around the origin. As a result Ω is not locally the epigraph of a function at the origin.

1

1

0

0 0 (a) quarter disk Ω0

1

0

1

(b) T (Ω0 ) spirals around 0

Figure 2.3. Domain Ω0 and its image T (Ω0 ) spiraling around the origin. Example 5.2. This second example can be found in F. Murat and J. Simon [1], where it is attributed to M. Zerner. Consider the Lipschitzian function λ defined on [0, 1] as follows: λ(0) = 0, and on each interval [1/3n+1 , 1/3n ]    2 1 1   ≤ s ≤ n+1 , 2 s − ,  n+1 n+1 3 3 3 def   λ(s) =  1 2 1  −2 s − , ≤ s ≤ n, n n+1 3 3 3

92

Chapter 2. Classical Descriptions of Geometries and Their Properties

where n ranges over all integers n ≥ 0. Associate with λ and a real δ > 0 the set def

Ω = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 − λ(x1 )| < δ x1 } (see Figure 2.4). The set Ω is the image of the triangle def

Ω0 = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 | < δ x1 } through the C 0,1 -homeomorphism def

T (x1 , x2 ) = (x1 , x2 + λ(x1 )),

T −1 (y1 , y2 ) = (y1 , y2 − λ(y1 )).

Since the triangle Ω0 is Lipschitzian, its image is a set of class C 0,1 . But Ω is not locally Lipschitzian in (0, 0) since Ω zigzags like a lightning bolt as it gets closer to the origin. Thus, however small the neighborhood around (0, 0), a direction eN (0, 0) cannot be found to make the domain locally the epigraph of a function.

1

1

0

0 0

1 (a) triangle Ω0

0

1 (b) T (Ω0 ) zigzags to 0

Figure 2.4. Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin.

5.4

Locally Lipschitzian Sets: Some Examples and Properties

The important subfamily of sets that are locally a Lipschitzian epigraph (that is, locally a C 0,1 epigraph) enjoys most of the properties of smooth domains. Convex sets and domains that are locally the epigraph of a C 1 - or smoother function are locally Lipschitzian. In a bounded path-connected Lipschitzian set, the geodesic distance between any two points is uniformly bounded. Their boundary has zero volume and locally finite boundary measure (cf. Theorem 5.7 below). Sobolev spaces defined on Ω have linear extension to RN . 5.4.1

Examples and Continuous Linear Extensions

According to this definition the whole space RN and the closed unit ball with its center or a small crack removed are not locally Lipschitzian since in the first case

5. Sets Locally Described by the Epigraph of a Function

93

the boundary is empty and in the second case the conditions cannot be satisfied at the center or along the crack, which is a part of the boundary. Similarly the set ∞ *

def

Ω =

 Ωn ,

def

Ωn =

n=1

 1 1 y ∈ RN : y − n < n+2 2 2

is not locally Lipschitzian since the conditions of Definition 5.2 (i) are not satisfied in 0 ∈ ∂Ω. However, the set def

Ω =

∞ * n=1

 Ωn ,

def

Ωn =

y ∈ R : |y − n| < N

1



2n+2

is locally Lipschitzian, but not Lipschitzian or equi-Lipschitzian in the sense of Definition 5.2. One of the important properties of a bounded (open) Lipschitzian domain Ω is the existence of a continuous linear extension of functions of the Sobolev space H k (Ω) from Ω to functions in H k (RN ) defined on RN , that is, (5.28) E : H k (Ω) → H k (RN ) and ∀ϕ ∈ H k (Ω), (Eϕ) Ω = ϕ (cf. the Calder´ on extension theorem in R. A. Adams [1, p. 83, Thm. 4.32, p. 91] and ˇas [1, Thm. 3.10, p. 80]). This property is also important in the existence of J. Nec optimal domains when it is uniform for a given family. For instance, this will occur for the family of domains satisfying the uniform cone property of section 6.4.1. 5.4.2

Convex Sets

Theorem 5.6. Any convex subset Ω of RN such that Ω = RN and int Ω = ∅ is locally Lipschitzian, and for each x ∈ ∂Ω the neighborhood Vx and the function ax can be chosen convex. Proof. (i) Construction of the function ax . Start with the unit vector eN and the hyperplane H orthogonal to eN . Pick any point x0 in the interior of Ω and choose ε > 0 such that B(x0 , 2ε) ⊂ int Ω. Since Ω = RN , ∂Ω = ∅ and we can associate with each x ∈ ∂Ω the direction d(x) = (x0 − x)/|x0 − x| and the matrix Ax ∈ O(N) such that d(x) = Ax eN . Note that the hyperplane Ax H is orthogonal to the vector / Ω and the minimum d(x) = Ax eN . By convexity, the point x− 0 = x − (x0 − x) ∈ distance δ from x− to Ω is such that 0 < δ < |x − x |. As a result there exists ρ, 0 0 0 < ρ < ε, such that for all ζ  ∈ H, |ζ  | < ρ, the line Lζ  = {x + Ax (ζ  + ζN eN ) : |ζN | ≤ |x − x0 |} def

 from the point x0 + Ax ζ  to x− 0 + Ax ζ in the direction Ax eN contains a point of Ω and a point of its complement. Hence there exists ζˆN , |ζˆN | ≤ |x − x0 |, such that yˆ = x + Ax (ζ  + ζˆN eN ) ∈ ∂Ω ∩ Lζ  minimizes ζN = Ax eN · (y − x) over all y ∈ Ω ∩ Lζ  . If yˆ1 and yˆ2 are two distinct minimizing points such that ζˆ1N = ζˆ2N ,

94

Chapter 2. Classical Descriptions of Geometries and Their Properties

then yˆ1 = x + Ax (ζ  + ζˆ1N eN ) = x + Ax (ζ  + ζˆ2N eN ) = yˆ2 . As a result, for all ζ  ∈ H, |ζ  | < ρ, the function ax (ζ  ) =

def

inf

Ax eN · (y − x)

y∈Ω∩Lζ 

is well-defined and finite and there exists a unique ζˆN , |ζˆN | < |x − x0 |, such that ax (ζ  ) = ζˆN and yˆ = x + Ax (ζ  + ζˆN eN ) is the unique minimizer. Choose the neighborhoods def

U(x) =

y∈R : N

&

|PH A−1 x (y − x)| < ρ |A−1 x (y − x) · eN | < |x − x0 |

,

def

Vx = {z ∈ H : |z| < ρ} .

Then condition (5.20) of Definition 5.2 is satisfied and it is easy to verify that the function ax satisfies conditions (5.21) and (5.22). (ii) Convexity of the function ax . By construction the neighborhoods Vx and U(x) are convex. The set U(x) ∩ Ω is convex. Then recall from (5.21)–(5.22) that U(x) ∩ {x + Ax (ζ  + ζN eN ) : ζ  ∈ Vx , ζN ≥ ax (ζ  )} = U(x) ∩ Ω. Thus the set on the left-hand side is convex. In particular, ∀y 1 , y 2 ∈ Ω ∩ U(x), ∀α ∈ [0, 1],

αy 1 + (1 − α)y 2 ∈ Ω ∩ U(x).

In particular, for any two y 1 and y 2 in ∂Ω ∩ U(x), 

j y j = x + Ax (ζ j + ζN eN ),



j ζN = ax (ζ j ), j = 1, 2,

and y α = αy 1 + (1 − α)y 2 = x + Ax [αζ 1 + (1 − α)ζ 2 ] ∈ Ω ∩ U(x)    1 2 + (1 − α)ζN ≥ ax αζ 1 + (1 − α)ζ 2 ⇒ αζN      ⇒ αax (ζ 1 ) + (1 − α)ax (ζ 2 ) ≥ ax αζ 1 + (1 − α)ζ 2 and ax is convex. (iii) The function ax is Lipschitzian. It is sufficient to show that ax is bounded in Vx . Then we conclude from I. Ekeland and R. Temam [1, Lem. 3.1, p. 11] that the function is continuous and convex and hence Lipschitz continuous in Vx . By construction the minimizing point yˆ belongs to U(x), which is bounded. 5.4.3

Boundary Measure and Integral for Lipschitzian Sets

The boundary measure on ∂Ω, which was defined in section 3.2 from the local C 1 -diffeomorphism, can also be defined from the local graph representation of the boundary and extended to the Lipschitzian case. Assuming that ∂Ω is compact, there is a finite subcover of open neighborhoods U(x) for ∂Ω that can be represented by a finite family of Lipschitzian graphs. Let

5. Sets Locally Described by the Epigraph of a Function

95

def

{Uj }m j=1 , Uj = U(xj ), be a finite open cover of ∂Ω corresponding to some finite m sequence {xj }m j=1 of points of ∂Ω. Denote by eN , H, {Vj }j=1 , Vj = Vxj , and {aj }, aj = axj , the associated elements of Definition 5.2. Introduce the notation Γ = ∂Ω,

Γj = Γ ∩ Uj , 1 ≤ j ≤ m,

(5.29)

and the map hj = hxj and its inverse gj = gxj as defined in (5.26)–(5.27): hx (ξ) = x + Ax (ξ  + [ξN + ax (ξ  )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

By Rademacher’s theorem the Lipschitzian   function ax is differentiable almost everywhere in Vx and belongs to W 1,∞ Vx (cf., for instance, L. C. Evans and R. F. Gariepy [1]). Since hj is defined from the Lipschitzian function aj , Dhj is defined almost everywhere in Vx ×R, but also HN −1 almost everywhere on Vx ×{0}. Thus the canonical density for C 1 -domains still makes sense and is given by the same formula (3.13): ωj (ζ  ) = ωxj (ζ  ) = | det Dhj (ζ  , 0)| | ∗(Dhj )−1 (ζ  , 0) eN |. def

(5.30)

It is easy to verify that for almost all ξ  ∈ Vj       Dhj (ξ , ξN ) = Aj    

 ... 0 ..  .  .. , det Dh (ξ  , ξ ) = 1, .. j N . .   .. . 0  . . . ∂N −1 aj (ξ ) 1

1

0 .. .

0 .. . 0 ∂1 aj (ξ  )∂2 aj (ξ  )

...

where ∂i aj is the partial derivative of aj with respect to the ith component of ζ  = (ζ1 , . . . , ζN −1 ) ∈ Vj . The matrix Dhj is invertible and its coefficients belong to L∞ and ∗(Aj )−1 = Aj . By direct computation 

1

 0   ∗ (Dhj )−1 (ξ  , ξN ) = Aj  ...  .  ..

0 .. .

...

−∂1 aj (ξ  )

0 .. .



.

0 .. .

...

0

    ,     −∂N −1 aj (ξ ) 1

= |(Aj )−1 ∗(Dhj )−1 (ζ  , 0) eN | =

1 + |∇aj (ζ  )|2 .

0

..

0

−∂2 aj (ξ  ) .. .

(5.31)

(5.32)

and finally ωj (ζ  ) = | ∗(Dhj )−1 (ζ  , 0) eN |

96

Chapter 2. Classical Descriptions of Geometries and Their Properties

Let {r1 , . . . , rm } be a partition of unity for the Uj ’s, that is,   r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1,   j m  rj (x) = 1 in a neighborhood U of Γ  

(5.33)

j=1

0 such that U ⊂ ∪m j=1 Uj . For any function f in C (Γ) the functions fj and fj ◦ hj defined as

fj (y) = f (y)rj (y), fj ◦ hj (ζ  , 0) = fj (x + Aj (ξ  + aj (ξ  ) eN ),

y ∈ Γj , ζ  ∈ Vj ,

(5.34)

respectively, belong to C 0 (Γj ) and C 0 (Vj ). The integral of f on Γ is then defined as 

def

f dΓ = Γ

m  j=1

 fj dΓj , Γj

def



fj (hj (ζ  , 0)) ωj (ζ  ) dζ  .

fj dΓ = Γj

(5.35)

Vj

Since ωj ∈ L∞ (Vj ), 1 ≤ j ≤ m, this integral is also well-defined for all f in L1 (Γ); that is, the function fj ◦ hj ωj belongs to L1 (Vj ) for all j, 1 ≤ j ≤ m. As in section 2, the tangent plane to Γj at y = hj (ζ  , 0) is defined by Dhj (ζ  , 0) Aj H, where Aj H is the hyperplane orthogonal to Aj eN . An outward normal field to Γj is given by   ∂1 aj (ζ  )   ..   . mx (ζ) = − ∗(Dhj )−1 (ζ  , ζN )eN = Aj   ∂N −1 aj (ζ  ) −1 and the unit outward normal to Γj at y = hj (ζ  , 0) ∈ Γj is given by   ∂1 aj (ζ  )   .. 1   . Aj  n(y) = n(hj (ζ  , 0)) = ' . ∂N −1 aj (ζ  ) 1 + |∇aj (ζ  )|2 −1

(5.36)

The matrix Aj can be removed by redefining the functions ax on the space Aj H orthogonal to Aj eN . The formulae also hold for C k,
-domains, k ≥ 1. We have seen that sets that are locally a C k,
epigraph, k ≥ 0, have nice properties. For instance, their boundary has zero “volume.” However, they generally have no locally finite “surface measure” (cf. M. C. Delfour, N. Doyon, and J.´sio [1, 2] for specific examples) as we shall see in section 6.5. Fortunately, P. Zole Lipschitzian sets have locally finite “surface measure.”

5. Sets Locally Described by the Epigraph of a Function Theorem 5.7. Let Ω ⊂ RN be locally Lipschitzian. Then   '   HN −1 U(x) ∩ ∂Ω ≤ 1 + c2x mN −1 Vx ,

97

(5.37)

where mN −1 and HN −1 are the respective (N − 1)-dimensional Lebesgue measure and (N − 1)-dimensional Hausdorff measure, and cx > 0 is the Lipschitz constant of ax in Vx . If, in addition, ∂Ω is compact, then HN −1 (∂Ω) < ∞.

(5.38)

Proof. From Theorem 5.4 and from formula (5.32),   

1/2  1/2   dζ ≤ 1 + c2x mN −1 Vx . 1 + |∇ax (ζ  )|2 HN −1 U(x) ∩ ∂Ω = V (x)

When the boundary of Ω is compact, then we can find a finite subcover of open neighborhoods {U(xj )}m j=1 . From the previous estimate, HN −1 (∂Ω) is bounded by a finite sum of bounded terms. 5.4.4

Geodesic Distance in a Domain and in Its Boundary

We can now justify property (2.29) in Theorem 2.6. Theorem 5.8. Let Ω ⊂ RN be bounded, open, and locally Lipschitzian. (i) If Ω is path-connected, then ∃cΩ , ∀x, y ∈ Ω,

distΩ (x, y) ≤ cΩ |x − y|.

(5.39)

dist∂Ω (x, y) ≤ c∂Ω |x − y|.

(5.40)

(ii) If ∂Ω is path-connected, then ∃c∂Ω , ∀x, y ∈ ∂Ω,

Remark 5.6. The boundary of a path-connected, bounded, open, and locally Lipschitzian set is generally not path-connected. As an example consider the set {x ∈ R2 : 1 < |x| < 2} whose boundary is made up of the two disjoint circles of radii 1 and 2. Remark 5.7. An open subset of RN with compact path-connected boundary is generally not pathconnected as can be seen from a domain in R2 made up of two tangent squares. Proof. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, there exist a finite sequence {xi }m i=1 of points of ∂Ω and a finite , B = B(x , R ), R = r . Moreover, from part (i) of the proof subcover {Bi }m i i i i x i i=1 of Theorem 5.2, ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi .

98

Chapter 2. Classical Descriptions of Geometries and Their Properties

Always from part (i) of the proof of Theorem 5.2, the neighborhoods U and V can be chosen as def

U = B(0, R)

and

def

V = BH (0, R),

and to each point x ∈ ∂Ω we can associate the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ  → ax (ζ  ) = axi (ζ  + ξi ) − axi (ξi ) : V → R . def

Since V = BH (0, R) is path-connected and for each ζ  ∈ V the map ζN → x + Ax [ζ  + ζN eN ] is continuous, Lζ  = B(x, R) ∩ {x + Ax [ζ  + ζN eN ] : ζN ∈ R} def

is a (path-connected) line segment. This is nice, but we need to further reduce the size of the neighborhoods U  (x) and Vx to make the piece of boundary ∂Ω ∩ U  (x) and hence Ω∩U  (x) path-connected. Since the family of functions {ax (ζ  ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous, there exists c > 0 such that ∀x ∈ ∂Ω,

∀ζ1 , ζ2 ∈ V,

|ax (ζ2 ) − ax (ζ1 )| ≤ c |ζ2 − ζ1 |.

Choose ρ, 0 < ρ ≤ R/(4c + 4). Then ∀ζ  ∈ BH (0, ρ), |ax (ζ  )| ≤ R/4 ⇒ |Ax (ζ  + ax (ζ  )eN )| < R ⇒ {x + Ax [ζ  + ax (ζ  )eN ] : ζ  ∈ BH (0, ρ)} ⊂ ∂Ω ∩ B(x, R).

∀x ∈ ∂Ω,

Since BH (0, ρ) is path-connected and the map ζ  → x + Ax [ζ  + ax (ζ  )eN ] is continuous, the image of BH (0, ρ) is path-connected. Redefine V  as BH (0, ρ) and U  as B(0, R) ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and take the restrictions of the functions ax to BH (0, ρ). From this construction, for all x ∈ ∂Ω the new sets U(x ) ∩ ∂Ω and U(x ) ∩ int Ω are path-connected. Using the same notation for the finite sequence xi ∈ ∂Ω, let {U  (xi ) : 1 ≤ i ≤ k} be the new finite subcovering of ∂Ω. For each x ∈ int Ω there exists 0 < rx < R/2 such that B(x, rx ) ⊂ int Ω. Consider the new set ΩR = Ω\ ∪ki=1 U  (xi ). Since ΩR is compact, there exists a finite open covering of ΩR by the sets {B(xi , ri ) : k + 1 ≤ i ≤ m}, xi ∈ ΩR , and ri = rxi . By construction B(xi , ri ) ⊂ int Ω. (i) We first show that distΩ (x, y) is uniformly bounded for any two points of Ω. Since Ω is covered by a finite number of balls, it is sufficient to show that the geodesic distance between any two points x and y in each ball is bounded. For k + 1 ≤ i ≤ m and x, y ∈ B(xi , ri ), distΩ (x, y) = |x − y| < R. For 1 ≤ i ≤ k, and x, y ∈ U  (xi ), there is a graph representation x = xi + Ai (ζ  + ζN eN ) and y = xi + Ai (ξ  + ξN eN ). Choose as first path the line from x = xi + Ai (ζ  + ζN eN ) to xi + Ai (ζ  + ai (ζ  )eN ), follow the boundary along the second path {λζ  +(1−λ)ξ  , ai (λζ  +(1−λ)ξ  ) : λ ∈ [0, 1]} from xi +Ai (ζ  +ai (ζ  )eN )

5. Sets Locally Described by the Epigraph of a Function

99

to xi +Ai (ξ  +ai (ξ  )eN ), and finally as third path the line from xi +Ai (ξ  +ai (ξ  )eN ) to y = xi + Ai (ξ  + ξN eN ). Its length is bounded by  1' |ζ  − ξ  |2 + |∇ai (λζ  + (1 − λ)ξ  ) · (ζ  − ξ  )|2 dλ + 2R, 2R + 0

√ which is bounded by 2R + 1 + c2 2ρ + 2R. Finally we proceed by contradiction. By definition of the geodesic distance distΩ (x, y)/|x − y| ≥ 1. If (5.39) is not true, there exist sequences {xn } and {yn } in Ω such that distΩ (xn , yn )/|xn − yn | goes to +∞, and hence, by boundedness of distΩ (xn , yn ), |xn − yn | goes to zero. Since Ω is bounded there exist subsequences, still denoted by {xn } and {yn }, and a point x ∈ Ω such that xn → x and yn → x. If x ∈ int Ω, there exists r > 0 such that B(x, r ) ⊂ int Ω and N such that for all n > N , xn , yn ∈ B(x, r ), where distΩ (xn , yn ) = |xn − yn | → 0 and we get a contradiction. If x ∈ ∂Ω, PH (B(x, ρ) − x) = BH (0, ρ) = V  . There exists N such that, for all n > N , xn , yn ∈ B(0, ρ) and there exist (ζn , ζnN ), and (ξn , ξnN ), ζn , ξn ∈ BH (0, ρ), ξnN ≥ ax (ξn ), and ζnN ≥ ax (ζn ) such that xn = x + Ax [ξn + ξnN eN ] and yn = x + Ax [ζn + ζnN eN ]. The path {p(λ); 0 ≤ λ ≤ 1} defined as   λζn + (1 − λ)ξn p(λ) = x + Ax ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] stays in B(x, R) ∩ Ω. Indeed λζn + (1 − λ)ξn ∈ BH (0, ρ), eN · A−1 x (p(λ) − x) = ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] ≥ ax (λζn + (1 − λ)ξn ) , and p(λ) ∈ B(x, R). This last property follows from the estimate ' |p(λ) − x| ≤ |λζn + (1 − λ)ξn |2 + |ax (λζn + (1 − λ)ξn ) |2 + |λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )]| ' ≤ 1 + c2 |λζn + (1 − λ)ξn | + λ |ζnN − ax (ζn )| + (1 − λ) |ξnN − ax (ξn )| ' ' ≤ 1 + c2 ρ + λ 2ρ + (1 − λ) 2ρ = (2 + 1 + c2 ) ρ < R. Since the path lies in B(x, R) ∩ Ω, its length can be estimated by computing   ζn − ξn  p (λ) = Ax ∇ax (λζn + (1 − λ)ξn ) · (ζn − ξn ) + [ζnN − ξnN ] + ax (ξn ) − ax (ζn ) ⇒ |p (λ)| ≤ |ζn − ξn | + 2c|ζn − ξn | ≤ (1 + 2c) |yn − xn |  1 ⇒ distΩ (xn , yn ) ≤ |p (λ)| dλ ≤ (1 + 2c) |yn − xn | ≤ (1 + 2c) 2ρ < R 0

and we again get a contradiction. (ii) The case where ∂Ω is path-connected follows by the same constructions and proof as in part (i).

100 5.4.5

Chapter 2. Classical Descriptions of Geometries and Their Properties Nonhomogeneous Neumann and Dirichlet Problems

With the help of the previous definition of boundary measure in section 5.4.3, we can now make sense of the nonhomogeneous Neumann and Dirichlet problems (for the Laplace equation) in Lipschitzian domains. Assuming that the functions aj belong to W 1,∞ (Vj ), it can be shown that the classical Stokes divergence theorem holds for such domains. Given a bounded smooth domain D in RN and a Lipschitzian domain Ω in D, then   ∀ ϕ ∈ C 1 (D, RN ), div ϕ

dx = ϕ

· n dΓ, (5.41) Ω

Γ

where the outward unit normal field n is defined by (5.36) for almost all ∈ Γj (with y = hx (ζ  , 0), ζ  ∈ Vj ) as (∂1 aj (ζ  ), . . . , ∂N −1 aj (ζ  ), −1) ' . 1 + |∇aj (ζ  )|2

∗

n(y) = Aj

The trace of a function g in W 1,1 (D) on Γ is defined through a W 1,∞ (D)N -extension N of the normal n as   def gϕ dΓ = div (gϕN ) dx (5.42) ∀ϕ ∈ D(RN ), Γ

Ω

(cf. S. Agmon, A. Douglis, and L. Nirenberg [1, 2]). The right-hand side is well-defined in the usual sense so that by the Stokes divergence theorem we obtain the trace g Γ defined on Γ. This trace is uniquely defined, denoted by γΓ g, and   (5.43) γΓ ∈ L W 1,1 (Ω), L1 (Γ) . Let Ω be a Lipschitzian domain in D. Given f ∈ L2 (D) and g ∈ H 1 (D), consider the following problem: ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 1 (Ω),    ∇y · ∇ϕ dx = f ϕ dx + g ϕ dΓ. Ω

If the condition

Ω



(5.44)

Γ

 f dx + Ω

g dΓ = 0

(5.45)

Γ

is satisfied, then by the Lax–Milgram theorem, problem (5.44)–(5.45) has a unique solution in H 1 (Ω)/ R: any two solutions of problem (5.44) can differ only by a constant. For instance, we can associate with the solution y = y(Ω) in H 1 (Ω)/ R of (5.44) the following objective function:   def  def 2 J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = (|∇ϕ(x)| − G) dx (5.46) Ω

for some G in L2 (D).

6. Sets Locally Described by a Geometric Property

101

For the nonhomogeneous Dirichlet problem let Ω be a Lipschitzian domain in D and consider the weak form of the problem ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 2 (Ω) ∩ H01 (Ω),    ∂ϕ y  ϕ dx = − g f ϕ dx − dΓ + ∂n Ω Γ Ω

(5.47)

for data (f, g) in L2 (D) × H 1 (D). If this problem has a solution y, then by Green’s theorem     ∂ϕ ∂ϕ dΓ = − g dΓ + ∇y · ∇ϕ dx − y f ϕ dx (5.48) ∂n Ω Γ Γ ∂n Ω and −y = f ∈ L2 (Ω),

1

y|Γ = g ∈ H 2 (Γ).

Here, for instance, we can associate with y the objective function   def  def 1 |ϕ − G|2 dx, G ∈ L2 (D). J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = 2 Ω

6

(5.49)

(5.50)

Sets Locally Described by a Geometric Property

A large class of sets Ω can be characterized by geometric segment properties. The basic segment property is equivalent to the property that the set is locally a C 0 epigraph (cf. section 6.2). For instance, it is sufficient to get the density of C k (Ω) in the Sobolev space W m,p (Ω) for any m ≥ 1 and k ≥ m. In this section, we establish the equivalence of the segment, the uniform segment, and the more recent9 uniform fat segment properties with the respective locally C 0 , the C 0 , and the equi-C 0 epigraph properties (cf. sections 6.2, 6.3, and 6.4). Under the uniform fat segment property the local functions all have the same modulus of continuity specified by the continuity of the dominating function at the origin (cf. section 6.3). For sets with a compact boundary the three segment properties are equivalent (cf. section 6.1). However, for sets with an unbounded boundary the uniform segment property is generally too meager to make the local epigraphs uniformly bounded and equicontinuous. Section 6.4 specializes the uniform fat segment property to the uniform cusp and cone properties, which imply that ∂Ω is, respectively, an equi-H¨olderian and equi-Lipschitzian epigraph. Section 6.5 discusses the existence of a locally finite boundary measure. We have already seen in section 5.4.3 that Lipschitzian sets have a locally finite boundary or surface (that is Hausdorff HN −1 ) measure. So we can make sense of boundary conditions associated with a partial differential equation on the domain. In general, this is not true for H¨ olderian domains. To illustrate that point, we construct examples of H¨ olderian sets for which the Hausdorff dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. 9 This was introduced starting with the uniform cusp property in M. C. Delfour and J.-P. Zo´sio [37] in 2001 and further refined and generalized in the sequence of papers by M. C. Delfour, le ´sio [1, 2] in 2005 and M. C. Delfour and J.-P. Zole ´sio [43] in 2007. N. Doyon, and J.-P. Zole

102

6.1

Chapter 2. Classical Descriptions of Geometries and Their Properties

Definitions and Main Results

In this section we use the notation introduced at the beginning of section 5: eN is a unit vector in RN and H = {eN }⊥ . The open segment between two distinct points x and y of RN will be denoted by def

(x, y) = {x + t(y − x) : ∀t, 0 < t < 1}. Definition 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the segment property if ∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃Ax ∈ O(N), such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform segment property if ∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(iii) Ω is said to satisfy the uniform fat segment property if there exist r > 0, λ > 0, and an open region O of RN containing the segment (0, λeN ) and not 0 such that for all x ∈ ∂Ω, ∃Ax ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O ⊂ int Ω.

(6.1)

Of course, when Ω is an open domain, Ω = int Ω, and we are back to the standard definition. Yet, as in Definition 3.1, those definitions involve only ∂Ω, Ω, and int Ω and the properties remain true for all sets in the class

 [Ω]b = A ⊂ RN : A = Ω and ∂A = ∂Ω . A set having the segment property must have an (N −1)-dimensional boundary and cannot simultaneously lie on both sides of any given part of its boundary. In fact, a domain satisfying the segment property is locally a C 0 epigraph in the sense of Definition 5.1, as we shall see in section 6.2. We first give a few general properties. Theorem 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If ∀x ∈ ∂Ω, ∃λ > 0, ∃Ax ∈ O(N),

(x, x + λAx eN ) ∈ int Ω,

then int Ω = ∅ and int Ω = Ω

and

∂(int Ω) = ∂Ω.

(6.2)

6. Sets Locally Described by a Geometric Property

103

(ii) Ω satisfies the segment property if and only if Ω satisfies the segment property. In particular int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Proof. (i) Pick any point x ∈ ∂Ω. By definition, for all µ ∈ ]0, 1[ , xµ = x+µ λ AeN ∈ int Ω and int Ω = ∅. As a result there exists a sequence µn > 0 → 0 such that int Ω  xµn → x

⇒ ∂Ω ⊂ int Ω ⊂ Ω

⇒ Ω ⊂ int Ω ⊂ Ω.

The second identity follows from Lemma 5.1 (i). (ii) Assume that Ω satisfies the segment property. For any x ∈ ∂Ω, there exist r > 0, λ > 0, and A ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, (y, y + λAeN ) ⊂ int Ω. We claim that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λ(−A)eN ) ⊂ int Ω.

Indeed, if there exist z ∈ B(x, r) ∩ Ω and α ∈ (0, λ) such that y − αAeN ∈ Ω, then y = (y − αAeN ) + αAeN ∈ int Ω, which leads to a contradiction. In the other direction we interchange Ω and Ω and repeat the proof. Finally, the two sets of properties follow from part (i) applied to Ω and Ω. We summarize the equivalences between the epigraph and segment properties that will be detailed in Theorems 6.5, 6.6, and 6.7 with the associated constructions and additional motivation. Theorem 6.2. Let Ω be a subset of RN such that ∂Ω = ∅. (i) (a) Ω is locally a C 0 epigraph if and only if Ω has the segment property; (b) Ω is a C 0 epigraph if and only if Ω has the uniform segment property; (c) Ω is an equi-C 0 epigraph if and only if Ω has the uniform fat segment property. (ii) If, in addition, ∂Ω is compact, the six properties are equivalent and there exists a dominating function h ∈ H that satisfies the properties of Theorems 5.1 and 5.2. Proof. (i) From Theorems 6.5, 6.6, and 6.7. (ii) When ∂Ω is compact, from part (i) and the equivalences in Theorem 5.2. We quote the following density results from R. A. Adams [1, Thm. 3.18, p. 54]. Theorem 6.3. If Ω has the segment property, then the set

 f |int Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to int Ω is dense in W m,p (int Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular, C k (int Ω) is dense in W m,p (int Ω) for any m ≥ 1 and k ≥ m.

104

6.2

Chapter 2. Classical Descriptions of Geometries and Their Properties

Equivalence of Geometric Segment and C 0 Epigraph Properties

As for sets that are locally a C 0 epigraph in Definition 5.1, the three cases of Definition 6.1 differ only when ∂Ω is unbounded. We first deal with the first two and will come back to the third one in section 6.3. Theorem 6.4. Let ∂Ω = ∅ be compact. Then the segment property and the uniform segment property of Definition 6.1 are equivalent. Proof. It is sufficient to show that the segment property implies the uniform segment property. Since ∂Ω is compact there exists a finite open subcover {Bi }m i=1 , Bi = of points of ∂Ω. From part (i) of B(xi , rxi ), of ∂Ω for some finite sequence {xi }m i=1 the proof of Theorem 5.2 ∃r > 0, ∀x ∈ ∂Ω,

∃i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi .

Define λ = min1≤i≤m λi > 0. Therefore for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi and hence, by choosing Ax = Axi ∈ O(N), we get ∀y ∈ Ω ∩ Bi ,

(y, y + λAx eN ) ⊂ (y, y + λi Axi eN ) ⊂ int Ω.

By restricting the above property to B(x, r) ⊂ Bi , ∀y ∈ Ω ∩ B(x, r),

(y, y + λAx eN ) ⊂ (y, y + λi Ax eN ) ⊂ int Ω.

Since r and λ are independent of x ∈ ∂Ω, Ω has the uniform segment property. Theorem 6.5. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If Ω satisfies the segment property, then Ω is locally a C 0 epigraph. For x ∈ ∂Ω, let B(x, rx ), λx , and Ax be the associated open ball, the height, and the rotation matrix. Then there exists ρx , def

0 < ρx ≤ rxλ = min {rx , λx /2} ,

(6.3)

which is the largest radius such that

 BH (0, ρx ) ⊂ PH (A−1 x (y − x)) : ∀y ∈ B(x, rxλ ) ∩ ∂Ω . The neighborhoods of Definition 5.1 can be chosen as def

Vx = BH (0, ρx ) and 

def U(x) = B(x, rxλ ) ∩ y ∈ RN : PH (A−1 x (y − x)) ∈ Vx ,

(6.4)

where10 BH (0, ρx ) denotes the open ball of radius ρx in the hyperplane H. For each ζ  ∈ Vx , there exists a unique yζ  ∈ ∂Ω∩U(x) such that PH A−1 x (yζ  −x) = ζ  and the function ζ  → ax (ζ  ) = (yζ  − x) · (Ax eN ) : Vx → R def

10 Note

that U(x) − {x} = Ax [B(0, rxλ ) ∩ {ζ : PH ζ ∈ Vx }].

6. Sets Locally Described by a Geometric Property

105

is well-defined, bounded, ∀ζ  ∈ Vx ,

|ax (ζ  )| < rxλ ,

(6.5)

and uniformly continuous in Vx , that is, ax ∈ C 0 (Vx ). (ii) Conversely, if Ω is locally a C 0 epigraph, both Ω and Ω satisfy the segment property. Proof. (i) By assumption for each x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃A ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAeN ) ⊂ int Ω.

Consider the set  def

W = PH A−1 (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω ,

rλ = min{r, λ/2}.

Therefore ∀ζ  ∈ W,

∃y ∈ B(x, r) ∩ ∂Ω,

PH A−1 (y − x) = ζ  .

If there exist y1 , y2 ∈ B(x, rλ )∩∂Ω such that PH A−1 (y1 −x) = PH A−1 (y2 −x) = ζ  , then y2 ∈ (y1 , y1 + AeN · (y2 − y1 ) AeN ) ⊂ (y1 , y1 + λAeN ) ∈ int Ω since |y2 − y1 | < 2rλ < λ and we get a contradiction. Therefore, the map ζ  → a(ζ  ) = AeN · (y − x) = eN · A−1 (y − x) : W → R def

is well-defined and a(0) = 0. (W is open and 0 ∈ W ). By definition, 0 ∈ W ⊂ BH (0, rλ ). Given any point ξ  ∈ W , the segment property is satisfied at the point yξ = x + A(ξ  + a(ξ  )eN ) ∈ ∂Ω ∩ B(x, rλ ), def

and (yξ , yξ + λAeN ) ⊂ int Ω and (yξ , yξ − λAeN ) ⊂ int Ω by Theorem 6.1 (ii): (yξ , yξ + λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ B(x, rλ ), (yξ , yξ − λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ ∩B(x, rλ ). Therefore, there exists a neighborhood BH (ξ  , R) of ξ  such that BH (ξ  , R) ⊂ BH (0, rλ ) and for all ζ  ∈ BH (ξ  , R), Lζ  ∩ int Ω ∩ B(x, rλ ) = ∅, where Lζ  = x + A(ζ  + R eN ) is the line through x + Aζ  in the direction AeN . If ξ  is not an interior point of W , there exists a sequence {ζn } ⊂ BH (ξ  , R),  ζn → ξ  , such that for all n, (Lζn ∩ B(x, rλ )) ∩ ∂Ω = ∅. Therefore, either Lζn ∩ B(x, rλ ) ⊂ int Ω or Lζn ∩ B(x, rλ ) ⊂ int Ω. Since for points ζn ∈ BH (ξ  , R) we already know that Lζn ∩ B(x, rλ ) ∩ int Ω = ∅, then Lζn ∩ B(x, rλ ) ⊂ int Ω. Choose   def 1 α = a(ξ  ) − rλ2 − |ξ  |2 2

106

Chapter 2. Classical Descriptions of Geometries and Their Properties

and the sequence zn = x + A(ζn + αeN ) → z = yξ − def

def

1 2



a(ξ  ) +

 rλ2 − |ξ  |2 AeN .

We get the following estimate: |z − x| <

1 (|yξ − x| + rλ ) < rλ , 2

1 1 (|yξ − x| + rλ ) = |ζn − ξ  | + (|yξ − x| + rλ ) . 2 2 ¯ ¯ There exists N such that, for all n ≥ N , |zn − x| < |zn − z| +

|ζn − ξ  | = |zn − z| <

1 (rλ − |yξ − x|) 2

⇒ |zn − x| < rλ

and zn ∈ B(x, rλ )∩Lζn ⊂ int Ω∩B(x, rλ ). The limit point z belongs to Ω∩B(x, rλ ). However, PH A−1 (z − x) = ξ  and   1  2  2 a(ξ ) + rλ − |ξ | < 0, AeN · (z − yξ ) = − 2 ' since, by construction of W and a, |a(ξ  )| < rλ2 − |ξ  |2 . This means that z ∈ (yξ , yξ − λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ ∩B(x, rλ ) and we get a contradiction with the fact that z ∈ Ω ∩ ∩B(x, rλ ). This proves that all points of W are interior points and, hence, that W is open. Since 0 ∈ W , there exists ρ, 0 < ρ ≤ rλ , such that ρ is the largest radius for which BH (0, ρ) ⊂ W . (Choice of the neighborhoods and conditions of Definition 5.1.) Finally the neighborhoods of Definition 5.1 can be chosen as

 def def V = BH (0, ρ) and U = B(x, rλ ) ∩ y : PH A−1 (y − x) ∈ BH (0, ρ) . By construction, condition (5.20) in Definition 5.2 is satisfied. It is then easy to check properties (5.21) and (5.22). (The function a is bounded and uniformly continuous in V .) Consider a point ξ  ∈ V and let yξ ∈ ∂Ω ∩ B(x, rλ ) be the associated unique point such that ξ  = PH A−1 (yξ − x) and a(ξ  ) = AeN · (yξ − x).

 By construction, since ξ  ∈ PH A−1 (y − x) : y ∈ B(x, rλ ) ∩ ∂Ω , |a(ξ  )| ≤ |yξ − x| < rλ and inf a(ζ  )  = lim   def

ζ →ξ

and

L = lim sup a(ζ  ) def

ζ  →ξ 

are finite. Hence the points y
= x + A(ζ  + eN ) ∈ ∂Ω ∩ B(x, rλ ), def

yL = x + x + A(ζ  + LeN ) ∈ ∂Ω ∩ B(x, rλ ) def

6. Sets Locally Described by a Geometric Property

107

belong to ∂Ω as limit points of points of ∂Ω. If y
= yξ , then |y
− yξ | ≤ |y
− x| + |yξ − x| < 2rλ ≤ λ and either yξ ∈ (y
, y
+ λd) or y
∈ (yξ , yξ + λd). By the segment property, this means that either yξ or y
belongs to int Ω, which contradicts the fact that both points belong to ∂Ω. Therefore y
= yξ and by the same argument yL = yξ . Hence a(ζ  ) =  = (y
− x) · AeN = (yξ − x) · AeN = a(ξ  ), lim inf   ζ →ξ

a(ξ  ) = (yξ − x) · AeN = (yL − x) · AeN = L = lim sup a(ζ  ), ζ  →ξ 

and a is continuous in ξ  ∈ W . In view of the fact that a is uniformly bounded, then a ∈ C(W ) and a is uniformly continuous in W . (ii) Pick a point x ∈ ∂Ω. By Definition 5.1, there exist neighborhoods U(x) and V (x), Ax ∈ O(N), and a continuous function ax : Vx → R with the appropriate properties. Choose rx > 0 such that B(x, 2rx ) ⊂ U(x). We want to show that for λx = rx ∀y ∈ B(x, rx ) ∩ Ω,

(y, y + λx Ax eN ) ⊂ int Ω.

From property (5.3) in Definition 5.1

 U(x) ⊂ y ∈ RN : PH A−1 x (y − x) ∈ Vx

⇒ BH (0, 2rx ) ⊂ Vx

def

and hence for all y ∈ B(x, rx ) ∩ Ω we have ζ  = PH A−1 (y − x) ∈ Vx . Moreover, y ∈ B(x, rx ) ∩ Ω ⊂ U(x) ∩ Ω implies that (y − x) · Ax eN ≥ ax (ζ  ). Consider the segment (y, y + λx Ax eN ). For each 0 < t < 1 (y + tλx Ax eN − x) · Ax eN = (y − x) · Ax eN + tλx ≥ ax (ζ  ) + tλx > ax (ζ  ) ⇒ (y, y + λx Ax eN ) ⊂ A+ , |y + tλx Ax eN − x| ≤ |y − x| + λx < rx + λx < 2rx ⇒ (y, y + λx Ax eN ) ⊂ B(x, 2rx ) ∩ A+ ⊂ U(x) ∩ int Ω ⇒ (y, y + λx Ax eN ) ⊂ int Ω and Ω satisfies the segment property. Furthermore, since Ω is locally a C 0 epigraph, so is Ω by changing Ax eN into −Ax eN . Hence Ω also satisfies the segment property. Theorem 6.6. Let Ω be a subset of RN such that ∂Ω = ∅. Ω has the uniform segment property if and only if Ω is a C 0 epigraph. In particular, the neighborhoods U(x) at x ∈ ∂Ω in RN and Vx at 0 in H can be chosen in the following way: given rλ = min{r, λ/2} and the largest radius ρ > 0 such that BH (0, ρ) ⊂ {PH z : ∀z ∈ B(x, rλ ) ∩ ∂Ω}, let def

V = BH (0, ρ)

and

def

U = B(0, rλ ) ∩ {y : PH z ∈ BH (0, ρ)}

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Chapter 2. Classical Descriptions of Geometries and Their Properties

and, for each x ∈ ∂Ω, define def

Vx = V

and

def

U(x) = x + Ax U.

Moreover, the functions ax : V → R can be chosen uniformly bounded, ∀x ∈ ∂Ω, ∀ζ  ∈ V,

|ax (ζ  )| ≤ rλ ,

and uniformly continuous in V , that is, ax ∈ C 0 (V ). Remark 6.1. It is important to notice that, when ∂Ω is unbounded, the uniform segment property does not generally imply that the family of functions {ax : x ∈ ∂Ω} can be chosen uniformly bounded and equicontinuous. To restore that property the segment will have to be fattened, as we shall see in the next section. Remark 6.2. The modulus of continuity in 0 of the function h of part (ii) determines the modulus of continuity of the family of functions {ax : x ∈ ∂Ω}. Proof. (i) By Theorem 6.5, Ω satisfies the segment property if and only if Ω is locally a C 0 epigraph. It remains to sharpen this result. (⇒) Ω satisfies a uniform segment property for some r > 0 and λ > 0, and we can now repeat the constructions in the proof of part (i) of Theorem 6.5. Let rλ = min{r, λ/2}. With each x ∈ ∂Ω can be associated

 PH A−1 x (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω = {PH z : ∀z ∈ B(x, rλ ) ∩ ∂Ω} since A−1 x B(x, rλ ) = B(x, rλ ). The set def

W = {PH z : ∀z ∈ B(x, rλ ) ∩ ∂Ω} is independent of x since rλ is now independent of x. Hence the largest radius ρ > 0 such that BH (0, ρ) ⊂ W is also independent of x. The family of bounded uniformly continuous functions ax defined on V = BH (0, ρ) such that ∂Ω is locally the graph function of ax in the neighborhood

 U(x) = B(x, rλ ) ∩ y : PH A−1 x (y − x) ∈ BH (0, ρ) = x + Ax U,

def

U = B(0, rλ ) ∩ {y : PH z ∈ BH (0, ρ)}

of x. The neighborhoods Vx and U(x) are now independent of x up to a rotation around the origin. By Definition 5.2 (ii), Ω is a C 0 epigraph. Moreover, by Theorem 6.5 (i) the functions ax : V → R can be chosen uniformly bounded by rλ (cf. (6.5)), and uniformly continuous in V , that is, ax ∈ C 0 (V ). (⇐) The proof is the same as the proof of part (ii) of Theorem 6.5. The uniform segment property follows from the fact that the neighborhoods U(x) − x of 0 are, up to a rotation around 0, independent of x ∈ ∂Ω. Therefore there exists r > 0 such that, for all x ∈ ∂Ω, B(0, 2r) ⊂ U(x) − x and hence B(x, 2r) ⊂ U(x). As a result we can repeat the proof with r in place of rx and λ = r, which are both independent of x.

6. Sets Locally Described by a Geometric Property d = Ax (0, eN )

109 d = Ax (0, eN )

fat set Ax O

fat set Ax O

y

y

Figure 2.5. Examples of arbitrary and axially symmetrical O around the direction d = Ax (0, eN ).

6.3

Equivalence of the Uniform Fat Segment and the Equi-C 0 Epigraph Properties

The uniform fat segment property strengthens the uniform segment property by fattening the segment of Definition 6.1 (ii). Before proving the equivalence with an equi-C 0 epigraph, we first show that the open set O of Definition 6.1 (iii) is equivalent to a parametrized axisymmetrical open region of the form  def

O(h, ρ, λ) = ζ  + ζN eN ∈ RN : ζ  ∈ BH (0, ρ), lim sup h(|ζ  )|) < ζN < λ (6.6) for some ρ > 0, λ > 0 and a dominating function h that belongs to the space of dominating functions introduced in (5.11) (see Figure 2.5): H = {h : [0, ∞[ → [0, ∞[ : h(0) = 0 and h is continuous in 0} .

(6.7)

The use of the lim sup h rather than h is necessary to make O(h, ρ, λ) open since we only assume that h is continuous in 0. Lemma 6.1. (i) Given ρ > 0, λ > 0, and h ∈ H, the region O(h, ρ, λ) contains the segment (0, λeN ), does not contain 0, and is open. (ii) Let λ > 0 be a real number and O be an open subset of RN containing (0, λeN ) and not 0. Then there exist ρ > 0 and a continuous function h : [0, ρ] → [0, ∞[ which is monotone strictly increasing such that the set O(h, ρ, λ/2) = {ξ : ξ  ∈ BH (0, ρ) and h(|ζ  |) < ξN < λ/2} def

is open and contains (0, λeN /2) and not 0, and O(h, ρ, λ/2) ⊂ O. Proof. (i) By definition    (0, λeN ) = ζN eN : 0 = lim h(|ξ |) < ζ < λ ⊂ O(h, ρ, λ), N  ξ →0

since h(0) = 0 and h is continuous at the origin. Also 0 = 0 + 0eN ∈ / O(h, ρ) since it would yield the contradiction 0 = limξ →0 h(|ξ  |) < ζN = 0. To show that O(h, ρ, λ)

110

Chapter 2. Classical Descriptions of Geometries and Their Properties

is open, it is sufficient to show that for each point ξ = ξ  + ξN eN ∈ O(h, ρ, λ) there exists a neighborhood of ξ contained in O(h, ρ, λ). But this is true by definition of the lim sup: given (ξ  , ξN ) ∈ O(h, ρ, λ), lim sup h(|ξ  |) < ξN < λ and there exists a neighborhood V (ξ  ) ⊂ BH (0, ρ) of ξ  . Hence 

ξ ∈ ζ  + ζN eN ∈ RN : ζ  ∈ V (ξ  ), lim sup h(|ζ  )|) < ζN < λ ⊂ O(h, ρ, λ), and O(h, ρ, λ) is open. (ii) For each integer n ≥ 0, let λn = λ/2n+1 . Since the segment [λ1 eN , λ0 eN ] is contained in O, there exists a largest radius r0 > 0 such that the cylinder C0 = {ζ : |ζ  | < r0 and λ1 < ζN < λ0 } def

is contained in O. Again, since the segment [λ2 eN , λ1 eN ] is contained in O, there exists a largest radius 0 < r1 ≤ r0 /2 such that the cylinder C1 = {ζ : |ζ  | < r1 and λ2 < ζN < λ1 } def

is contained in O. Similarly, for n ≥ 2, there exists a largest radius 0 < rn ≤ rn−1 /2 such that the cylinder Cn = {ζ : |ζ  | < r1 and λn < ζN < λn−1 } def

is contained in O. By construction, 2n rn ≤ r0 , rn  0, ∪n∈N Cn ⊂ O is open and contains (0, λeN /2) and not 0. Since the boundary of that set is piecewise constant, we make a few adjustments to make it piecewise linear and continuous. First construct the set   ζN − λ1 λ0 − ζN def + r1 . O1 = ζ : λ1 < ζN < λ0 and |ζ  | < r0 λ0 − λ1 λ0 − λ1 For n ≥ 2, choose   ζN − λn λn−1 − ζN def . On = ζ : λn < ζN ≤ λn−1 and |ζ  | < rn−1 + rn λn−1 − λn λn−1 − λn From the condition 0 < rn ≤ rn−1 /2, the following union is open: O  = ∪n≥1 On , def

(0, λeN /2) ⊂ O ⊂ O, and 0 ∈ / O .

Set ρ = r0 and define the function h : [0, ρ] → [0, ∞[ on each interval [rn−1 , rn [ as follows: def

h(θ) = rn−1

θ − λn λn−1 − θ + rn , λn−1 − λn λn−1 − λn

rn−1 ≤ θ < rn .

It is continuous and monotone strictly increasing since rn−1 < rn and λn−1 > λn . Moreover, it is easy to check that O = O(h, ρ, λ/2).

6. Sets Locally Described by a Geometric Property

111

We now complete the equivalences between the epigraph and the segment properties of Theorems 6.5 and 6.6. Theorem 6.7. Let Ω be a subset of RN such that ∂Ω = ∅. Ω is an equi-C 0 epigraph if and only if Ω has the uniform fat segment property. Proof. (⇐) Assume that the uniform fat segment property is verified for r > 0, λ > 0, and the open set O. By Lemma 6.1 (ii), there exist ρ > 0 and a continuous function h : [0, ρ] → [0, ∞[ which is monotone strictly increasing, the set O(h, ρ, λ/2) = {ξ : ξ  ∈ BH (0, ρ) and h(|ζ  |) < ξN < λ/2} def

is open and contains (0, λeN /2) and not 0, and O(h, ρ, λ/2) ⊂ O. So the uniform fat segment property is verified for r > 0, λ/2 > 0, and the new open set O(h, ρ, λ/2). In particular, Ω satisfies the uniform segment property with λ/2 instead of λ. By Theorem 6.6, the neighborhoods at x ∈ ∂Ω can be chosen in the following way: given rλ = min{r, λ/4} and the largest radius ρ > 0 such that BH (0, ρ ) ⊂ {PH z : ∀z ∈ B(x, rλ ) ∩ ∂Ω}, let V = BH (0, ρ ) def

and U = B(0, rλ ) ∩ {y : PH z ∈ BH (0, ρ )} def

and, for each x ∈ ∂Ω, define def

Vx = V

and

def

U(x) = x + Ax U.

By Theorem 6.5 (i) with λ/2 instead of λ, the function ax : V → R is uniformly continuous in V and ∀ζ  ∈ V,

|ax (ζ  )| < rλ .

Given ζ  and ξ  in BH (0, ρ ) such that ξ  − ζ  ∈ BH (0, ρ), consider the points yζ  = x + Ax (ζ  + ax (ζ  )eN ) ∈ B(x, rλ ) ∩ ∂Ω, def

yξ = x + Ax (ξ  + ax (ξ  )eN ) ∈ B(x, rλ ) ∩ ∂Ω. def

By subtracting, yξ = yζ  + Ax (ξ  − ζ  + (ax (ξ  ) − ax (ζ  ))eN ). If ax (ξ  ) − ax (ζ  ) > h(|ξ  − ζ  |), then h(|ξ  − ζ  |) < ax (ξ  ) − ax (ζ  ) < λ/4 + λ/4 = λ/2 ⇒ yξ = yζ  + Ax (ξ  − ζ  + (ax (ξ  ) − ax (ζ  ))eN ) ∈ yζ  + Ax O(h, ρ, λ/2) ⊂ int Ω. But this contradicts the fact that yξ ∈ ∂Ω. Since the argument can be repeated with ξ  in place of ζ  , for all x ∈ ∂Ω, ∀ζ  , ξ  ∈ BH (0, ρ ) such that |ξ  − ζ  | < ρ ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |). (6.8)

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Since h is continuous in 0, for all ε > 0, there exists δ, 0 < δ < ρ, such that θ < δ implies h(θ) < ε. Hence, for all x ∈ ∂Ω, ∀ζ  , ξ  ∈ BH (0, ρ ) such that |ξ  − ζ  | < δ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |) < ε

and the family {ax } is equicontinuous with respect to x ∈ ∂Ω. (⇒) We go back to the proof of (i) ⇒ (ii) in Theorem 5.1 that we copy below for the benefit of the reader in order to keep track of all the parameters. By Definition 5.1 (iii), Ω is an equi-C 0 epigraph and the family of functions {ax ∈ C 0 (V ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous. In particular, U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that ∀x ∈ ∂Ω,

def

Vx = V

def

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Definition 5.1 

⇒ BH (0, 3r) ⊂ V. U(x) ⊂ y ∈ RN : PH A−1 x (y − x) ∈ V Use the modulus of continuity defined in (5.14): ∀θ ∈ [0, 3r],

h(θ) = sup sup

sup

ζ  ∈H x∈∂Ω ξ ∈V |ζ  |=1 ξ  +θζ  ∈V

|ax (θζ  + ξ  ) − ax (ξ  )|.

(6.9)

The function h(θ), as the sup of the family θ → |ax (θζ  + ξ  ) − ax (ξ  )| of continuous functions with respect to (ζ  , ξ  ), is lower semicontinuous with respect to θ and bounded in [0, 3r] since the ax ’s are uniformly bounded. By construction h(0) = 0 and by equicontinuity of the ax , for all ε > 0, there exists δ > 0 such that for all θ ∈ [0, 3r], |θ| < δ, ∀x ∈ ∂Ω, ∀ζ  ∈ H, |ζ  | = 1,

|ax (θζ  + ξ  ) − ax (ξ  )| < ε

⇒ |h(θ)| < ε,

where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the new neighborhoods V  = BH (0, ρ) and U  = {ξ ∈ B(0, ρ) : PH ξ ∈ V  } = B(0, ρ) and extend h by the constant h(ρ) to [ρ, ∞[ . Therefore h ∈ H and, by definition of h, for all x ∈ ∂Ω and for all ξ  and ζ  in BH (0, ρ) such that |ξ  − ζ  | < ρ   ξ − ζ      |ax (ξ  ) − ax (ζ  )| = ax ζ  + |ξ  − ζ  |  − a (ζ ) x ≤ h(|ξ − ζ |)  |ξ − ζ | ⇒ ∀x ∈ ∂Ω, ∀ξ  , ζ  ∈ BH (0, ρ) |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |).

Now that we have recalled the proof of Theorem 5.1 we complete the proof by showing that the uniform fat segment property of Definition 6.1 is verified with λ = r > 0, r > 0, and  def

O(h, r, r) = ζ  + ζN eN ∈ RN : ζ  ∈ BH (0, r), lim sup h(|ζ  |) < ζN < r .

6. Sets Locally Described by a Geometric Property

113

By Lemma 6.1 (i), the region O(h, r, r) contains the segment (0, r eN ), does not contain 0, and is open. Given a point x ∈ ∂Ω, we want to show that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O(h, r, r) ⊂ int Ω.

For y ∈ B(x, r) ∩ Ω ⊂ U(x) ∩ Ω ζ  = PH A−1 x (y − x) ∈ V def

and

ζN = (y − x) · Ax eN ≥ ax (ζ  ). def

Associate with ξ ∈ O(h, r, r) the point z = y + Ax ξ = x + Ax (ζ  + ξ  + (ζN + ξN )eN ). def

By construction

ζN + ξN

  PH A−1 x (z − x) = ζ + ξ and Ax eN · (z − x) = ζN + ξN , > ax (ζ  ) + lim sup h(|ξ  |) ≥ ax (ζ  ) + lim inf h(|ξ  |) ≥ ax (ζ  ) + h(|ξ  |)

⇒ ζN + ξN > ax (ζ  ) + h(|ξ  |) > ax (ζ  ) + ax (ζ  + ξ  ) − ax (ζ  ) = ax (ζ  + ξ  ) and z ∈ A+ x . In addition |z − x| ≤ |z − y| + |y − x| ≤

√ 2r + r < 3r

⇒ z ∈ U(x) ∩ A+ x = U(x) ∩ int Ω

and y + Ax O(h, r, r) ⊂ int Ω. This proves that Ω satisfies the uniform fat segment property of Definition 6.1 with λ = r > 0, r > 0, and O(h, r, r).

6.4

Uniform Cone/Cusp Properties and H¨ olderian/ Lipschitzian Sets

In this section we make the connection between the notions of H¨olderian and Lipschitzian sets of Definition 5.2 and the dominating function h generating the set O(h, ρ, λ) of the uniform fat segment property. We first recall some elements of Definition 5.2. Definition 6.2. Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is locally Lipschitzian if Ω is locally a C 0,1 epigraph. Ω is locally H¨ olderian if Ω is locally a C 0,
epigraph for some , 0 <  < 1. (ii) Ω is Lipschitzian if Ω is a C 0,1 epigraph. Ω is H¨ olderian if Ω is a C 0,
epigraph for some , 0 <  < 1. (iii) Ω is equi-Lipschitzian if Ω is an equi-C 0,1 epigraph. Ω is equi-H¨ olderian if Ω is an equi-C 0.
epigraph for some , 0 <  < 1.

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It is natural to associate with Definition 6.2 the following family of dominating functions:


h
(θ) = λ (θ/ρ) ,

0 <  ≤ 1,

in H. For  = 1, the region O(h1 , ρ, λ) is the open cone of height λ and aperture ω given by tan ω = ρ/λ, 0 < ω < π/2, h1 (θ) = λ

1 θ = θ. ρ tan ω

For 0 <  < 1, O(h
, ρ, λ) defines a cuspidal region around λeN of height λ. 6.4.1

Uniform Cone Property and Lipschitzian Sets

Lipschitzian sets have also been equivalently characterized by a purely geometric uniform cone property which seems to have been originally introduced by S. Agmon [1]. In this section we establish the equivalence of the two sets of definitions and express the previous properties and formulae of Definition 5.2 in terms of the parameters of the cone. One of the important properties of a family of open domains satisfying the uniform cone property is that the extension operator from the domains to RN is uniformly continuous. Notation 6.1. Given λ > 0 and 0 < ω < π/2, denote by C(λ, ω) the open cone   1 def C(λ, ω) = y ∈ RN : PH (y) < y · eN < λ , tan ω where PH is the orthogonal projection onto the hyperplane H = {eN }⊥ orthogonal to the direction eN (cf. Figure 2.6). Further, associate with an arbitrary point

direction Ax eN

aperture ω

height λ vertex x

Figure 2.6. The cone x + Ax C(λ, ω) in the direction Ax eN .

6. Sets Locally Described by a Geometric Property

115

x ∈ RN , an Ax ∈ O(N), a direction Ax eN , and the translated and rotated cone x + Ax C(λ, ω). It is readily seen that C(λ, ω) = O(h, ρ, λ) with h(θ) = θ/ tan ω and ρ = λ tan ω. Definition 6.3. Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the local uniform cone property if ∀x ∈ ∂Ω, ∃λ > 0, ∃ω > 0, ∃r > 0, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax C(λ, ω) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform cone property if ∃λ > 0, ∃ω > 0, ∃r > 0, ∀x ∈ ∂Ω, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax C(λ, ω) ⊂ int Ω.

Theorem 6.8. Let Ω be a subset of RN such that ∂Ω = ∅. Ω is equi-Lipschitzian if and only if Ω satisfies the uniform cone property. The proof will be the same as the one of Theorem 6.9 in the next section. 6.4.2

Uniform Cusp Property and H¨ olderian Sets

Definition 6.4. Let Ω be a subset of RN such that ∂Ω = ∅, 0 <  < 1, and h
(θ) = λ(θ/ρ)
. (i) Ω is said to satisfy the cusp property of index , 0 <  < 1, if ∀x ∈ ∂Ω, ∃h ∈ H, ∃λ > 0, ∃r > 0, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O(h
, ρ, λ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform cusp property of index , 0 <  < 1, if ∃λ > 0, ∃h ∈ H, ∃r > 0, ∀x ∈ ∂Ω, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O(h
, ρ, λ) ⊂ int Ω.

The Lipschitzian case of Definitions 6.2 and 6.3 corresponds to  = 1 and ρ = λ tan ω in the above definition, but we wanted to keep the two terminologies distinct. Remark 6.3. In some applications, it might be interesting to relax the uniform cusp property by permitting the axis of the cuspidal region to bend: this makes the region look

116

Chapter 2. Classical Descriptions of Geometries and Their Properties

like a horn, and the corresponding property becomes a horn condition or property. Horn-shaped domains have been studied in several contexts in the literature. In particular conditions on domains have been introduced in the context of extension operators and embedding theorems: the domains of F. John [1]; the (ε, δ)-domains of P. W. John [1]; and the domains satisfying a flexible horn condition (which is a broader notion than the previous two) of O. V. Besov [1, 2]. Theorem 6.9. Let Ω be a subset of RN such that ∂Ω = ∅ and let 0 <  < 1. Ω is equi-H¨ olderian of index  if and only if Ω has the uniform cusp property of index . Proof. By Theorem 6.7, Ω has the uniform fat segment property if and only if Ω is an equi-C 0 epigraph. It remains to sharpen the equivalence. (⇐) In addition, Ω has the uniform cusp property of index . From inequality (6.8) in the proof of Theorem 6.7 for all x ∈ ∂Ω and ∀ζ  , ξ  ∈ BH (0, ρ ) such that |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h
(|ξ  − ζ  |)

and since h
(θ) = λ(θ/ρ)
, the functions ax are equi-C 0,
and Ω is equi-H¨olderian of index . (⇒) In addition, Ω is equi-H¨ olderian of index . In the proof of Theorem 6.7, we have constructed the modulus of continuity h ∈ H in (6.9): ∀θ ∈ [0, 3r],

h(θ) = sup sup

sup

ζ  ∈H x∈∂Ω ξ ∈V |ζ  |=1 ξ  +θζ  ∈V

|ax (θζ  + ξ  ) − ax (ξ  )|.

Since the family {ax : x ∈ ∂Ω} is equi-C 0,
, the modulus h in (6.9) is also C 0,
and there exists c such that |h(θ)| ≤ c |θ|
. As a result for all x ∈ ∂Ω and all ζ  , ξ  ∈ V such that |ξ  − ζ  | < 3r |ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |) ≤ c|ξ  − ζ  |θ . Let h
(θ) = c θ
. Then O(h
, r, r) ⊂ O(h, r, r) and Ω satisfies the uniform fat segment property for O(h
, r, r), but this is precisely the uniform cusp property of index .

6.5

Hausdorff Measure and Dimension of the Boundary

We have already seen in section 5.4.3 that Lipschitzian sets have a locally finite boundary measure HN −1 (∂Ω). In particular, when ∂Ω is compact, HN −1 (∂Ω) is finite. We now construct examples of H¨ olderian sets of index α, 0 < α < 1, with compact boundary (verifying the uniform cusp property) for which the Hausdorff dimension of the boundary is strictly greater than N − 1 and hence their boundary measure HN −1 (∂Ω) = +∞. We first give an upper bound. Theorem 6.10. Let Ω be a subset of RN with compact boundary. If Ω satisfies the uniform cusp property of Definition 6.4 (ii) associated with the function h(θ) = θ α , 0 < α < 1, then the Hausdorff dimension of ∂Ω is less than or equal to N − α.

6. Sets Locally Described by a Geometric Property

117

Proof. From Theorem 6.9, Ω is equi-H¨olderian of index α, that is, an equi-C 0,α epigraph. Let r > 0, ρ > 0, and λ > 0 be the parameters of Theorem 6.6, and let Ax eN , Ax ∈ O(N), be the direction associated with the point x ∈ ∂Ω. By assumption the functions ax : V → R satisfy the condition ∀ζ1 , ζ2 ∈ V,

|ax (ζ2 ) − ax (ζ1 )| ≤ c |ζ2 − ζ1 | . α

(6.10)

Since ∂Ω is compact, there exists a finite number of points {xi ∈ ∂Ω : 1 ≤ i ≤ m} such that ∂Ω ⊂ ∪m i=1 U(xi ). Given ε, 0 < ε < ρ, let NΩ (ε) be the number of hypercubes of dimension N and side ε required to cover ∂Ω and let NΩ,i (ε) be the number of hypercubes of dimension N and side ε required to cover ∂Ω ∩ U(xi ). We have the following estimate:  r N −1 c(√N − 1 ε)α λ . NΩ,i (ε) ≤ ε ε Indeed the neighborhood V = BH (0, ρ) ⊂ BH (0, rλ ) can be covered by [rλ /ε]N −1 (N − 1)-dimensional hypercubes of side ε. On each (N − 1)-dimensional hypercube of side ε the variation between the minimum and the maximum of the function ax is bounded by ' α √ α c (N − 1)ε2 = c N − 1ε . So the number of N -dimensional hypercubes of side ε necessary to cover the hypersurface above each (N − 1)-dimensional hypercube of side ε is α / . c √ . N − 1ε ε Finally

α  N −1  c √ N − 1ε + 1 +1 ε ε α  √  1 1 ≤ N −1 1−α (rλ + ε)N −1 c N − 1 + ε1−α ε ε  √  α 1 N −1 c N − 1 + ε1−α . ≤ N −α (rλ + ε) ε

NΩ,i (ε) ≤

r

λ

As a result for all β > N − α NΩ,i (ε) ≤

m i=1

NΩ,i (ε) ≤ m

1 εN −α

(rλ + ε)

⇒ NΩ (ε)εβ ≤ εβ−N +α m (rλ + ε) ⇒ ∀β > N − α,

N −1

N −1

α  √  N − 1 + ε1−α c

α  √  N − 1 + ε1−α c

Hβ (∂Ω) = 0.

By definition, the Hausdorff dimension of ∂Ω is less than or equal to N − α.

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Chapter 2. Classical Descriptions of Geometries and Their Properties

It is possible to construct examples of sets verifying the uniform cusp property for which the Hausdorff dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. Example 6.1. This following two-dimensional example of an open domain with compact boundary satisfying the uniform cusp condition for the function h(θ) = θ α , 0 < α < 1, can easily be generalized to an N -dimensional example. Consider the open domain def

Ω = {(x, y) : −1 < x ≤ 0 and 0 < y < 2} ∩ {(x, y) : 0 < x < 1 and f (x) < y < 2} ∩ {(x, y) : 1 ≤ x < 2 and 0 < y < 2} in R2 (see Figure 2.7), where f : [0, 1] → R is defined as follows: def

f (x) = dC (x)α ,

0 ≤ x ≤ 1,

and C is the Cantor set on the interval [0, 1] and dC (x) is the distance function from the point x to the set C (dC is uniformly Lipschitzian of constant 1). This function is equal to 0 on C. Any point in [0, 1]\C belongs to one of the intervals of length 3−k , k ≥ 1, which has been deleted from [0, 1] in the sequential construction of the Cantor set. Therefore the distance function dC (x) is equal to the distance function to the two end points of that interval. In view of this special structure it can be shown that ∀x, y ∈ [0, 1],

|dC (y)α − dC (x)α | ≤ |y − x|α .

Denote by Γ the piece of the boundary ∂Ω specified by the function f = dC . On Γ the uniform cusp condition is verified with ρ = 1/6, λ = (1/6)α , and h(θ) = θ α (see

2 y

1

domain Ω

C(λ, h, ρ, e2 )

f (x) 0

−1

x 0

1

2

Figure 2.7. Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θα .

6. Sets Locally Described by a Geometric Property

0

1/9

2/9

1/3

119

2/3

7/9

8/9

1

Figure 2.8. f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. Figure 2.8). Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓ (ε) of hypercubes of dimension N and side ε required to cover Γ. The construction of the Cantor set is done by sequentially deleting intervals. At step k = 0 the interval (1/3, 2/3) of width 3−1 is removed. At step k a total of 2k intervals of width 3−(k+1) are removed. Thus if we pick ε = 3−(k+1) , the interval [0, 1] can be covered with exactly 3(k+1) intervals. Here we are interested in finding a lower bound to the total number of squares of side ε necessary to cover Γ. For this purpose we keep only the 2k intervals removed at step k. Vertically it takes  α   −1 −(k+1) α 2−1 3−(k+1) 2 3 ≥ − 1. 3−(k+1) 3−(k+1) Then we have for β ≥ 0 0 NΩ (ε) ≥ NΓ (ε)≥ 2k

2−1 3−(k+1) 3−(k+1)

1

α −1

 k ≥ 2k−α 3(k+1)(1−α) − 2k = 3(1−α) 2 2−α 3(1−α) − 2k and NΩ (ε)(3

−k 1+β

)

−k(1+β)

≥3



≥ 3



−(α+β)

3

(1−α)

k 2

2

k 2

2

−α (1−α)

3

−α (1−α)

3

 −

 −2

2 3(1+β)

k

k .

The second term goes to zero as k goes to infinity. The first term goes to infinity as k goes to infinity if 3−(α+β) 2 > 1, that is, 0 < α+β < ln 2/ ln 3. Under this condition, H1+β (∂Ω) = H1+β (Γ) = +∞ for all 0 < α < ln 2/ ln 3 and all 0 ≤ β < ln 2/ ln 3 − α.

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Therefore given 0 < α < ln 2/ ln 3 ∀β, 0 ≤ β + α < ln 2/ ln 3,

H1+β (∂Ω) = +∞

and the Hausdorff dimension of ∂Ω is strictly greater than 1. Given 0 < α < 1, it is possible to construct an optimal example of a set verifying the uniform cusp property for which the Hausdorff dimension of the boundary is exactly N − α and hence HN −1 (∂Ω) = +∞. Example 6.2 (Optimal example of a set that verifies the uniform cusp property with h(θ) = |θ|α , 0 < α < 1, and whose boundary has Hausdorff dimension exactly equal to N − α.). For that purpose, we need a generalization of the Cantor set. Denote by C1 the Cantor set. Recall that each x, 0 ≤ x ≤ 1, can be written uniquely (if we make a certain convention) as ∞ aj (3, x) x= , 3j j=1 where aj (3, x) can be regarded as the jth digit of x written in basis 3. From this the Cantor set is characterized as follows: x ∈ C1 ⇐⇒ ∀j, aj (3, x) = 1. Similarly for an arbitrary integer k ≥ 1, each x ∈ [0, 1] can be uniquely written in the form ∞ aj (2k + 1, x) x= (2k + 1)j j=1 and we can define the set Ck as x ∈ Ck ⇐⇒ ∀j, aj (2k + 1, x) = k. In a certain sense, if k1 > k2 , Ck1 contains more points than Ck2 . We now use these sets to construct the family of set Dk as follows: x ∈ D1 ⇐⇒ 2x ∈ C1 and for k > 1, x ∈ Dk ⇐⇒ 2k+1 (x − 2k ) ∈ Ck . 2 Note that if k1 = k2 , Dk1 Dk2 = ∅ since the Dk ’s contain only points from the def interval [1 − 2k−1 , 1 − 2k ]. Consider now the set D = ∪∞ k=1 Dk and go back to def

Example 6.1 with the function f replaced by the function f (x) = dD (x)α . Again it can be shown that for all x, y ∈ [0, 1], |dD (y)α − dD (x)α | ≤ |y − x|α . Note that on the interval [1 − 2k−1 , 1 − 2k ] we have dD (x)α = dDk (x)α . Denote by Γ the piece of boundary ∂Ω specified by the function f = dD and by Γk the part of boundary ∂Ω specified by the function f = dD (= dDk ) on the interval [1 − 2k−1 , 1 − 2k ]. Once again on Γ the uniform cusp property is verified

6. Sets Locally Described by a Geometric Property

121

with ρ = 1/6, λ = (1/6)α , and h(θ) = θα . Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓk (ε) of hypercubes of dimension N and side ε required to cover Γk . The construction of the set Ck is also done sequentially by deleting intervals. At step j = 0 the interval ]k/(2k + 1), (k + 1)/(2k + 1)[ of width (2k + 1)−1 is removed. At step j a total of 2j intervals of width (2j + 1)−(j+1) are removed. If we consider the intervals that remain at step j, a total of 2j+1 nonempty disjoint intervals of width (k/(2k + 1))j+1 remain in the set Ck . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k + 1) created at step j + 1. If we construct the set Dk in the same way, at step j a total of 2j nonempty disjoint intervals of width (k/(2k + 1))j+1 1/2k remain in the set Dk . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k (2k + 1)). Pick  j+1 1 k ε= k 2 2k + 1 and look for a lower bound on the number of squares of side ε necessary to cover Γk . For this purpose, consider only the 2j+1 nonempty disjoint intervals remaining at step j. As they each contain a gap of length j+1  1 k 2k + 1 2k (2k + 1) vertically it takes 1α  0 j+1 j+1  k 1 2k + 1 k 2 2k + 1 2k+1 (2k + 1) k 1α  0 j+1 j+1 1 2k + 1 k k 2 −1 ≥ 2k + 1 2k+1 (2k + 1) k ε-cubes. Then we have for β ≥ 0 00 1α  1 j+1 j+1 k 1 2k + 1 j+1 k NΩ (ε) ≥ NΓ (ε) ≥ 2 2 −1 2k + 1 2k+1 (2k + 1) k  j+1   2(2k + 1)kα 2k ≥ − 2j+1 k(2k + 1)α 2α(k+1) (2k + 1)α   j+1 2k(1−α)−α  − 2j+1 = 2(2k + 1)1−α k α−1 (2k + 1)α and hence ε1+β NΩ (ε) 0  j+1 11+β  k(1−α)−α     1 k 1−α α−1 j+1 2 j+1 ≥ k 2(2k + 1) − 2 2k 2k + 1 (2k + 1)α 0  0 α+β 1j+1 j+1 11+β 2k(1−α)−α 1 k k j+1 2 −2 . ≥ 2k + 1 2k 2k + 1 2k(1+β) (2k + 1)α

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Chapter 2. Classical Descriptions of Geometries and Their Properties

The second term goes to zero as j goes to infinity. The first term goes to infinity as j goes to infinity if  ∀k,

k 2k + 1

α+β 2 > 1,

that is ∀k,

0<α+β <

log 2 . log ((2k + 1)/k)

As k can be chosen arbitrarily large, the former inequality reduces to 0 < α + β < 1. Under this condition there exists an integer k for which H1+β (∂Ω) = H1+β (Γk ) = +∞ for all 0 < α < 1 and all 0 ≤ β < 1 − α. Therefore, given 0 < α < 1 ∀β, 0 ≤ β < 1 − α,

H1+β (∂Ω) = +∞.

This implies that the Hausdorff dimension of ∂Ω is greater than or equal to 2 − α, which is the upper bound we obtained in Theorem 6.10.

Chapter 3

Courant Metrics on Images of a Set 1

Introduction

A natural way to construct a family of variable domains is to consider the images of a fixed subset of RN by some family of transformations of RN . The structure and the topology of the images can be specified via the natural algebraic and topological structures of the space of transformations or equivalence classes of transformations for which the full power of function analytic methods is available. There are many ways to do that, and specific constructions and choices are again very much problem dependent. In 1972 A. M. Micheletti [1] introduced what may be one of the first complete metric topologies on a family of domains of class C k , k ≥ 1, that are the images of a fixed open domain through a family of C k -diffeomorphisms of RN . There the natural underlying algebraic structure is the group structure of the composition of transformations with the identity transformation as the neutral element. Her analysis culminates with the construction of a complete metric on the quotient of the group by an appropriate closed subgroup of transformations leaving the fixed subset unaltered. She called it the Courant metric 1 because it is proved in the book of R. Courant and D. Hilbert [1, p. 420] that the nth eigenvalue of the Laplace operator depends continuously on the domain Ω, where Ω = (I + f )Ω0 is the image of a fixed domain Ω0 by I + f and f is a smooth mapping. But there is no notion of a metric in that book. Her constructions naturally extend to other families of transformations of RN or of fixed holdalls D. In section 2 we first extend her generic constructions associated with the space C0k (RN , RN ) of mappings from RN into RN to a larger family of Banach spaces of mappings such as C k (RN , RN ), C k,1 (RN , RN ), or B k (RN , RN ), and beyond to Fr´echet spaces such as B(RN , RN ) or C0∞ (RN , RN ) of infinitely continuously 1 “Nello studiare la continuit` a dell’n-esimo autovalore dell’operatore di Laplace - ∆Ω relativo ad un aperto limitato Ω con dati di Dirichlet nulli, considerato comme funzione dell’aperto Ω, Courant introduce una nozione di vicinanza tra due domini basata su un diffeomorphismo del tipo I + ψ con ψ ∈ C 1 (Rm ), che transforma l’uno nell’altro.” (cf. A. M. Micheletti [1]).

123

124

Chapter 3. Courant Metrics on Images of a Set

differentiables mappings. We emphasize the geodesic character of the construction of the metric and its interpretation as trajectories of bounded variation on the group. The next step in the construction is the choice of the closed subgroup of transformations of RN that is very much problem dependent. Originally, it was chosen as the set of transformations that leave the underlying set or pattern unaltered. However, in some applications, it could be unaltered up to a translation, a rotation, or a flip. The underlying set or pattern can be a closed set or an open crack-free set.2 This includes closed submanifolds of RN . It is shown that, as long as the subgroup is closed, we get a complete Courant metric on the quotient group. In this section we also characterize the tangent space to the group of transformations of RN that leads to the Courant metric. It is an example of an infinite-dimensional manifold (cf. Theorems 2.14 and 2.17 in sections 2.5 and 2.6). In section 3, we free the constructions from the framework of bounded continuously differentiable transformations to reach the spaces of all homeomorphisms or C k -diffeomorphisms of RN or an open subset D of RN . Again it is shown that they are complete metric spaces. Hence, from section 2, their quotient by a closed subgroup yields a Courant metric and a complete metric topology. With such larger spaces, it now becomes possible to consider subgroups involving not only translations but also isometries, symmetries, or flips in RN or D.

2

Generic Constructions of Micheletti

The original constructions of A. M. Micheletti [1] were carried out for the Banach space C0k (RN , RN ), k ≥ 0.3 In this section, we generalize them to Banach spaces Θ of mappings from RN into RN under fairly general assumptions.

Space F (Θ) of Transformations of RN

2.1

Associate with a real vector space Θ of mappings from RN to RN the following space of transformations of RN :  def

F(Θ) = I + f : f ∈ Θ, (I + f ) bijective and (I + f )−1 − I ∈ Θ ,

(2.1)

def

where x → I(x) = x : RN → RN is the identity mapping.4 It will be shown that def

F(Θ) is a group for the composition (F ◦ G)(x) = F (G(x)) of transformations of RN . Equivalent right-invariant metrics on F(Θ) will be introduced under Assumptions 2.1 and 2.2 and related to some notion of geodesics in the group F(Θ). The completeness of F(Θ) will require Assumption 2.3 and either 2 Cf.

Definition 7.1 in Chapter 8. space of k times continuously differentiable functions from RN to at infinity. It is also denoted by B0k (RN , RN ) in the literature. 4 For Θ = C k (RN , RN ), this definition is equivalent to the one of A. M. Micheletti [1]: 0

3 C k (RN , RN ) denotes the 0 RN , whose derivatives vanish

def

F (C0k (RN , RN )) =





F : RN → RN : F − I ∈ C0k (RN , RN ) and F −1 ∈ C k (RN , RN ) .

2. Generic Constructions of Micheletti

125

(i) the assumption that Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous that makes the group a complete metric space, or (ii) Assumption 2.4 that makes the group a complete (metric) topological group. Finally, the right-invariant Courant metric will be defined as the quotient metric ∀F, G ∈ F(Θ),

def

dG ([F ], [H]) =

˜ inf d(F ◦ G, H ◦ G),

˜ G,G∈G

def

[F ] = F ◦ G

(2.2)

associated with the quotient group F(Θ)/G of F(Θ) by a closed subgroup G of F(Θ). The subgroup considered by A. M. Micheletti [1] was def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 } to quotient out all F whose image of the fixed subset Ω0 of RN is Ω0 and make the quotient group F(Θ)/G(Ω0 ) isomorphic to the set of images def

X (Ω0 ) = {F (Ω0 ) : ∀F ∈ F(Θ)}

(2.3)

of Ω0 by the elements of F(Θ). The abstract results will be applied with Θ equal to Banach spaces such as C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, and, through special constructions, to the Fr´echet spaces C0∞ (RN , RN ) ⊂ B(RN , RN ) = ∩k≥0 Bk (RN , RN ). We proceed by introducing the assumptions step by step. Assumption 2.1. Θ is a real vector space of mappings from RN into RN and ∀g ∈ Θ, ∀ I + f ∈ F(Θ),

g ◦ (I + f ) ∈ Θ.

Theorem 2.1. Let Θ be a real vector space of mappings from RN into RN . F(Θ) is a group for the composition ◦ if and only if Assumption 2.1 is verified. In particular (I + f )−1 − I = −f ◦ (I + f )−1 ∈ Θ. Example 2.1. (1) Θ = B0 (RN , RN ), space of bounded continuous functions. For g ∈ B0 (RN , RN ) and I + f ∈ F(B 0 (RN , RN )), the composition g ◦ (I + f ) is continuous and g ◦ (I + f )C 0 = gC 0 < ∞. (2) Θ = C 0 (RN , RN ), space of bounded uniformly continuous functions. For g ∈ C 0 (RN , RN ) and I + f ∈ F(C 0 (RN , RN )), g ◦ (I + f )C 0 = gC 0 < ∞, and since g and I + f are uniformly continuous, so is the composition g ◦ (I + f ). (3) Θ = C00 (RN , RN ), subspace of C 0 (RN , RN ). From (2) g ◦ (I + f )C 0 = gC 0 < ∞ and g ◦ (I + f ) ∈ C 0 (RN , RN ). It remains to show that g ◦ (I + f ) ∈ C00 (RN , RN ). Since g ∈ C00 (RN , RN ), for all ε > 0, there exists ρ0 > 0 such that for all x such that |x| > ρ0 , |g(x)| < ε. But f also belongs to C00 (RN , RN ) and

126

Chapter 3. Courant Metrics on Images of a Set

there exists ρ ≥ 2ρ0 such that for all x such that |x| > ρ, |f (x)| < ρ0 . In particular, |x + f (x)| ≥ |x| − |f (x)| > ρ − ρ0 ≥ ρ0 and hence |g(x + f (x))| < ε. In conclusion for all ε > 0, there exists ρ > 0 such that for all x such that |x| > ρ, |g(x + f (x))| < ε and g ◦ (I + f ) ∈ C00 (RN , RN ). (4) Θ = C 0,1 (RN , RN ). This is the space of bounded Lipschitz continuous functions on RN that is contained in C 0 (RN , RN ). For f, g ∈ C 0,1 (RN , RN ), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ) (1 + c(g)). Proof of Theorem 2.1. I ∈ F(Θ) for f = 0. If I + f ∈ F(Θ), (I + f )−1 is bijective, (I + f )−1 − I ∈ Θ, (F −1 )−1 = F = I + f , and (F −1 )−1 − I = f ∈ Θ. Hence (I + f )−1 ∈ F(Θ) and I = (I + f ) ◦ (I + f )−1 = (I + f )−1 + f ◦ (I + f )−1 ⇒ (I + f )−1 − I = −f ◦ (I + f )−1 . The last property to check is the composition (I + f ) ◦ (I + g). It is bijective as the composition of two bijective transformations. Moreover, by linearity of Θ and Assumption 2.1 (I + f ) ◦ (I + g) − I = g + f ◦ (I + g) ∈ Θ.

(2.4)

From (2.4) and the invertibility of the composition [(I + f ) ◦ (I + g)]−1 − I = −[g + f ◦ (I + g)] ◦ (I + g)−1 ◦ (I + f )−1 ∈ Θ by using Assumption 2.1 twice. Conversely, if (2.4) is true for all f ∈ Θ and I + g ∈ F(Θ), by linearity of Θ, Assumption 2.1 is verified. Associate with F ∈ F(Θ) the following function of I and F : def

d(I, F ) =

inf

n

F =F1 ◦···◦Fn Fi ∈F (Θ) i=1

Fi − IΘ + Fi−1 − IΘ ,

(2.5)

where the infimum is taken over all finite factorizations of F in F(Θ) of the form F = F1 ◦ · · · ◦ Fn ,

Fi ∈ F(Θ).

In particular d(I, F ) = d(I, F −1 ). Extend this function to all F and G in F(Θ): d(F, G) = d(I, G ◦ F −1 ). def

By definition, d is right-invariant since for all F , G, and H in F(Θ) d(F, G) = d(F ◦ H, G ◦ H).

(2.6)

2. Generic Constructions of Micheletti

127

To show that d is a metric5 on F(Θ) necessitates a second assumption. Assumption 2.2. (Θ,  · ) is a normed vector space of mappings from RN into RN and for each r > 0 there exists a continuous function c0 (r) such that ∀{fi }ni=1 ⊂ Θ such that

n

fi  < α < r;

(2.7)

(I + f1 ) ◦ · · · ◦ (I + fn ) − I < α c0 (r).

(2.8)

i=1

then

Theorem 2.2. Under Assumptions 2.1 and 2.2, d is a right-invariant metric on F(Θ). Example 2.2. (1), (2), (3) Since the norm on the three spaces B0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is sufficient to check it for B 0 (RN , RN ). Consider {fi }ni=1 ⊂ B0 (RN , RN ) such that n fi C 0 < α < r. (2.9) i=1

Using the convention (I + fn ) ◦ (I + fn ) = (I + fn ) in the summation (I + f1 ) ◦ · · · ◦ (I + fn ) − I = (I + fn ) − I +

n−1

(I + fi ) ◦ · · · ◦ (I + fn ) − (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

= fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α < r

⇒ c0 (r) = r.

i=1

(4) Θ = C 0,1 (RN , RN ). Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max {fi C 0 , c(fi )} < α < r.

(2.10)

i=1 5A

function d : X × X → R is said to be a metric on X if (cf. J. Dugundji [1])

(i) d(F.G) ≥ 0, for all F, G, (ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G, (iv) d(F, H) ≤ d(F, G) + d(G, H), for all F, G, H. A metric d on a group (F , ◦) is said to be right-invariant if for all H ∈ F , d(F ◦H, G◦H) = d(F, G).

128

Chapter 3. Courant Metrics on Images of a Set

From the previous example (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) +

n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 ))

i=1

≤ c(fn ) + ≤

 n

n−1

)i+1 c(fi ) e

i=1



)n

c(fi ) e[

i=1

k=1

c(fi )]

c(fk )

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er . Proof of Theorem 2.2. Properties (i) and (iii) are verified by definition. For the triangle inequality (iv), F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a finite factorization of F ◦ H −1 . Consider arbitrary finite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = (I + α1 ) ◦ · · · ◦ (I + αm ), G◦H

−1

= (I + β1 ) ◦ · · · ◦ (I + βn ),

I + αi ∈ F(Θ), I + βj ∈ F(Θ).

This yields a new finite factorization of F ◦ H −1 . By definition of d, d(I, F ◦ H −1 ) ≤

m

αi  + αi ◦ (I + αi )−1  +

i=1

n

βj  + βj ◦ (I + βj )−1 ,

i=j

and by taking infima over each factorization, d(F, H) = d(I, F ◦ H −1 ) ≤ d(I, F ◦ G−1 ) + d(I, G ◦ H −1 ) = d(F, G) + d(G, H) using the symmetry property (iii). To verify (ii) it is sufficient to show that F = I ⇐⇒ d(I, F ) = 0. Clearly I ◦ I is a factorization of F = I and 0 ≤ d(I, F ) ≤ 0. Conversely, if d(I, F ) = 0, for ε, 0 < ε < 1, there exists a finite factorization F = F1 ◦ · · · ◦ Fn such that n

Fi − I + Fi−1 − I < ε < 1.

i=1

By Assumption 2.2, F − I + F −1 − I < ε c0 (1), which implies that F − I = 0 and F −1 − I = 0. This completes the proof.

2. Generic Constructions of Micheletti

129

An important aspect of the previous construction was to build in the triangle inequality to make d a (right-invariant) metric. However, there are other ways to introduce a topology on F(Θ). The interest of this specific choice will become apparent later on. For instance, the original Courant metric of A. M. Micheletti [1] in 1972 was constructed from the following slightly different right-invariant metric on F(Θ): given H ∈ F(Θ), consider finite factorizations H1 ◦ · · · ◦ Hm , Hi ∈ F(Θ), 1 ≤ i ≤ m, of H and finite factorizations G1 ◦ · · · ◦ Gn , Gj ∈ F(Θ), 1 ≤ j ≤ n, of H −1 , and define m n def d1 (I, H) = inf Hi − I + inf Gi − I, (2.11) H=H1 ◦···◦Hm i=1 m≥1

H −1 =G1 ◦···◦Gn i=1 n≥1

where the infima are taken over all finite factorizations of H and H −1 in F(Θ) and ∀F, G ∈ F(Θ),

d1 (G, F ) = d1 (I, F ◦ G−1 ). def

(2.12)

Another example is the topology generated by the right-invariant semimetric 6 ∀H ∈ F(Θ),

d0 (I, H) = H − I + H −1 − I, def

∀F, G ∈ F(Θ),

d0 (G, F ) = d0 (I, F ◦ G−1 ). def

(2.13)

Given r > 0 and F ∈ F(Θ), let Br (F ) = {G ∈ F(Θ) : d0 (F, G) < r}. Let τ0 be the weakest topology on F(Θ) such that the family {Br (F ) : 0 ≤ r < ∞} is the base for τ0 . By definition d1 (F, G) ≤ d(F, G) ≤ d0 (F, G) and the injections (F, d1 ) → (F, d) → (F, d0 ) are continuous. Theorem 2.3. Under Assumptions 2.1 and 2.2, the topologies generated by d1 , d, and d0 are equivalent. Proof. Since d0 , d, and d1 are right-invariant on the group F(Θ), it is sufficient to prove the equivalence for d0 and d1 around I. For all ε > 0 with δ = ε ∀H ∈ F(Θ), d0 (I, H) < δ,

d1 (I, H) < ε,

since d1 (I, H) ≤ d0 (I, H). Conversely, by Assumption 2.2, for any δ, 0 < δ < 1, d1 (I, H) < δ < 1

⇒ d0 (I − H) = H − I + H −1 − I < 2δ c(1).

So for any ε > 0, pick δ = min{1, ε/c(1)}/2. Hence ∀ε > 0, ∃δ > 0,

∀H ∈ F(Θ), d1 (I, H) < δ,

d0 (I, H) < ε.

This concludes the proof of the equivalences. 6 Given a space X, a function d : X × X → R is said to be a semimetric if (i) d(F.G) ≥ 0, for all F, G,

(ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G. This notion goes back to Fr´echet and Menger.

130

Chapter 3. Courant Metrics on Images of a Set F. Murat and J. Simon [1] in 1976 considered as candidates for Θ the spaces W k+1,c (RN , RN )   def = f ∈ W k,∞ (RN , RN ) : ∀ 0 ≤ |α| ≤ k + 1, ∂ α f ∈ C(RN , RN )

and W k+1,∞ (RN , RN ), k ≥ 0. The first space W k+1,c (RN , RN ) is equivalent to the space C k+1 (RN , RN ), k ≥ 0, algebraically and topologically. The second space W k+1,∞ (RN , RN ) coincides with C k,1 (RN , RN ), k ≥ 0. For the spaces C k+1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, they obtained the following pseudo triangle inequality for the semimetric d0 : d0 (F1 ,F3 ) ≤d0 (F1 , F2 ) + d0 (F2 , F3 ) + d0 (F1 , F2 ) d0 (F2 , F3 ) P [d0 (F1 , F2 ) + d0 (F2 , F3 )] ,

(2.14) ∀F1 , F2 , F3 ,

where P : R+ → R+ is a continuous increasing function. With that additional property, they called d0 a pseudodistance 7 and showed that they could construct a metric from it. Lemma 2.1. Assume that the semimetric d0 satisfies the pseudo triangle inequality (2.14). Then, for all α, 0 < α < 1, there exists a constant ηα > 0 such that the (α) function d0 : F(Θ) × F(Θ) → R+ defined as def

(α)

d0 (F1 , F2 ) = inf{d0 (F1 , F2 ), ηα }α is a metric on E. The Micheletti construction with the metric d1 was given in 2001 for the spaces C k (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, in M. C. Delfour and J.-P. Zo´sio [37] so that pseudodistances can be completely bypassed. The metrics d and le d1 constructed from the semimetric d0 under the general Assumptions 2.1 and 2.2 are both more general and interesting since they are of geodesic type by the use of infima over finite factorizations (I + θ1 ) ◦ · · · ◦ (I + θn ) of I + θ when the semimetric d0 is interpreted as an energy n

d0 (I + θi , I) =

i=1

n

θi  +  − θi ◦ (I + θi )−1 

(2.15)

i=1

7 This terminology is not standard. In the literature a pseudometric or pseudodistance is usually reserved for a function d : X × X → R satisfying

(i) d(F.G) ≥ 0,

for all F, G,

(ii’) d(F, G) = 0 ⇐= F = G, (iii) d(F, G) = d(G, F ),

for all F, G.

(iv) d(F, H) ≤ d(F, G) + d(G, H),

for all F, G, H.

Condition (ii) for a metric is weakened to condition (ii’) for a pseudodistance.

2. Generic Constructions of Micheletti

131

over families of piecewise constant trajectories T : [0, 1] → F(Θ) starting from I at time 0 to I + θ at time 1 in the group F(Θ) by assigning times ti , 0 = t0 < t1 < · · · < tn < tn+1 = 1, at each jump:  I, 0 ≤ t < t1 ,      (I + θ1 ), t 1 < t < t2 ,      t 2 < t < t3 ,   (I + θ2 ) ◦ (I + θ1 ), def ... T (t) = (2.16)    (I + θi ) ◦ · · · ◦ (I + θ1 ), ti < t < ti+1 ,      ...     (I + θn ) ◦ · · · ◦ (I + θ1 ), tn < t ≤ 1. The jump in the group F(Θ) at time ti is given by − −1 [T (t)]ti = T (t+ −I i ) ◦ T (ti ) def

= [(I + θi ) ◦ · · · ◦ (I + θ1 )] ◦ [(I + θi−1 ) ◦ · · · ◦ (I + θ1 )]−1 − I = θi as an element of Θ that could be viewed as the tangent space to F(Θ) ⊂ I + Θ. The choice of d in a geodesic context is more interesting than the choice of d1 that would involve a double infima in the energy (2.15). At this stage, in view of the equivalence Theorem 2.3, working with the metric d or d1 is completely equivalent. Assumption 2.3. (Θ,  · ) is a normed vector space of mappings from RN into RN and there exists a continuous function c1 such that for all I + f ∈ F(Θ) and g ∈ Θ g ◦ (I + f ) ≤ g c1 (f ).

(2.17)

The next assumption is a kind of uniform continuity. Assumption 2.4. (Θ,  · ) is a normed vector space of mappings from RN into RN . For each g ∈ Θ, for all ε > 0, there exists δ > 0 such that ∀γ ∈ Θ such that γ < δ,

g ◦ (I + γ) − g < ε.

(2.18)

Theorem 2.4. Under Assumptions 2.1 to 2.4, F(Θ) is a topological (metric) group. Example 2.3 (Checking Assumption 2.3). (1), (2), (3) Since the norm on the three spaces B 0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is sufficient to check it for B 0 (RN , RN ). Consider g ∈ B0 (RN , RN ) and I +f ∈ F(B 0 (RN , RN )). Since I + f is a bijection g ◦ (I + f )C 0 ≤ gC 0 and the constant function c1 (r) = 1 satisfies Assumption 2.3.

(2.19)

132

Chapter 3. Courant Metrics on Images of a Set

(4) Θ = C 0,1 (RN , RN ). Consider g ∈ C 0,1 (RN , RN ) and I + f ∈ F(C 0,1 (RN , RN )). From (1) g ◦ (I + f )C 0 ≤ gC 0 and since all the functions are Lipschitz c(g ◦ (I + f )) ≤ c(g) (1 + c(f )) ⇒ g ◦ (I + f )C 0,1 ≤ gC 0,1 (1 + c(f )) ≤ gC 0,1 (1 + f C 0,1 ) and the continuous function c1 (r) = 1 + r satisfies Assumption 2.3. Assumption 2.4 is a little trickier. It involves not only norms but also a “uniform continuity” of the function g that is not verified for g ∈ B0 (RN , RN ) and for the Lipschitz part c(g) of the norm of C 0,1 (RN , RN ). Example 2.4 (Checking Assumption 2.4). (1) Θ = B0 (RN , RN ). In that space there exists some g that is not uniformly continuous. For that g there exists ε > 0 such that for all n > 0, there exists xn , yn , |yn − xn | < 1/n such that |g(yn ) − g(xn )| ≥ ε. Define the function γn (x) = yn − xn . Therefore, there exists ε > 0 such that for all n with γn C 0 < 1/n, g ◦ (I + γn ) − gC 0 ≥ |g(xn + γn (xn )) − g(xn )| = |g(xn + yn − xn ) − g(xn )| ≥ ε and Assumption 2.4 is not verified. However, B0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : B 0 (RN , RN ) → R is continuous. (2) and (3) Θ = C 0 (RN , RN ). By the uniform continuity of g ∈ C 0 (RN , RN ), for all ε > 0 there exists δ > 0 such that ∀x, y ∈ RN ,

|g(y) − g(x)| < ε/2.

For γC 0 < δ ∀x ∈ RN , ∀x ∈ RN ,

|(I + γ)(x) − x| = |γ(x)| ≤ γC 0 < δ,

|g(x + γ(x)) − g(x)| < ε/2

⇒ g ◦ (I + γ) − gC 0 ≤ ε/2 < ε

and Assumption 2.4 is verified. In addition, C 0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0 (RN , RN ) → R is continuous. (4) Θ = C 0,1 (RN , RN ). Since g ∈ C 0,1 (RN , RN ) ∀x, y ∈ RN ,

|g(y) − g(x)| < c(g) |y − x|.

Therefore for def

γC 0,1 = max{γC 0 , c(γ)} < δ, γC 0 < δ and we get g ◦ (I + γ) − gC 0 ≤ c(g) γC 0 < c(g) δ.

2. Generic Constructions of Micheletti

133

Unfortunately, for the part c(g ◦ (I + γ)) of the C 0,1 -norm, for all x = y ∈ RN |g(y + γ(y)) − g(y) − (g(x + γ(x)) − g(x))| ≤ c(g) + c(g) c(I + γ), |y − x| c(g ◦ (I + γ) − g) ≤ c(g) [1 + c(γ)] < c(g) [1 + δ] and we cannot satisfy Assumption 2.4. However, C 0,1 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0,1 (RN , RN ) → R is continuous. Remark 2.1. Since the spaces C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) have the same norm and Assumptions 2.2 and 2.3 involve only the norms, they will always be verified for C0k (RN , RN ) and C k (RN , RN ) if they are verified for B k (RN , RN ), k ≥ 0. Solely Assumption 2.1 will require special attention. The fact that Assumption 2.4 is not always verified will not prevent us from proving that the space (F(Θ), d) is complete. The completeness will be proved in Theorem 2.6 under Assumptions 2.1 to 2.3, where Assumption 2.4 will be replaced by the condition Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. The following theorem will be proved in Theorems 2.11, 2.12, and 2.13 of section 2.5 and Theorems 2.15 and 2.16 of section 2.6. Theorem 2.5. Let k ≥ 0 be an integer. (i) C0k (RN , RN ) and C k (RN , RN ) satisfy Assumptions 2.1 to 2.4, C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : C k (RN , RN ) → RN is continuous. (ii) B k (RN , RN ) and C k,1 (RN , RN ) verify Assumptions 2.1 to 2.3, C k,1 (RN , RN ) ⊂ B k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : B k (RN , RN ) → RN is continuous. Proof of Theorem 2.4. From J. L. Kelley [1], it is sufficient to prove that ∀F ∈ F(Θ), ∀H ∈ F(Θ) such that d(I, H) → 0,

d(I, F −1 ◦ H ◦ F ) → 0.

Rewrite F −1 ◦ H ◦ F as follows: F −1 ◦ H ◦ F − I = F −1 ◦ [I + H − I] ◦ F − F −1 ◦ F 

= F −1 ◦ (I + γ) − F −1 ◦ F

 = (F −1 − I) ◦ (I + γ) − (F −1 − I) ◦ F + γ ◦ F, γ = H − I and g = (F −1 − I) ◦ (I + γ) − (F −1 − I). def

def

By Assumption 2.3, γ ◦ F  ≤ γ c1 (F − I),

g ◦ F  ≤ g c1 (F − I);

134

Chapter 3. Courant Metrics on Images of a Set

by Assumption 2.4, for all ε > 0, there exists δ, 0 < δ < min{1, ε/(4c0 (1) c1 (F − I))}, such that γ = H − I < δ c0 (1)

⇒

g =(F −1 − I) ◦ (I + γ) − (F −1 − I) < ε/(4 c1 (F − I)).

Combining the above properties ∀H such that d(I, H) < δ, ⇒ F

−1

H − I < δ c0 (1)

◦ H ◦ F − I ≤ γ c1 (F − I) + ε/2 ≤ δ c0 (1) c1 (F − I) + ε/4 ≤ ε/2.

By Assumption 2.2, for all H such that d(I, H) < δ < 1, we have H − I < δ c0 (1) and H −1 − I < δ c0 (1). Finally, for all ε > 0, there exists δ > 0 such that d(I, H) < δ implies F −1 ◦ H ◦ F − I ≤ ε/2. By a similar argument, for all ε > 0, there exists δ  > 0 such that d(I, H) < δ  implies F −1 ◦ H −1 ◦ F − I ≤ ε/2. Therefore, for all ε > 0 there exists δ0 = min{δ, δ  } > 0 such that for all H, d(I, H) < δ0 , d(I, F −1 ◦ H ◦ F ) ≤ F −1 ◦ H ◦ F − I + F −1 ◦ H −1 ◦ F − I < ε and F(Θ) is a topological group for the metric d. Theorem 2.6. (i) Let Assumptions 2.1 to 2.3 be verified for a Banach space Θ contained in C 0 (RN , RN ) such that, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. Then (F(Θ), d) is a complete metric space. (ii) Let Assumptions 2.1 to 2.4 be verified for a Banach space Θ. Then F(Θ) is a complete topological (metric) group. Remark 2.2. F(C 0,1 (RN , RN )) is a complete metric space since the assumptions of Theorem 2.6 are verified, but it is not a topological group since Assumption 2.4 is not verified. Proof of Theorem 2.6. (i) Let {Fn } be a Cauchy sequence in F(Θ). (a) Boundedness of {Fn − I} and {Fn−1 − I} in Θ. For each ε, 0 < ε < 1, there exists N such that, for all m, n > N , d(Fm , Fn ) < ε. By the triangle inequality |d(I, Fm ) − d(I, Fn )| ≤ d(Fm , Fn ) < ε, {d(I, Fn } is Cauchy, and, a posteriori, is bounded by some constant L. For each n, n ) such that there exists a factorization Fn = (I + f1n ) ◦ · · · ◦ (I + fν(n)

ν(n)

0≤

fin  + fin ◦ (I + f1n )−1  − d(I, Fn ) < ε < 1

i=1



ν(n)

⇒

i=1

fin  + fin ◦ (I + fin )−1  < L + ε < L + 1.

2. Generic Constructions of Micheletti

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By Assumption 2.2 Fn − I + Fn−1 − I < 2(L + ε) c0 (L + 1) and the sequences {Fn − I} and {Fn−1 − I} are bounded in Θ. (b) Convergence of {Fn − I} and {Fn−1 − I} in Θ. For any m, n −1 Fn − Fm = (Fn ◦ Fm − I) ◦ Fm

and by Assumption 2.3 −1 −1 Fm − Fn  = (Fn ◦ Fm − I) ◦ Fm  ≤ Fn ◦ Fm − I c1 (Fm − I) −1 ⇒ Fm − Fn  ≤ c+ 1 Fn ◦ Fm − I,

def

c+ 1 = sup c1 (Fm − I) < ∞ m

since {Fm − I} is bounded. For all ε, 0 < ε < c0 (1) c+ 1 , there exists N such that ∀n, m, > N,

−1 d(I, Fn ◦ Fm ) = d(Fm , Fn ) < ε/(c0 (1) c+ 1 ) < 1.

By Assumption 2.2 + −1 Fn ◦ Fm − I < c0 (1)ε/(c0 (1) c+ 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − I − (Fn − I) ≤ c+ 1 Fn ◦ Fm − I < ε

and {Fn − I} is a Cauchy sequence in Θ that converges to some F − I ∈ Θ since Θ is a Banach space. Similarly, we get that {Fn−1 − I} is a Cauchy sequence in Θ that converges to some G − I ∈ Θ. (c) F is bijective, G = F −1 , and F ∈ F(Θ). For all n, Fn , Fn−1 , F , and G are continuous since Θ ⊂ C 0 (RN , RN ). By continuity of f → f (x) : Θ → RN , f → x + f (x) : Θ → RN is continuous and [Fn−1 (x) − x| + x → [G(x) − x] + x. By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F − I, F −1 − I ∈ Θ, and F ∈ F(Θ). (d) Convergence of Fn → F in F(Θ). To check that Fn → F in F(Θ) recall that by Assumption 2.3 d(Fn , F ) = d(I, F ◦ Fn−1 ) ≤I − F ◦ Fn−1  + I − Fn ◦ F −1  = (Fn − F ) ◦ Fn−1  + (F − Fn ) ◦ F −1  ≤ c1 (Fn−1 − I)Fn − F  + c1 (F −1 − I)F − Fn    ≤ max c1 (Fn−1 − I) + c1 (F −1 − I) F − Fn , n

which goes to zero as Fn − I goes to F − I in Θ. (ii) The proof (ii) differs from the proof of (i) only in part (c). Consider G ◦ F − I = G ◦ F − G ◦ Fn + (G − Fn−1 ) ◦ Fn .

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Since {Fn − I} is bounded in Θ and Fn−1 → G, the term (G − Fn−1 ) ◦ Fn goes to 0 as n → ∞ by Assumption 2.3. The second terms can be rewritten as G ◦ F − G ◦ Fn = F − Fn + (G − I) ◦ F − (G − I) ◦ Fn = F − Fn + g ◦ F − g ◦ Fn , where g = G − I ∈ Θ. The term F − Fn goes to zero. As for the second terms

 g ◦ F − g ◦ Fn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g ◦ Fn . Let gn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g and let γn = (F − Fn ) ◦ Fn−1 . By boundedness of {Fn − I} and {Fn−1 − I} in Θ and Assumption 2.3, γn  = (F − Fn ) ◦ Fn−1  ≤ F − Fn  c1 (Fn−1 − I) ≤ c+ 1 F − Fn , gn ◦ Fn  ≤ gn  c1 (Fn − I) ≤ c+ 1 gn , −1 c+ 1 = sup{c1 (Fn − I), c1 (Fn − I)} < ∞. def

n

By Assumption 2.4, for all ε > 0 there exists δ > 0 such that γn  < δ c+ 1

⇒ gn  = g ◦ (I + γn ) − g < ε/c+ 1.

By combining for all ε > 0, there exists δ > 0 such that F − Fn  < δ

+ ⇒ γn  ≤ c+ 1 F − Fn  < c1 δ + ⇒ gn ◦ Fn  < c+ 1 ε/c1 = ε ⇒ F − Fn  < δ ⇒ g ◦ F − g ◦ Fn  < ε.

As a result G ◦ F − I is bounded by an expression that goes to zero as n → ∞ and G ◦ F = I. By an analogous argument, we can prove that F ◦ G − I = 0, that is, F −1 = G. So F is bijective, F − I, F −1 − I ∈ Θ. Hence F ∈ F(Θ).

2.2

Diffeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

Consider as other candidates for Θ the spaces k N N ∞ k N N B(RN , RN ) = ∩∞ k=0 B (R , R ) = ∩k=0 C (R , R ),

(2.20)

k N N C0∞ (RN , RN ) = ∩∞ k=0 C0 (R , R )

(2.21)

def

def

and the resulting groups F(B(RN , RN )) 

I + f : f ∈ B(RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ B(RN , RN ) and F(C0∞ (RN , RN )) 

I + f : f ∈ C0∞ (RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ C0∞ (RN , RN )

2. Generic Constructions of Micheletti

137

that satisfy Assumption 2.1 but not the other assumptions since B(RN , RN ) and C0∞ (RN , RN ) are Fr´echet but not Banach spaces. To get around this, observe that k N N ∞ k N N F(B(RN , RN )) = ∩∞ k=0 F(B (R , R )) = ∩k=0 F(C (R , R )), k N N F(C0∞ (RN , RN )) = ∩∞ k=0 F(C0 (R , R )),

where (F(C k (RN , RN )), dk ) and (F(C0k (RN , RN )), dk ) are topological (metric) groups with the same metrics {dk } and the following monotony property: dk (F, G) ≤ dk+1 (F, G) resulting from the monotony of the norms f C k = max f C i ≤ 0≤i≤k

max f C i = f C k+1 .

0≤i≤k+1

Theorem 2.7. The spaces F(B(RN , RN )) and F(C0∞ (RN , RN )) are complete topological (metric) groups for the distance def

d∞ (F, G) =

∞ 1 dk (F, G) , 2k 1 + dk (F, G)

(2.22)

k=0

where dk is the metric associated with C k (RN , RN ) and C0k (RN , RN ).8 This theorem is a consequence of the following general lemma. Lemma 2.2. Assume that (Fk , dk ), k ≥ 0, is a family of groups each complete with respect to their metric dk and that ∀k ≥ 0,

Fk+1 ⊂ Fk

and

dk (F, G) ≤ dk+1 (F, G), ∀F, G ∈ ∩∞ k=0 Fk .

(2.23)

Then the intersection F∞ = ∩∞ k=0 Fk is a group that is complete with respect to the metric d∞ defined in (2.22). If, in addition, each (Fk , dk ) is a topological (metric) group, then F∞ is a topological (metric) group. Proof. By definition, F∞ is clearly a group. (i) d∞ is a metric on F∞ . d∞ (F, G) = 0 implies that dk (F, G) = 0 for all k and F = G and conversely. Also d∞ (F, G) = d∞ (G, F ). The function x/(1 + x) is monotone strictly increasing with x ≥ 0. Therefore, dk (F, H) ≤ dk (F, G)+dk (G, H) implies dk (F, H) dk (F, G) dk (G, H) ≤ + 1 + dk (F, H) 1 + dk (F, G) + dk (G, H) 1 + dk (F, G) + dk (G, H) dk (G, H) dk (F, G) + , ≤ 1 + dk (F, G) 1 + dk (G, H) ∞ ∞ ∞ 1 dk (F, H) 1 dk (F, G) 1 dk (G, H) ≤ + 2k 1 + dk (F, H) 2k 1 + dk (F, G) 2k 1 + dk (G, H) k=0

k=0

k=0

and d∞ (F, H) ≤ d∞ (F, G) + d∞ (G, H). 8 In view of (2.20), it is important to note that F (B(RN , RN )) is a topological group by using the family (F (C k (RN , RN )), dk ) in Lemma 2.2 and not the family (F (B k (RN , RN )), dk ).

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Chapter 3. Courant Metrics on Images of a Set

(ii) (F∞ .d∞ ) is complete. Let {Fn } be a Cauchy sequence in F∞ ; that is, for all ε, 0 < ε < 1/2, there exists N0 such that for n, m > N0 , d∞ (Fn , Fm ) < ε. In particular ∞

1 dk (Fn , Fm ) d0 (Fn , Fm ) ≤ <ε 1 + d0 (Fn , Fm ) 2k 1 + dk (Fn , Fm )

⇒ d0 (Fn , Fm ) < ε/(1 − ε) < 2ε

k=0

and {Fn } is a Cauchy sequence in F0 that converges to some F in F0 . Choose a subsequence of {Fn } such that there exists N1 ≥ N0 such that for n, m > N1 , d∞ (Fn , Fm ) < ε/22 . Then ∞

1 dk (Fn , Fm ) 1 1 d1 (Fn , Fm ) ≤ < 2 ε ⇒ d1 (Fn , Fm ) < ε/(1 − ε) < 2ε 2 1 + d1 (Fn , Fm ) 2k 1 + dk (Fn , Fm ) 2 k=0

and {Fn } is a Cauchy sequence in F1 that converges to some G1 in F1 . By continuous injection of F1 into F0 , G1 = F ∈ F0 ∩ F1 . Proceeding in that way, we get that there exists F ∈ F∞ = ∩∞ k=0 Fk such that Fn → F in Fk for all k ≥ 0. Therefore, F∞ is complete.

2.3

Closed Subgroups G

Given an arbitrary nonempty subset Ω0 of RN , consider the family  def

X (Ω0 ) = F (Ω0 ) ⊂ RN : ∀F ∈ F(Θ)

(2.24)

of images of Ω0 by the elements of F(Θ). By introducing the subgroup def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 } ,

(2.25)

[F ] → F (Ω0 ) : F(Θ)/G(Ω0 ) → X (Ω0 )

(2.26)

we obtain a bijection

between the set of images X (Ω0 ) and the quotient space F(Θ)/G(Ω0 ). Using this bijection, the topological structure of X (Ω0 ) will be identified with the topological structure of the quotient space. Lemma 2.3. Let Assumptions 2.1 to 2.3 hold with Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , let the mapping f → f (x) : Θ → RN be continuous.9 If the nonempty subset Ω0 of RN is closed or if Ω0 = int Ω0 ,10 the family def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.27)

is a closed subgroup of F(Θ). 9 This 10 Such

means that F(Θ) ⊂ I + Θ ⊂ Hom(RN , RN ). a set will be referred to as a crack-free set in Definition 7.1 (ii) of Chapter 8. Indeed,

by definition Ω is crack-free if Ω = Ω. If, in addition, Ω is open, then Ω = Ω = Ω and Ω = Ω = int Ω.

2. Generic Constructions of Micheletti

139

Proof. G(Ω0 ) is clearly a subgroup of F(Θ). It is sufficient to show that for any sequence {Fn } in G(Ω0 ) that converges to F in F(Θ), then F ∈ G(Ω0 ). The other properties are straightforward. If {Fn } converges to F in F(Θ), then by Assumption 2.3 we have Fn ◦ F −1 − I + F ◦ Fn−1 − I → 0. From F ◦ Fn−1 − I → 0, for each x ∈ Ω0 the sequence {F (Fn−1 (x))} ⊂ F (Ω0 ) converges to x ∈ Ω0 and hence Ω0 ⊂ F (Ω0 ). From the convergence Fn ◦F −1 −I → 0 for each x ∈ F (Ω0 ), F −1 (x) ∈ Ω0 and the sequence {Fn (F −1 (x))} ⊂ Ω0 converges to x ∈ F (Ω0 ) and hence F (Ω0 ) ⊂ Ω0 . Therefore, Ω0 ⊂ F (Ω0 ) ⊂ Ω0 and Ω0 = F (Ω0 ). Since F is a homeomorphism, for all A, F (A) = F (A) and F (A) = F (A). As a result int F (A) = F (int A). In view of the identity Ω0 = F (Ω0 ), F (Ω0 ) = F (Ω0 ) = Ω0

and

int Ω0 = int F (Ω0 ) = F (int Ω0 ).

From this, if Ω0 is closed, Ω0 = Ω0 and F (Ω0 ) = Ω0 . If Ω0 = int Ω0 , we also get F (Ω0 ) = Ω0 . In both cases F ∈ G(Ω0 ). Remark 2.3. A. M. Micheletti [1] assumes that Ω0 is a bounded connected open domain of class C 3 in order to make all the images F (Ω0 ) bounded connected open domains of class C 3 with Θ = C03 (RN , RN ). Open domains of class C k , k ≥ 1, are locally C k epigraphs and domains that are locally C 0 epigraphs are crack-free11 by Theorem 5.4 (ii) of Chapter 2. Remark 2.4. F. Murat and J. Simon [1] were mainly interested in families of images of locally Lipschitzian (epigraph) domains. Recall that F(C 1 (RN , RN )) transports locally Lipschitzian (epigraph) domains onto locally Lipschitzian (epigraph) domains (cf. A. Bendali [1] and A. Djadane [1]), but that F(C 0,1 (RN , RN )) does not, as shown in Examples 5.1 and 5.2 of section 5.3 in Chapter 2. Of special interest are the closed submanifolds of RN of codimension greater than or equal to 1. For instance, we can choose Ω0 = S n , 2 ≤ n < N , the n-sphere of radius 1 centered in 0. We get a family of closed curves in R3 for n = 2 and of closed surfaces in R3 for n = 3. Similarly, by choosing Ω0 = B1n (0), the closed unit ball of dimension n, we get a family of curves for n = 1 and a family of surfaces for n = 2. The choice of [0, 1]n yields a family of surfaces with corners for n = 2. Explicit expressions can be obtained for normals, curvatures, and density measures as in Chapter 2. Example 2.5 (Closed surfaces in R3 ). For Ω0 = S 2 in R3 , f ∈ C 1 (R3 , R3 ), and F = I + f , the tangent field at the point ξ ∈ S 2 is Tξ S 2 = {ξ}⊥ and the unit normal is ξ. At a point x = F (ξ) of the image 11 Cf.

Definition 7.1 (ii) of Chapter 8.

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Chapter 3. Courant Metrics on Images of a Set

F (S 2 ), the tangent space and the unit normal are given by TF (ξ) F (S 2 ) = DF (ξ){ξ}⊥ and n(F (x)) =

∗

DF (ξ)−1 ξ . | ∗DF (ξ)−1 ξ|

Example 2.6 (Curves in RN ). Let eN be a unit vector in RN , let H = {eN }⊥ , and let def

Ω0 = {ζN eN : ζN ∈ [0, 1]} ⊂ RN . For f ∈ C 1 (RN , RN ), and F = I + f , the tangent field at the point ξ ∈ Ω0 is Tξ Ω0 = R eN and the normal space (Tξ Ω0 )⊥ is H. At a point x = F (ξ) of the curve F (Ω0 ), the tangent and normal spaces are given by TF (ξ) F (Ω0 ) = R DF (ξ)eN

TF (ξ) F (Ω0 )⊥ = {DF (ξ)eN }⊥ .

and

In practice, one might want to use other subgroups, such as all the translations by some vector a ∈ RN : def

Fa (x) = a + x, ⇒

Fa−1 (x)

x ∈ RN ,

− a + x = F−a (x)

⇒ Fa − I = a ∈ Θ ⇒

Fa−1

− I = −a ∈ Θ

(2.28) (2.29)

for all Θ that contain the constant functions such as Bk (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ), but not C0k (RN , RN ). Therefore  def

Gτ = Fa : ∀a ∈ RN

(2.30)

is a subgroup of F(Θ). In that case, it is also a closed subgroup of F(Θ): Fa ◦ Fb (x) = a + b + x = Fa+b (x),

F0 = I

and (Gτ , ◦) is isomorphic to the one-dimensional closed abelian group (RN , +). This means that all the topologies on Gτ are equivalent and (Gτ , ◦) is closed in F(Θ).

2.4

Courant Metric on the Quotient Group F (Θ)/G

When F(Θ) is a topological right-invariant metric group and G is a closed subgroup of F(Θ), the quotient metric def

dG (F ◦ G, H ◦ G) =

˜ inf d(F ◦ G, H ◦ G)

˜ G,G∈G

(2.31)

induces a complete metric topology on the quotient group F(Θ)/G.12 The quotient metric dG(Ω0 ) with Θ = C0k (RN , RN ) was referred to as the Courant metric by 12 Cf.,

for instance, D. Montgomery and L. Zippin [1 sect. 1.22, p. 34, sect. 1.23, p. 36])

(i) If G is a topological group whose open sets at e have a countable basis, then G is metrizable and, moreover, there exists a metric which is right-invariant. (ii) If G is a closed subgroup of a metric group F, then F/G is metrizable.

2. Generic Constructions of Micheletti

141

A. M. Micheletti [1]. In that case F(Θ) is a topological (metric) group. However, we have seen that for Θ = C 0,1 (RN , RN ) and B0 (RN , RN ), F(Θ) is not a topological group. Yet, the completeness of (F (Θ), d) and the right-invariance of d are sufficient to recover the result for an arbitrary closed subgroup G of F. It is important to have the weakest possible assumptions on G to secure the widest range of applications. Theorem 2.8. Assume that (F, d) is a group with right-invariant metric d that is complete for the topology generated by d. For any closed subgroup G of F, the function dG : F × F → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.32)

is a right-invariant metric on F(Θ)/G and the space (F/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F/G. Proof. (i) F/G is clearly a group with unit element G under the composition law def

[F ◦ G] ◦ [H ◦ G] = [F ◦ H] ◦ G. (ii) (dG is a metric). Since d is right-invariant, the definition (2.31) is equivalent to def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G). G∈G

(2.33)

Since d is symmetrical, so is dG . For the triangle inequality let g1 , g2 ∈ G: dG (F ◦ G, H ◦ G) ≤ d(F, H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G ◦ g2 , H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G, H ◦ g1 ) by right-invariance. The triangle inequality for dG follows by taking the infima over g1 , g2 ∈ G. Since F/G is a group, it remains to show that dG (I ◦ G, H ◦ G) = 0 if and only if H ∈ G. Clearly dG (I ◦G, I ◦G) ≤ d(I, I) = 0. Conversely, let dG (I ◦G, H ◦G) = 0. For all n > 0, there exists gn ∈ G such that d(gn−1 , H) = d(I, H ◦ gn ) < 1/n. Therefore gn−1 → H in F(Θ). But since G is closed limn→∞ gn−1 ∈ G and H ∈ G. In particular, the canonical mapping F → π(F ) = F ◦ G : F(Θ) → F(Θ)/G is continuous since dG (I ◦ G, F ◦ G) ≤ d(I, F ) for all F ∈ F(Θ). (iii) (Completeness). Consider a Cauchy sequence {Fn ◦ G} in F/G. It is sufficient to show that there exists a subsequence which converges to some F ◦ G ∈ F(Θ)/G. Construct the subsequence {Fν ◦ G} such that dG (Fν ◦ G, Fν+1 ◦ G) < 1/(2ν ). Then there exists a sequence {Hν } in F such that (i) Hν ∈ Fν ◦ G and (ii) d(Hν , Hν+1 ) < 1/(2ν ). We proceed by induction. By definition of dG , if dG (F1 ◦ G, F2 ◦G) < 1/2, there exist H1 ∈ F1 ◦G and H2 ∈ F2 ◦G such that d(H1 , H2 ) < 1/2. Similarly, if Hν ∈ Fν ◦ G, it follows that there exists Hν+1 with the required property. Because dG (Fν ◦ G, Fν+1 ◦ G) < 1/2ν , there exist G1 and G2 in G such that d(Fν ◦ G1 , Fν+1 ◦ G2 ) < 1/2ν . If G3 is such that Fν ◦ G1 = Hν ◦ G3 , then we can

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Chapter 3. Courant Metrics on Images of a Set

choose Hν+1 = Fν+1 ◦ G2 ◦ G−1 3 since d(Hν , Fν+1 ◦ G2 ◦ G−1 3 ) = d(Hν ◦ G3 , Fν+1 ◦ G2 ) = d(Fν ◦ G1 , Fν+1 ◦ G2 ). It is easy to check that {Hν } is a Cauchy sequence in F. Indeed ∀i < j,

d(Hi , Hj ) ≤

j−1

d(Hn , Hn+1 ) ≤

n=i

1 1 1 + · · · + j ≤ i−1 . 2i 2 2

By completeness of F, {Hν } converges to some H in F. From (ii) the canonical map π : F → F/G is continuous. So the sequence {π(Hν ) = Fν ◦ G} converges to π(H) = H ◦ G in F/G. R (iv) Associate with G the equivalence relation F ∼ H if F ◦ H −1 ∈ G. By definition, the quotient topology on F/R is the finest topology T on F/R for which the canonical map F → π(F ) : F → F/R is continuous. But, by definition, F/R = F/G and, from part (ii), π is also continuous for the right-invariant quotient metric. Therefore, the identity map i : (F/R, T ) → (F/G, dG ) is continuous. The topology T in [I] is defined by the fundamental system of neighborhoods {π(F ) : d(F, I) < ε}. But dG (π(F ), π(I)) ≤ d(F, I) < ε and i({π(F ) : d(F, I) < ε}) ⊂ {π(F ) : dG ([F ], [I]) < ε} that is a neighborhood of [I] in the (F/G, dG ) topology. Therefore i−1 is continuous and the two topologies coincide. We now have all the ingredients to conclude. Theorem 2.9. Let Θ be any of the spaces Bk (RN , RN ), C k (RN , RN ), C0k (RN , RN ), C k,1 (RN , RN ), k ≥ 0, or C ∞ (RN , RN ), C0∞ (RN , RN ). (i) The group (F(Θ), d) is a complete right-invariant metric space. For Θ equal to C k (RN , RN ) or C0k (RN , RN ), 0 ≤ k ≤ ∞, (F(Θ), d) is also a topological group. (ii) For any closed subgroup G of F(Θ), the function dG : F(Θ) × F(Θ) → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.34)

is a right-invariant metric on F(Θ)/G and the space (F(Θ)/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F(Θ)/G. (iii) Let Ω0 , ∅ = Ω0 ⊂ RN , be closed or let Ω0 = int Ω0 . Then def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.35)

is a closed subgroup of F(Θ). Finally, it is interesting to recall that A. M. Micheletti [1] used the quotient metric to prove the following theorem.

2. Generic Constructions of Micheletti

143

Theorem 2.10. Fix k = 3, the Courant metric of F(C03 (RN , RN ))/G(Ω0 ), and a bounded open connected domain Ω0 of class C 3 in RN . Let X (Ω0 ) be the family of images of Ω0 by elements of F(C03 (RN , RN ))/G(Ω0 ). The subset of all bounded open domains Ω in X (Ω0 ) such that the spectrum of the Laplace operator −∆ on Ω with homogeneous Dirichlet conditions on the boundary ∂Ω does not have all its eigenvalues simple is of the first category.13 This theorem says that, up to an arbitrarily small perturbation of the domain, the eigenvalues of the Laplace operator of a C 3 -domain can be made simple.

2.5

Assumptions for Bk (RN , RN ), C k (RN , RN ), and C0k (RN , RN )

2.5.1

Checking the Assumptions

By definition, for all integers k ≥ 0 C0k (RN , RN )) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ) ∀x ∈ RN ,

f → f (x) : B0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. The spaces are all Banach spaces endowed with the same norm:14 def

f Ck =

max ∂ α f C .

0≤|α|≤k

As a consequence, the evaluation map f → f (x) is continuous for all the spaces. The following results are quoted directly from A. M. Micheletti [1] and apply to all spaces Θ ⊂ C k (RN , RN ). Denote by S(x)[y1 ][y2 ], . . . , [yk ] at the point x ∈ RN a k-linear form with arguments y1 , . . . , yk . Lemma 2.4. Assume that F and G are two mappings from RN to RN such that F is k-times differentiable in an open neighborhood of x and G is k-times differentiable in an open neighborhood of F (x). Then the kth derivative of G ◦ F in x is the sum of a finite number of k-linear applications on RN of the form . / (h1 , h2 , . . . , hk ) → G(
) (F (x)) F (λ1 ) (x)[h1 ][h1 ] . . . [hλ1 ] / . (2.36) . . . F (λ ) (x)[hk−λ +1 ] . . . [hk ] , where  = 1, . . . , k and λ1 + · · · + λ
= k. 13 A set B is said to be nowhere dense if its closure has no interior or, alternatively, if B is dense. A set is said to be of the first category if it is the countable union of nowhere dense sets (cf. J. Dugundji [1, Def. 10.4, p. 250]). 14 Here

RN is endowed with the norm |x| =

)N

i=1

x2i

1/2

.

144

Chapter 3. Courant Metrics on Images of a Set

Proof. We proceed by induction on k. The result is trivially true for k = 1. Assuming that it is true for k − 1, we prove that it is also true for k. This is an obvious consequence of the observation that for a mapping from RN into L((RN )k−1 , RN ) of the form x → G(
) (F (x))[F (λ1 ) (x)] . . . [F (λ ) (x)], where  = 1, . . . , k − 1 and λ1 + · · · + λ
= k − 1, the differential in x is G(
+1) (F (x))[F (1) ][F (λ1 ) (x)] . . . [F (λ ) (x)] + G(
) (F (x))[F (λ1 +1) (x)] . . . [F (λ ) (x)] + · · · + G(
) (F (x))[F (λ1 ) (x)] . . . [F (λ +1) (x)], (2.37) and the result is true for k. Lemma 2.5. Given g and f in C k (RN , RN ), let ψ = g ◦ (I + f ). Then for each x ∈ RN |ψ(x)| = |g(x + f (x))|, |ψ

(1)

(x)| ≤ |g (1) (x + f (x))| [1 + |f (1) (x)|],

|ψ (i) (x)| ≤ |g(1) (x + f (x))| |f (i) (x)| +

i

|g (j) (x + f (x))| aj (|f (1) (x)|, . . . , |f (i−1) (x)|)

j=2

for i = 2, . . . , k, where aj is a polynomial with positive coefficients. Proof. This is an obvious consequence of Lemma 2.4. Theorem 2.11. Assumptions 2.1 and 2.3 are verified for the spaces C0k (RN , RN ), C k (RN , RN ), and B k (RN , RN ), k ≥ 0. Proof. (i) Assumption 2.3. From Lemma 2.5 it readily follows that f ◦ (I + g) ∈ Bk (RN , RN ) for all f ∈ B k (RN , RN ) and I + g in F(B k (RN , RN )). Moreover, ψC = f C , ψ

(1)

C ≤ f (1) C (1 + g (1) C ),

ψ (i) C ≤ f (1) C g (i) C +

i

f (j) C aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k = max ψ i C ≤ f C k c1 (gC k ) i=0,...,k

for some polynomial function c1 that depends only on k.

2. Generic Constructions of Micheletti

145

(ii) Assumption 2.1. The case of Bk (RN , RN ) is a consequence of (i). For C (RN , RN ) and each i, ψ(i) is uniformly continuous as the finite sum and product of compositions of uniformly continuous functions by Lemma 2.4. As for C0k (RN , RN ), for all I + g ∈ F(C0k (RN , RN )) and f ∈ C0k (RN , RN ), h = f ◦ (I + g) ∈ C k (RN , RN ) from the previous case. It remains to show that h ∈ C0k (RN , RN ). This amounts to showing that |h(x)| and |h(i) (x)| go to zero as |x| → ∞, i = 1, . . . , k. We again proceed by induction on k. Since |g(x)| → 0 as |x| → ∞, |x + g(x)| ≥ |x| − |g(x)| → ∞ as |x| → ∞ and |h(x)| = |f (x + g(x))| → 0 as |x| → ∞ and Assumption 2.1 is true for k = 0. Always, from Lemma 2.5 and the identity h(x) = f (x + g(x)), k

|h(1) (x)| ≤ |f (1) (x)| [1 + |g (1) (x)|] → 0 as |x| → 0 and Assumption 2.1 is true for k = 1. For k ≥ 2 and 2 ≤ i ≤ k, again, from Lemma 2.5 |h(i) (x)| ≤ |f (1) (x + g(x))| g (i) C +

i

|f (j) (x + g(x))| aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k. Since for each j, |f (j) (x)| → 0 as |x| → ∞, by the same argument, |f (j) (x + g(x))| → 0 as |x| → ∞ and hence the whole right-hand side goes to 0. Theorem 2.12. Assumption 2.2 is verified for the spaces C0k (RN , RN ), C k (RN , RN ), and Bk (RN , RN ), k ≥ 0. It is a consequence of the following lemma of A. M. Micheletti [1]. Lemma 2.6. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: if the sequence f1 , . . . , fn in C r (RN , RN ) is such that n fi C r < α, 0 < α < s, (2.38) i=1

then for the map F = (I + fn ) ◦ · · · ◦ (I + f1 ), F − IC r ≤ α c(r, s).

(2.39)

Proof. We again proceed by induction on r. For simplicity define def

Fi = (I + fi ) ◦ · · · ◦ (I + f1 ),

Fn = F.

By the definition of F we have F − I = f1 + f2 ◦ (I + f1 ) + · · · + fn ◦ (I + fn−1 ) ◦ · · · ◦ (I + f1 ) = f1 + f2 ◦ F1 + · · · + fn ◦ Fn−1 .

(2.40)

146

Chapter 3. Courant Metrics on Images of a Set

)n Then F − IC 0 ≤ i=1 fi C 0 ≤ α and c(0, s) = 1 for all integers s > 0. From (2.40) and Lemma 2.5 we further get sup |(F − I)(1) (x)| ≤ f1 C 1 + f2 C 1 [1 + f1 C 1 ] x

≤

n i=1

fi C 1

n (

+ · · · + fn C 1 [1 + fn−1 C 1 ] . . . [1 + f1 C 1 ] (2.41) 0 n 1 )n (1 + fi C 1 ) ≤ fi C 1 e( i=1 fi C 1 ) ≤ α eα ≤ α es

i=1

i=1

and F − IC 1 ≤ α eα . Hence c(1, s) = es for all integers s > 0. We now show that if the result is true for r − 1, it is true for r. We have to evaluate |(Fn − I)(r) (x)|. It is obvious that Fn − I = (I + fn ) ◦ Fn−1 − I = Fn−1 − I + fn ◦ Fn−1 and hence (Fn − I)(r) (x) = (Fn−1 − I)(r) (x) + (fn ◦ Fn−1 )(r) (x).

(2.42)

For r ≥ 2 from Lemma 2.5 we have |(fn ◦ Fn−1 )(r) (x)| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| r |fn(j) (Fn−1 (x))| aj (|(Fn−1 − I)(1) (x)|, . . . , |(Fn−1 − I)(r−1) (x)|). + j=2

By the induction assumption |(Fn−1 − I)(i) (x)| ≤ (Fn−1 − I)C r−1 ≤ α c(r − 1, s), i = 1, . . . , r − 1, and the fact that aj is a polynomial dependent on r for j = 2, . . . , r, there exists a constant L(r, s) such that |(fn ◦ Fn−1 )(r) (x)| r |fn(j) (Fn−1 (x))| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| + L(r, s) j=2

(2.43)

≤ fn C r |(Fn−1 − I)(r) (x)| + (r − 1)L(r, s)fn C r . Define M (r, s) = (r − 1)L(r, s). From (2.42) and (2.43), we get |(Fn − I)(r) (x)| ≤ [1 + fn C r ] |(Fn−1 − I)(r) (x)| + M (r, s)fn C r . After repeating this procedure n − 1 times we have |(Fn − I)(r) (x)| ≤ [1 + fn C r ] . . . [1 + f2 C r ]f1 C r + [1 + fn C r ] . . . [1 + f2 C r ]M (r, s)f2 C r + · · · + M (r, s)fn C r ≤ max{M (r, s), 1} eα α. Henceforth c(r, s) = max {M (r, s), 1}es . Theorem 2.13. Assumption 2.4 is verified for C k (RN , RN ) and C0k (RN , RN ). It is a consequence of the following slightly modified versions of lemmas of A. M. Micheletti [1].

2. Generic Constructions of Micheletti

147

Lemma 2.7. Given f ∈ C k (RN , RN ) and γ ∈ C 0 (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ, γC 0 < δ,

sup f (i) ◦ (I + γ) − f (i) C 0 < ε.

(2.44)

0≤i≤k

Proof. Since f and its derivatives f (i) all belong to C 0 (RN , RN ), they are uniformly continuous functions. Hence, for each ε > 0 there exists δ > 0 such that ∀x, y ∈ RN , |x − y| < δ,

sup |f (i) (y) − f (i) (x)| < ε. 0≤i≤k

In particular for γ ∈ C 0 (RN , RN ) such that γC 0 < δ |x + γ(x) − x| = |γ(x)| ≤ γC 0 < δ

⇒ sup |f (i) (x + γ(x)) − f (i) (x)| < ε 0≤i≤k

and we get the result. Lemma 2.8. For each g ∈ C k (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ ∈ C k (RN , RN ) such that γC k < δ,

g ◦ (I + γ) − gC k < ε.

(2.45)

Proof. Again proceed by induction on k. For k = 0 the result is a consequence of Lemma 2.7. Then check that if it is true for k − 1 it is true for k. From Lemma 2.4 the kth derivative of g ◦(I +γ)−g is the sum of a finite number of k-linear mappings of the form g (r) (x + γ(x))[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)]   = g (r) (x + γ(x)) − g (r) (x) [(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] + g (r) (x)[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)], where r = 1, . . . , k and λ1 + · · · + λr = k. Since |(I + γ)(λ1 ) (x)| |(I + γ)(λ2 ) (x)| . . . |(I + γ)(λr ) (x)| ≤ (γC k + 1)r , the norm of the mapping can be bounded above by the following expression: sup |g (r) (x + γ(x)) − g (r) (x)| (γC k + 1)r + γC k p(γC k , gC k ), x

(2.46)

where p is a polynomial. From this upper bound and Lemma 2.7 the result of the lemma is true for k. 2.5.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ

148

Chapter 3. Courant Metrics on Images of a Set

and the tangent space in each point of the affine space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a sufficiently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.14. Let Θ be equal to C0k (RN , RN ), C k (RN , RN ), or B k (RN , RN ), k ≥ 1. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.5. The generic framework of Micheletti is similar to an infinite-dimensional Riemannian manifold. Proof. (i) For k ≥ 1 and f sufficiently small, F is bijective and has a unique inverse. Given y ∈ RN , consider the mapping def

S(x) = y − f (x),

x ∈ RN .

Then for any x1 and x2 , S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )] ⇒ |S(x2 ) − S(x1 )| ≤ |f (x2 ) − f (x1 )| ≤ f (1) C 0 |x2 − x1 |. For f (1) C 1 < 1, S is a contraction and, for each y, there exists a unique x such that y − f (x) = S(x) = x, [I + f ](x) = y. Therefore I + f is bijective. But in order to show that F ∈ F(B k (RN , RN )) we also need to prove that [I + f ]−1 ∈ B k (RN , RN )). The Jacobian matrix of F (x) is equal to I + f (1) (x) and ∀x ∈ RN ,

|F (1) (x) − I| ≤ F (1) − IC 0 = f (1) C 0 .

If there exists x ∈ RN such that F (1) (x) is not invertible, then for some 0 = y ∈ RN , F (1) (x)y = 0 and |y| = |F (1) (x)y − y| ≤ |F (1) (x) − I||y| ≤ f (1) C 0 |y|

⇒ 1 ≤ f (1) C 0 < 1,

which is a contradiction. Therefore, for all x ∈ RN , F (1) (x) is invertible. As a result the conditions of the implicit function theorem are met and from L. Schwartz [3, Vol. 1, Thm. 29, p. 294, Thm. 31, p. 299] we have that F −1 = [I + f ]−1 ∈ C k (RN , RN ). It remains to prove that F −1 = [I+f ]−1 ∈ Bk (RN , RN ) and the continuity. We again proceed by induction on k. Set y = F (x). By definition g(y) = F −1 (y) − y = x − F (x) = −f (x) and gC 0 = f C0 . Always, from Lemma 2.5 and the identity g(y) = −f (x) = −f (F −1 (y)) = −f (y + g(y)), |g (1) (y)| ≤ |f (1) (x)| [1 + |g(1) (y)|]

⇒ |g (1) (y)| [1 − |f (1) (x)|] ≤ |f (1) (x)|.

2. Generic Constructions of Micheletti

149

Since |f (1) (x)| ≤ f C0 < 1 0 ≤ |g (1) (y)| ≤

|f (1) (x)| 1 − f C0

⇒ g(1) C 0 ≤

|f (1) (x)| f (1) C0 ≤ . 1 − f C0 1 − f C0

(2.47)

For k ≥ 2, again from Lemma 2.5, we have )k 0 ≤ |g

(k)

j=2

(y)| ≤

)k j=2

≤

|f (j) (x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − |f (1) (x)| |f

(j)

(x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − f (1) C 0

(2.48) .

So there exists a constant ck such that F −1 − ICk ≤ ck F − IC k = ck f C k . Therefore, for k ≥ 1, ∀f ∈ Bk (RN , RN ) such that f (1) C 0 < 1,

F = I + f ∈ F(B k (RN , RN )).

Note that the condition is on f (1) C 0 and not f C k , but by definition f (1) C 0 ≤ f C 1 ≤ f C k and the condition f C k < 1 yields the same result. In particular, the map f → I + f : B(0, 1) ⊂ Bk (RN , RN ) → F ( B k (RN , RN )) is continuous. All this can now be specialized to the spaces C0k (RN , RN ) and C k (RN , RN ) by using the identity (I + f )−1 − I = −f ◦ (I + f )−1 . (ii) Since F(Θ) is contained in the affine space I + Θ, any tangent element to F(Θ) will be contained in Θ. From part (i) for k ≥ 1 and for all θ ∈ Θ there exists τ > 0 such that for all t, 0 ≤ t ≤ τ , I + tΘ ∈ F(Θ). For any F ∈ F(Θ), t → (I + tθ) ◦ F : [0, τ ] → F(Θ) is a continuous path in F(Θ) and the limit of (I + tθ) − I [(I + tθ) ◦ F ] ◦ F −1 − I = =θ t t will be an element of the tangent space TF F(Θ) in F to F(Θ). Therefore for all F ∈ F(Θ), TF F(Θ) = Θ.

2.6 2.6.1

Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) Checking the Assumptions

By definition, for all integers k ≥ 0 C

k,1

N

def

(RN , R ) =

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω f ∈ C (RN , R ) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y| k

N

&

150

Chapter 3. Courant Metrics on Images of a Set

endowed with the same norm:15   def f C k,1 = max max ∂ α f C , ck (f ) = max {f C k , ck (f )} ,

(2.49)

0≤|α|≤k

def

|f (y) − f (x)| and ∀k ≥ 1, |y − x| y=x

c(f ) = sup



def

ck (f ) =

c(∂ α f ),

(2.50)

|α|=k

and the convention c0 (f ) = c(f ). C0k,1 (RN , RN ) is the space of functions f ∈ C k,1 (RN , RN ) such that f and all its derivatives go to zero at infinity. They are all Banach spaces. By definition C0k,1 (RN , RN ) ⊂ C k,1 (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ), ∀x ∈ RN ,

f → f (x) : B 0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. As a consequence, the evaluation map f → f (x) is continuous for all the spaces. We deal only with the case C k,1 (RN , RN ). The case C0k,1 (RN , RN ) is obtained from the case C k,1 (RN , RN ) as in the previous section. Theorem 2.15. Assumptions 2.1 and 2.3 are verified for the spaces C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0. Proof. The composition and multiplication of bounded and Lipschitz functions is bounded and Lipschitz. For f ∈ C 0,1 (RN , RN ) and I + g ∈ F(C 0,1 (RN , RN )), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ◦ (I + g)) ≤ c(f ) (1 + c(g)). Hence f ◦ (I + g)C 0,1 ≤ f C 0,1 (1 + c(g)) ≤ f C 0,1 (1 + gC 0,1 ) and the continuous function c1 (r) = 1 + r satisfies Assumption 2.3. For k = 1, D(f ◦ (I + g)) = Df ◦ (I + g) (I + Dg) and D(f ◦ (I + g))C ≤ Df C (1 + DgC ), c(D(f ◦ (I + g))) ≤ c(Df ) (1 + c(g)) (1 + DgC ) + Df C (1 + c(Dg)) ⇒ D(f ◦ (I + g))C 0,1 ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + c(g)) ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ⇒ f ◦ (I + g)C 1,1 ≤ f C 1,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ≤ f C 1,1 (1 + gC 1,1 )2 . The general case is obtained by induction over k. Theorem 2.16. Assumption 2.2 is verified for C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0.

)N 2 1/2 15 N Here R

is endowed with the norm |x| =

i=1

xi

.

2. Generic Constructions of Micheletti

151

Proof. Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max{fi C 0 , c(fi )} < α < r.

(2.51)

i=1

From Theorem 2.12 (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) + ≤ c(fn ) + ≤

 n

n−1 i=1 n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 )) )i+1 c(fi ) e

k=1

c(fk )

i=1



)n

c(fi ) e[

i=1

c(fi )]

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er for k = 0. We use the same technique for k ≥ 1 but with heavier computations. 2.6.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ and the tangent space in each point of the affine space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a sufficiently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.17. Let Θ be equal to C0k,1 (RN , RN ) or C k,1 (RN , RN ), k ≥ 0. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.6. The generic framework of Micheletti is similar to an infinite-dimensional Riemannian manifold.

152

Chapter 3. Courant Metrics on Images of a Set

Proof. (i) Consider perturbations of the identity of the form F = I + f , f ∈ C 0,1 (RN , RN ). Consider for any y ∈ RN the map def

x → S(x) = y − f (x) : RN → RN . For all x1 and x2 S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )], |S(x2 ) − S(x1 )| = |f (x2 ) − f (x1 )| ≤ c(f ) |x2 − x1 |, and f is contracting for c(f ) < 1. Hence for each y ∈ RN there exists a unique x ∈ RN such that ⇒ y − f (x) = x

S(x) = x

⇒ F (x) = y def

and F is bijective. It remains to prove that g = F −1 − I ∈ C 0,1 (RN , RN ) to conclude that I + f ∈ F(C 0,1 (RN , RN )). For any y ∈ RN and x = F −1 (y) F −1 (y) − y = x − F (x), gC 0 = F −1 − IC 0 = sup |F −1 (y) − y| = sup |(I − F )(F −1 (y))| y∈RN

y∈RN

= sup |(I − F )(x)| = F − IC 0 < ∞ x∈RN

since F : RN → RN is bijective and gC 0 = f C 0 . For all y1 and y2 and xi = F −1 (yi ), |g(y2 ) − g(y1 )| = |F −1 (y2 ) − y2 − (F −1 (y1 ) − y1 )| ≤ |x2 − F (x2 ) − (x1 − F (x1 )| = |f (x2 ) − f (x1 )| ≤ c(f )|x2 − x1 | ≤ c(f )|F −1 (y2 ) − F −1 (y1 )| = c(f )|g(y2 ) − g(y1 ) + (y2 − y1 )| ⇒ |g(y2 ) − g(y1 )| ≤

c(f ) |y2 − y1 | 1 − c(f )

⇒ c(g) ≤

c(f ) < ∞. 1 − c(f )

Therefore ∀f ∈ C 0,1 (RN , RN ) such that c(f ) < 1,

F = I + f ∈ F(C 0,1 (RN , RN )).

Moreover since c(f ) ≤ f C 0,1 , the condition f C 0,1 < 1 yields c(f ) < 1 and I +f ∈ F(C 0,1 (RN , RN )). Furthermore, the map f → I + f : B(0, 1) ⊂ C 0,1 (RN , RN ) → F(C 0,1 (RN , RN )) is continuous. We use the same technique for k ≥ 1 plus the arguments of Theorem 2.14 (i). (ii) Same proof as Theorem 2.14 (ii).

3. Generalization to All Homeomorphisms and C k -Diffeomorphisms

3

153

Generalization to All Homeomorphisms and C k -Diffeomorphisms

With a variation of the generic constructions associated with the Banach space Θ of section 2, it is possible to construct a metric on the whole space of homeomorphisms Hom(RN , RN ) or on the whole space Diff k (RN , RN ) of C k -diffeomorphisms of RN , k ≥ 1, or of an open subset D of RN . First, recall that, for an open subset D of RN , the topology of the vector space C k (D, RN ) can be specified by the family of seminorms def

∀K compact ⊂ D,

nK (F ) = F C k (K) .

(3.1)

This topology is equivalent to the one generated by the monotone increasing subfamily {Ki }i≥1 of compact sets   1 def Ki = x ∈ D : d
D (x) ≥ and |x| ≤ i , i ≥ 1, (3.2) i where d
D (x) = inf y∈
D |y − x|. This leads to the construction of the metric def

δk (F, G) =

∞ 1 nKi (F − G) i 1 + n (F − G) 2 Ki i=1

(3.3)

that generates the same topology as the initial family of seminorms and makes C k (D, RN ) a Fr´echet space with metric δk . Given an open subset D of RN , consider the group Homk (D)  def

= F ∈ C k (D, RN ) : F : D → D is bijective and F −1 ∈ C k (D, RN )

(3.4)

of transformations of D. For k = 0, Hom0 (D) = Hom(D) and for k ≥ 1, Homk (D) = Diff k (D), where the condition F −1 ∈ C k (D, RN ) is redundant. One possible choice of a topology on Homk (D) is the topology induced by C k (D, RN ). This is the so-called weak topology on Homk (D) (cf., for instance, M. Hirsch [1, Chap. 2]). However, Homk (D) is not necessarily complete for that topology. In order to get completeness, we adapt the constructions of A. M. Micheletti [1]. Associate with F ∈ Homk (D) and each compact subset K ⊂ D the following function of I and F : def

qK (I, F ) =

inf

n

F =F1 ◦···◦Fn F ∈Homk (D)
=1

nK (I, F
) + nK (I, F
−1 ),

(3.5)

where the infimum is taken over all finite factorizations F = F1 ◦ · · · ◦ Fn , Fi ∈ Homk (D), of F in Homk (D). By construction, qK (I, F −1 ) = qK (I, F ). Extend this definition to all pairs F and G in Homk (D): qK (F, G) = qK (I, G ◦ F −1 ). def

(3.6)

154

Chapter 3. Courant Metrics on Images of a Set

The function qK is a pseudometric 16 and the family {qK : K compact ⊂ D} defines a topology on Homk (D). This topology is metrizable for the metric def

dk (F, G) =

∞ 1 qKi (F, G) , 2i 1 + qKi (F, G) i=1

(3.7)

constructed from the monotone increasing subfamily {Ki }i≥1 of compact sets defined in (3.2). By definition, dk (F, G) = dk (G, F ) and dk is symmetrical. It is right-invariant since for all F , G, and H in Homk (D)) ∀F, G, and H ∈ Homk (D), ∀F ∈ Homk (D),

dk (F, G) = dk (F ◦ H, G ◦ H), dk (I, F ) = dk (I, F −1 ).

Theorem 3.1. Homk (D), k ≥ 0, is a group under composition, dk is a rightinvariant metric on Homk (D), and (Homk (D), dk ) is a complete metric space. We need Lemmas 2.4 and 2.5 and the analogues of Lemma 2.6 and Theorem 2.11 for the spaces C k (K, RN ), K a compact subset of D, whose elements are bounded and uniformly continuous. Lemma 3.1. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: for any compact K ⊂ D and a sequence F1 , . . . , Fn in C r (K, RN ) is such that n

F
− IC r (K) < α,

0 < α < s;

(3.8)


=1

then for the map F = Fn ◦ · · · ◦ F1 , F − IC r (K) ≤ α c(r, s).

(3.9)

Theorem 3.2. Given k ≥ 0, there exists a continuous function c1 such that for all F, G ∈ Homk (D) and any compact K ⊂ D F ◦ GC k (K) ≤ F C k (K) c1 (GC k (K) ). 16 A

(3.10)

map d : X × X → R on a set X is called a pseudometric (or ´ ecart, or gauge) whenever

(i) d(x, y) ≥ 0, for all x and y, (ii) x = y ⇒ d(x, y) = 0, (iii) d(x, y) = d(y, x), (iv) d(x, z) ≤ d(x, y) + d(y, z) (J. Dugundji [1, p. 198]). This notion is analogous to the one of seminorm for topological vector spaces, but for a group the terminology semimetric (cf. footnote 7) is used for a metric without the triangle inequality.

3. Generalization to All Homeomorphisms and C k -Diffeomorphisms

155

Proof. From Lemma 2.5 it readily follows that ψ = F ◦ G ∈ Homk (D). Moreover, ψC 0 (K) = F C 0 (K) , ψ

(1)

C 0 (K) ≤ F (1) C 0 (K) G(1) C 0 (K) ,

ψ (i) C 0 (K) ≤ F (1) C 0 (K) G(i) C 0 (K) +

i

F (j) C 0 (K) aj (G(1) C 0 (K) , . . . , G(i−1) C 0 (K) )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k (K) = max ψ i C(K) ≤ F C k (K) c1 (GC k (K) ) i=0,...,k

for some polynomial function c1 that depends only on k. Proof of Theorem 3.1. (i) Homk (D) is a group. The composition of C k k diffeomorphisms is again a C -diffeomorphism. By definition, each element has an inverse and the neutral element is I. (ii) dk is a metric. dk (F, G) ≥ 0, dk (F, G) = dk (G, F ). For the triangle inequality F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a finite factorization of F ◦ H −1 . Consider arbitrary finite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = H1 ◦ · · · ◦ Hm ,

Hi ∈ C k (D, D),

G ◦ H −1 = L1 ◦ · · · ◦ Ln ,

Lj ∈ C k (D, D).

This yields a new finite factorization of F ◦ H −1 . By definition of qKi , qKi (I, F ◦ H

−1

)≤

m

nKi (I, H
) +

nKi (I, H
−1 )

+

n

nKi (I, L
 ) + nKi (I, L−1
 )


 =1


=1

and by taking infima over each factorization, qKi (F, H) = qKi (I, F ◦ H −1 ) ≤qKi (I, F ◦ G−1 ) + qKi (I, G ◦ H −1 ) = qKi (F, G) + qKi (G, H). By definition of dk , since {Ki } is a monotonically increasing sequence of compact sets and {qKi } is a monotonically increasing sequence of pseudometrics, we then get the triangle inequality dk (F, H) ≤ dk (F, G) + dk (G, H). Since Homk (D) is a group and dk is right-invariant, it remains to show that dk (I, F ) = 0 if and only if F = I. Clearly if F = I, qKi (I, F ) = 0 since the qKi ’s are pseudometrics and dk (I, F ) = 0. Conversely, if def

dk (I, F ) =

∞ 1 qKi (I, F ) = 0, i 1 + q (I, F ) 2 Ki i=1

156

Chapter 3. Courant Metrics on Images of a Set

then for all Ki , qKi (I, F ) = 0. For each Ki and all ε, 0 < ε < 1, there exists a finite factorization F = F1 ◦ · · · ◦ Fn such that n

nKi (F
− I) + nKi (F
−1 − I) < ε < 1


=1

⇒ nKi (F − I) ≤



nKi (F
◦ · · · ◦ Fn − F
+1 ◦ · · · ◦ Fn ).




From Lemma 3.1 ∀ε > 0,

nKi (F − I) < ε c(k, 1)

and (F − I)|Ki = 0. Since this is true for all i, F − I = 0 on D and dk is a metric. (iii) Completeness. Let {Fn } be a Cauchy sequence in Homk (D). (a) Boundedness of {Fn } and {Fn−1 } in C k (D, RN ). For all ε, 0 < ε < 1, there exists N such that, for all m, n > N , dk (Fm , Fn ) < ε. By the triangle inequality |dk (I, Fm ) − dk (I, Fn )| ≤ dk (Fm , Fn ) < ε. For all i, {qKi (I, Fn } is Cauchy and, a posteriori, bounded by some constant L. For ν(n) each n, there exists a factorization Fn = Fn1 ◦ · · · ◦ Fn such that

ν(n)

0≤

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) − dk (I, Fn ) < ε < 1

i=1



ν(n)

⇒

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) < L + ε < L + 1.

i=1

By Lemma 3.1 Fn − IC k (Ki ) + Fn−1 − IC k (Ki ) < 2(L + ε) c(k, L + 1) and the sequences {Fn − I} and {Fn−1 − I} and, a fortiori, {Fn } and {Fn−1 } are bounded in C k (Ki , RN ). (b) Convergence of {Fn } and {Fn−1 } in C k (D, RN ). For any m, n −1 − I) ◦ Fm Fn − Fm = (Fn ◦ Fm

and by Theorem 3.2 −1 − I) ◦ Fm C k (Ki ) Fm − Fn C k (Ki ) = (Fn ◦ Fm −1 ≤ Fn ◦ Fm − IC k (Ki ) c1 (Fm C k (Ki ) ) −1 ⇒ Fm − Fn C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) , def

c+ 1 = sup c1 (Fm C k (Ki ) ) < ∞ m

3. Generalization to All Homeomorphisms and C k -Diffeomorphisms

157

since {Fm } is bounded in C k (Ki , RN ). For all ε, 0 < ε < c(k, 1) c+ 1 (where c(k, 1) is the constant of Lemma 3.1), ∃N, ∀n, m > N,

−1 dk (I, Fn ◦ Fm ) = dk (Fm , Fn ) < ε/(c(k, 1) c+ 1 ) < 1.

By Lemma 3.1 + −1 − IC k (Ki ) < c(k, 1)ε/(c(k, 1) c+ Fn ◦ Fm 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − Fn )C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) < ε

and, since this is true for all Ki , {Fn } is a Cauchy sequence in C k (D, RN ) that converges to some F ∈ C k (D, RN ) since C k (D, RN ) is a Fr´echet space for the family of seminorms {qKi }. Similarly, we get that {Fn−1 } is a Cauchy sequence in C k (D, RN ) that converges to some G ∈ C k (D, RN ). (c) F is bijective, G = F −1 , and F ∈ Homk (D). For all n, Fn , Fn−1 , F and G are continuous since C k (D, RN ) ⊂ C 0 (D, RN ). By continuity of F → F (x) : C k (D, RN ) → RN , Fn−1 (x) → G(x). By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F, F −1 ∈ C k (D, RN ), and F ∈ Homk (D). (d) Convergence of Fn → F in Homk (D). To check that Fn → F in Homk (D) recall that by Theorem 3.2 for each Ki qKi (Fn , F ) = qKi (I, F ◦ Fn−1 ) ≤ I − F ◦ Fn−1 C k (Ki ) + I − Fn ◦ F −1 C k (Ki ) = (Fn − F ) ◦ Fn−1 C k (Ki ) + (F − Fn ) ◦ F −1 C k (Ki ) ≤ c1 (Fn−1 C k (Ki ) )Fn − F C k (Ki ) + c1 (F −1 C k (Ki ) )F − Fn C k (Ki )   ≤ max c1 (Fn−1 C k (Ki ) ) + c1 (F −1 C k (Ki ) ) F − Fn C k (Ki ) n

that goes to zero as Fn goes to F in C k (D, RN ). This completes the proof. Remark 3.1. In section 2 it was possible to consider in F(Θ) a subgroup G of translations since Fa (x) = a + x

⇒ (Fa − I)(x) = a ∈ Θ

for spaces Θ that contain constants. Yet, since all of them were spaces of bounded functions in RN , it was not possible to include rotations or flips. But this can be done now in Homk (RN ), and we can quotient out not only by the subgroup of translations but also by the subgroups of isometries or rotation that are important in image processing.

Chapter 4

Transformations Generated by Velocities 1

Introduction

In Chapter 3 we have constructed quotient groups of transformations F(Θ)/G and their associated complete Courant metrics. Such spaces are neither linear nor convex. In this chapter, we specialize the results of Chapter 3 to spaces of transformations that are generated at time t = 1 by the flow of a velocity field over a generic time interval [0, 1] with values in the tangent space Θ. The main motivation is to introduce a notion of semiderivatives in the direction θ ∈ Θ on such groups as well as a tractable criterion for continuity via C 1 or continuous paths in the quotient group endowed with the Courant metric. This point of view was adopted by J.´sio [2, 7] as early as 1973 and considerably expanded in his th`ese d’´etat in P. Zole 1979. One of his motivations was to solve a shape differential equation of the type A V (t) + G(Ωt (V )) = 0, t > 0, where G is the shape gradient of a functional and A a duality operator.1 At that time most people were using a simple perturbation of the identity to compute shape derivatives. The first comprehensive book systematically promoting the velocity method was published in 1992 by J. Soko
lowski and ´sio [9]. Structural theorems for the Eulerian shape derivative of smooth J.-P. Zole ´sio [7] and generalized to nonsmooth domains were first given in 1979 by J.-P. Zole ´sio [14]. The velocity point domains in 1992 by M. C. Delfour and J.-P. Zole ´ [2], of view was also adopted in 1994 by R. Azencott [1], in 1995 by A. Trouve ´ and in 1998 by A. Trouve [3] and L. Younes [2] to construct complete metrics and geodesic paths in spaces of diffeomorphisms generated by a velocity field with a broad spectrum of applications to imaging.2 The reader is referred to the forthcoming book of L. Younes [6] for a comprehensive exposition of this work and

1 Cf.

´sio [7] and the recent book by M. Moubachir and J.-P. Zole ´sio [1]. J.-P. Zole first author would like to thank Robert Azencott, who pointed out his work and the contributions of his team during a visit in Houston early in 2010. Special thanks also to Alain Trouv´ e for an enlightening discussion shortly after in Paris. 2 The

159

160

Chapter 4. Transformations Generated by Velocities

to related papers such as the ones of P. W. Michor and D. Mumford [1, 2, 3] and L. Younes, P. W. Michor, I. Shah, and D. Mumford [1]. In view of the above motivations, this chapter begins with section 2, which specializes the results of Chapter 3 to transformations generated by velocity fields. It also explores the connections between the constructions of Azencott and Micheletti that implicitly use a notion of geodesic path with discontinuities. Section 3 motivates and adapts the definitions of Gateaux and Hadamard semiderivatives in topological vector spaces (cf. Chapter 9) to shape functionals defined on shape spaces. The analogue of the Gateaux semiderivative for sets is obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative comes from the velocity (speed ) method. The first notion does not extend to submanifolds and does not incorporate the chain rule for the semiderivative of the composition of functions (for instance, to get the semiderivative with respect to the parameters of an a priori parametrized geometry), while Hadamard does. In Chapter 9, flows of velocity fields will be adopted as the natural framework for defining shape semiderivatives. In section 3.3 the velocity and transformation viewpoints will be emphasized through a series of examples of commonly used families of transformations of sets. They include C k -domains, Cartesian graphs, polar coordinates, and level sets. The following two sections give technical results that will be used to characterize continuity and semidifferentiability along local paths without restricting the analysis to the subgroup GΘ of F(Θ) of section 2. In section 4 we establish the equivalence between deformations obtained from a family of C 1 -paths and deformations obtained from the flow of a velocity field. Section 4.1 gives the equivalence under relatively general conditions. Section 4.2 shows that Lipschitzian perturbations of the identity operator can be generated by the flow of a nonautonomous velocity field. In section 4.3 the conditions of section 4.2 are sharpened for the special families of velocity fields in C0k (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ). The constrained case where the family of domains are subsets of a fixed holdall is studied in section 5. In both sections 4 and 5 we show that, under appropriate conditions, starting from a family of transformations is locally equivalent to starting from a family of velocity fields. This key result bridges the two points of view. Section 6 builds on the results of section 4 to establish that the continuity of a shape function with respect to the Courant metric on F(Θ)/G is equivalent to its continuity along the flows of all velocity fields V in C 0 ([0, τ ]; Θ) for Θ equal to C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0. This result is of both intrinsic and practical interest since it is generally easier to check the continuity along paths than directly with respect to the Courant metric. Finally, from the discussion in section 3, the technical results of section 4 will again be used in Chapter 9 to construct local C 1 -paths generated by velocity fields V in C 0 ([0, τ ]; Θ) to define the shape semiderivatives.

2. Metrics on Transformations Generated by Velocities

2

161

Metrics on Transformations Generated by Velocities

2.1

Subgroup GΘ of Transformations Generated by Velocities

From Chapter 3 recall the definition of the group  def

F(Θ) = I + θ : θ ∈ Θ, (I + θ)−1 ∃, and (I + θ)−1 − I ∈ Θ for the Banach space Θ = C 0,1 (RN , RN ) with norm   |θ(y) − θ(x)| def def , θΘ = max sup |θ(x)|, c(θ) , c(θ) = sup |y − x| N y=x x∈R

(2.1)

and the associated complete metric d on F(Θ). This section specializes to a subgroup of transformations I + θ of F(Θ) that are specified via a vector field V on a generic interval [0, 1] with the following properties: (i) for all x ∈ RN , the function t → V (t, x) : [0, 1] → RN belongs to L1 (0, 1; RN ) and  1  1 V (t)C dt = sup |V (t, x)| dt < ∞; (2.2) 0

0 x∈RN

(ii) for almost all t ∈ [0, 1], the function x → V (t, x) : RN → RN is Lipschitzian and  1  1 |V (t, y) − V (t, x)| c(V (t)) dt = sup dt < ∞. (2.3) |y − x| 0 0 x=y In view of the assumptions on V , it is natural to introduce the norm  1 def V L1 (0,1;Θ) = V (t)Θ dt.

(2.4)

0

Associate with V ∈ L1 (0, 1; Θ) the flow dTt (V ) = V (t) ◦ Tt (V ), dt

T0 (V ) = I.

(2.5)

In view of the assumptions on V , for each X ∈ RN , the differential equation x (t; X) = V (t, x(t; X)),

x(0; X) = X,

(2.6)

has a unique solution in W 1,1 (0, 1; RN ) ⊂ C([0, 1], RN ), the mapping X → x(·; X) : RN → C([0, 1], RN ) is continuous, and for all t ∈ [0, 1] the mapping X → Tt (V ) def (X) = x(t; X) : RN → RN is bijective. Define  def

GΘ = T1 (V ) : ∀V ∈ L1 ([0, 1]; Θ) .

(2.7)

162

Chapter 4. Transformations Generated by Velocities

Theorem 2.1.

(i) The set GΘ with the composition ◦, ∀V1 , V2 ∈ L1 (0, 1; Θ),

T1 (V1 ) ◦ T1 (V2 ),

(2.8)

is a subgroup of F(Θ) and for all V ∈ L1 (0, 1; Θ), sup Tt (V ) − IΘ = sup θV (t)Θ ≤ V L1 exp2(1+V L1 ) , 0≤t≤1

(2.9)

0≤t≤1 def

where θV (t) = Tt (V ) − I and L1 stands for L1 (0, 1; Θ). In particular, for all t, Tt (V ) ∈ F(Θ) and t → Tt (V ) : [0, 1] → F(Θ) is a continuous path. (ii) For all V ∈ L1 (0, 1; Θ), (T1 (V ))−1 = T1 (V − ) for V − (t, x) = −V (1 − t, x), −θV ◦ (I + θV )−1 = (T1 (V ))−1 − I = T1 (V − ) − I = θV − , −

V L1 (0,1;Θ) = V L1 (0,1;Θ) ,

(2.10) (2.11)

and sup Tt (V )−1 − IΘ = sup θV− (t)Θ ≤ V L1 exp2(1+V L1 ) .

0≤t≤1

(2.12)

0≤t≤1

(iii) For all V ∈ L1 (0, 1; Θ), T1 (V ) − IΘ + (T1 (V ))−1 − IΘ ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) (2.13) ⇒ d(T1 (V ), I) ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) .

(2.14)

(iv) For all V, W ∈ L1 (0, 1; Θ),   max sup θW (t) − θV (t)Θ , sup θW − (t) − θV − (t)Θ 0≤t≤1

≤V L1 exp

(1+V L1 )

0≤t≤1

(2.15)  1 + W L1 exp2(1+W L1 ) W (s) − V (s)L1 ,



where L1 stands for L1 (0, 1; Θ). Remark 2.1. Note that the condition (I − θ)−1 exists and (I − θ)−1 ∈ Θ in the definition of F(Θ) is automatically verified for the transformations I + θV generated by a velocity field V ∈ L1 (0, 1; Θ). Proof. (i) Choosing V = 0, the identity I is in GΘ . To show that GΘ is closed under the composition, it is sufficient to construct a W ∈ L1 (0, 1; Θ) such that T1 (W ) = T1 (V1 ) ◦ T1 (V2 ). Given τ ∈ (0, 1) define the concatenation    t 1   V , x , 0 ≤ t ≤ τ, τ 2 τ def   (V1 ∗ V2 )(t, x) = (2.16)  t−τ 1   V1 ,x , τ < t ≤ 1. 1−τ 1−τ

2. Metrics on Transformations Generated by Velocities

163

It is easy to check that W = V1 ∗ V2 ∈ L1 (0, 1; Θ) as the concatenation of the L1 -functions V1 and V2 so that the equation dTt (W ) = W (t) ◦ Tt (W ), dt

T0 (W ) = I

has a unique solution and T1 (W ) ∈ GΘ . On the interval [0, τ ] 

 s  t/τ V2 (s , Ts τ (W )(X)) ds , Ts (W )(X) d = τ τ 0 0  t  ⇒ Tt τ (W )(X) − X = V2 (s , Ts τ (W )(X)) ds , 0 ≤ t ≤ 1

Tt (W )(X) − X =

t

V2

s

0

⇒ Tt τ (W )(X) = Tt (V2 )(X)

and Tτ (W ) = T1 (V2 ).

On the interval [τ, 1] 



t

Tt (W )(X) − T1 (V2 ) =

V1 

τ t−τ 1−τ

= 



s−τ , Ts (W )(X) 1−τ

d

s−τ 1−τ

  V2 s , Tτ +s (1−τ ) (W )(X) ds

0 t

⇒ Tτ +t (1−τ ) t(W )(X) − T1 (V2 ) =

  V2 s , Tτ +s (1−τ ) (W )(X) ds ,

0 ≤ t ≤ 1,

0

Tτ +t (1−τ ) (W )(X) = Tt (V1 ) ◦ T1 (V2 ) and T1 (W )(X) = T1 (V1 ) ◦ T1 (V2 ). To show that each T1 (V ) has an inverse in GΘ , consider the new function y(t; X) = [T1−t (V ) ◦ (T1 (V ))−1 ](X) = x(1 − t; (T1 (V ))−1 (X)), def

0 ≤ t ≤ 1,

for which y(0; X) = [T1 (V ) ◦ (T1 (V ))−1 ](X) = X and y(1; X) = (T1 (V ))−1 (X). Its equation is given by dy (t; X) = −V (1 − t, y(t; X)), dt def

y(0; X) = X,

and the associated vector field in V − (t, x) = −V (1 − t, x) L1 (0, 1; Θ). Therefore GΘ is a group of homeomorphisms Hom(RN ). To show that GΘ ⊂ F(Θ), it is necessary to show T1 (V ) − I ∈ Θ. Given X ∈ RN , consider the functions

that clearly belongs to of RN , that is, GΘ ⊂ that for each V , θ = x(t) = Tt (V )(X) and

164

Chapter 4. Transformations Generated by Velocities

θ(t; X) = x(t) − X. By definition  t V (s, X + θ(s; X)) ds, θ(t; X) = x(t) − X = 0  t  t |V (s, X + θ(s; X)) − V (s, X)| ds + |V (s, X)| ds |θ(t; X)| ≤ 0 0  t  t ≤ c(V (s)) |θ(s; X)| ds + |V (s, X)| ds, 0 0  t  t c(V (s)) sup |θ(s; X)| ds + sup |V (s, X)| ds. sup |θ(t; X)| ≤ X

0

X

0

X

t Given 0 < α < 1 let gα (t) = exp( 0 c(V (s)) ds/α). Then it is easy to show that supX |θ(t; X)| 1 sup ≤ g (t) 1 − α α t∈[0,1] ⇒ sup θ(t)C ≤ t∈[0,1]



gα (1) 1−α

1

sup |V (s, X)| ds X

0



1

V (s)C ds. 0

To prove that θ is Lipschitzian, associate with X, Y ∈ RN , X = Y , the functions x(t) = Tt (V )(X), y(t) = Tt (V )(Y ), θ(t; X) = x(t) − X, and θ(t; Y ) = y(t) − Y . By definition  t θ(t; X) − θ(t; Y ) = V (s, Y + θ(s; Y )) − V (s, X + θ(s; X)) ds, 0  t  t c(V (s)) |θ(s; Y )) − θ(s; X)| ds + c(V (s)) |Y − X| ds |θ(t; X) − θ(t; Y )| ≤ 0 0  t  t |θ(t; X) − θ(t; Y )| |θ(s; Y )) − θ(s; X)| ⇒ sup ≤ ds + c(V (s)) sup c(V (s)) ds |Y − X| |Y − X| X=Y X=Y 0 0  gα (1) 1 c(V (s)) ds. ⇒ sup c(θ(t)) ≤ 1−α 0 t∈[0,1] Combining the last inequality with the one of the first part, for all α, 0 < α < 1, sup θ(t)Θ ≤ t∈[0,1]

gα (1) 1−α



1

V (t)Θ dt. 0

1 1 For V = 0, θ = 0; for V = 0, choose α = 0 V (t)Θ dt/( 0 V (t)Θ dt + 1). This yields (2.9). Since (I + θ)−1 = (T1 (V ))−1 = T1 (V − ) is of the same form for the velocity field V − , (I + θ)−1 − I = (T1 (V ))−1 − I = T1 (V − ) − I ∈ Θ. Therefore GΘ is a subgroup of F(Θ). (ii) From the construction of the inverse in part (i) and the fact that (T1 (V ))−1 − I = −θV ◦ (I + θV )−1 . (iii) From inequality (2.9) applied to θV and θV − and identity (2.11).

2. Metrics on Transformations Generated by Velocities (iv) Let θV (t) = Tt (V ) − I and let θW (t) = Tt (W ) − I:  t θW (t) − θV (t) = W (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds 0  t = (W (s) − V (s)) ◦ Ts (W ) ds 0  t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0  t W (s) − V (s)C ds θW (t) − θV (t)C ≤ 0  t + c(V (s)) θW (s) − θV (s)C ds 0

gα (1) 1−α

⇒ sup θW (t) − θV (t)C ≤ 0≤t≤1

for 0 < α < 1 and gα (t) = exp

t

1

W (s) − V (s)C ds 0

c(V (s) ds/α. Similarly,

0





t

θW (t) − θV (t) =

(W (s) − V (s)) ◦ Ts (W ) ds  t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0  t c(θW (t) − θV (t)) ≤ c(W (s) − V (s)) (1 + c(θW (s))) ds 0  t c(V (s)) c(θW (s) − θV (s)) ds + 0

0

⇒ sup c(θW (t) − θV (t)) ≤ 0≤t≤1

gα (1) 1−α



1

c(W (s) − V (s)) (1 + c(θW (s))) ds. 0

From part (i) sup c(θW (s)) ≤ W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) 0≤s≤1

and finally sup θW (t) − θV (t)Θ 0≤t≤1

≤

gα (1) (1 + W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) ) W (s) − V (s)L1 (0,1;Θ) ds. 1−α

By choosing α = V L1 (0,1;Θ) /(1 + V L1 (0,1;Θ) ) sup θW (t) − θV (t)Θ 0≤t≤1

≤ V L1 exp(1+V L1 ) (1 + W L1 (0,1;Θ) exp2(1+W L1 ) ) W (s) − V (s)L1 ds.

165

166

2.2

Chapter 4. Transformations Generated by Velocities

Complete Metrics on GΘ and Geodesics

Since GΘ is a subgroup of the complete group F(Θ), its closure GΘ with respect to the metric d of F(Θ) is a closed subgroup whose elements can be “approximated” by elements constructed from the flow of a velocity field. Is (GΘ , d) complete? Can a velocity field be associated with a limit element? It is very unlikely that (GΘ , d) can be complete unless we strengthen the metric d to make the associated Cauchy sequence of velocities {Vn } converge in L1 (0, 1; Θ). We have seen that for each V ∈ L1 (0, 1; Θ), we have a continuous path t → Tt (V ) in GΘ . Since the part of the metric on I +θV is defined as an infimum over all finite factorizations (I + θV1 ) ◦ · · · ◦ (I + θVn ) of I + θV and (I + θV − ) ◦ · · · ◦ (I + θVn− ) 1 of (I + θV )−1 , T1 (V ) = T1 (V1 ) ◦ T1 (V2 ) ◦ · · · ◦ T1 (Vn ), n inf θVi Θ + θV − Θ , d(I, T1 (V )) = T1 (V )= T1 (V1 )◦···◦T1 (Vn ) i=1

i

is there a velocity V ∗ ∈ L1 (0, 1; Θ) that achieves that infimum? Do we have a geodesic path between I and T1 (V ) that would be achieved by V ∗ ∈ L1 (0, 1; Θ)? Given a velocity field we have constructed a continuous path t → Tt (V ) : [0, 1] → F(Θ). At each t, there exists δ > 0 sufficiently small such that the path t → Tt+s (V ) ◦ Tt−1 (V ) : [0, δ] → F(Θ) is differentiable in t in the group F(Θ):  Tt+s (V ) ◦ Tt−1 (V ) − I 1 t+s V (r) ◦ Tr (V ) dr ◦ Tt−1 (V ) = s s t (2.17) −1 → V (t) ◦ Tt (V ) ◦ Tt (V ) = V (t) dF Tt ⇒ = V (t) a.e. in Θ. (2.18) dt With this definition a jump from I + θ1 to [I + θ2 ] ◦ [I + θ1 ] at time 1/2 becomes −1 (V ) − I = [I + θ2 ] ◦ [I + θ1 ] ◦ [I + θ1 ]−1 − I = I + θ2 − I = θ2 , T1/2+s (V ) ◦ T1/2

that is, a Dirac delta function θ2 in the tangent space Θ to F(Θ) at 1/2. Now the norm of the velocity V = θ2 δ1/2 in the space of measures becomes  V M1 ((0,1);Θ) =

1

θ2 Θ δ1/2 dt = θ2 Θ . 0

If we have n such jumps, V M1 ((0,1);Θ) =



θi Θ .

1=1,...,n

The metric of Micheletti is the infimum of this norm over all finite factorizations. A consequence of this analysis is that a factorization of an element I + θ of the form (I +θ1 )◦(I +θ2 )◦· · ·◦(I +θn ) is in fact a path t → Tt : [0, 1] → F(Θ) of bounded

2. Metrics on Transformations Generated by Velocities

167

variations in F(Θ) between I and I +θ by assigning times 0 < t1 < t2 < · · · < tn < 1 to each jump in the group F(Θ):  0 ≤ t < t1 ,   I,    , t1 ≤ t < t2 , I + θ  1  def t2 ≤ t < t3 , Tt (V ) = (I + θ1 ) ◦ (I + θ2 ),    ...     (I + θ1 ) ◦ (I + θ2 ) ◦ · · · ◦ (I + θn ), tn ≤ t ≤ 1, n dF Tt def = V (t), V M1 ((0,1);Θ) = θi δ(ti ) ⇒ θi Θ , V (t) = dt 1=1,...,n i=1

where the time-derivative is taken in the group sense (2.17) and V is a bounded measure. With this interpretation in mind, for V ∈ L1 (0, 1; Θ) the metric of Micheletti should reduce to 3  1  13 3 dF Tt (v) 3 3 3 d(I, T1 (V )) = inf v(t)Θ dt = inf 3 dt 3 dt, v∈L1 (0,1;Θ) 0 v∈L1 (0,1;Θ) 0 Θ T1 (v)=T1 (V )

T1 (v)=T1 (V )

where the derivative is in the group F(Θ). As a result we could talk of a geodesic path in F(Θ) between I and T1 (V ). If this could be justified, the next question is whether GΘ is complete in (F(Θ), d) or complete with respect to the metric constructed from the function  1 def dG (I, T1 (V )) = inf v(t)Θ dt, (2.19) 1 v∈L (0,1;Θ) T1 (v)=T1 (V )

0

dG (T1 (V ), T1 (W )) = d(T1 (V ) ◦ T1 (W )−1 , I). def

(2.20)

By definition dG (T1 (V )−1 , I) = dG (T1 (V ), I),

dG (T1 (V ), T1 (W )) = dG (T1 (W ), T1 (V ))

and dG is right-invariant. The infimum is necessary since the velocity taking I to T1 (V ) is not unique. Theorem 2.2. Let Θ be equal to C k,1 (RN , RN ), C k+1 (RN , RN ), C0k+1 (RN , RN ), or Bk+1 (RN , RN ), k ≥ 0. Then dG is a right-invariant metric on GΘ . Remark 2.2. The theorem is also true for the right-invariant metric dG (T1 (V ), I) = d(T1 (W ), T1 (V )) + def

inf

W ∈L1 (0,1;Θ) T1 (W )=T1 (V )

W L1 (0,1;Θ) ,

dG (T1 (V ), T1 (W )) = dG (T1 (V ) ◦ T1 (W )−1 , I). def

(2.21)

(2.22)

168

Chapter 4. Transformations Generated by Velocities

Proof. We give only the proof for C 0,1 (RN , RN ). It remains to show that the triangle inequality holds and that dG (T1 (W ), T1 (V )) = 0 if and only if T1 (W ) = T1 (V ). Given U, V, W in L1 (0, 1; Θ), T1 (W ) ◦ T1 (U )−1 = T1 (W ) ◦ T1 (V )−1 ◦ T1 (V ) ◦ T1 (U )−1 . For all v ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (W ) ◦ T1 (V )−1 and for all u ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (V ) ◦ T1 (U )−1 , the concatenation v ∗ u is such that T1 (v ∗ u) = T1 (W ) ◦ T1 (U )−1 . By definition dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ v ∗ uL1 = vL1 + uL1 . By taking infima with respect to v and u we get the triangle inequality dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ dG (T1 (W ) ◦ T1 (V )−1 , I) + dG (T1 (V ) ◦ T1 (U )−1 , I). If T1 (V ) = I, then inf w∈L1 (0,1;Θ) wL1 (0,1;Θ) = 0 and dG (T1 (V ), I) = 0. ConT1 (w)=I

versely, if dG (T1 (V ), I) = 0, then inf v∈L1 (0,1;Θ) vL1 (0,1;Θ) = 0 and there exists a T1 (v)=T1 (V )

sequence vn ∈ L1 (0, 1; Θ) such that vn → 0 in L1 (0, 1; Θ) and T1 (vn ) = T1 (V ). By continuity of T1 (vn ) − I with respect to vn from Theorem 2.1 (iv), T1 (V ) − I = T1 (vn ) − I → T1 (0) − I = 0 and T1 (V ) = I. Getting the completeness with the metric dG and even dG is not obvious. For a Cauchy sequence {T1 (Vn )}, the L1 -convergence of the velocities {Vn } to some V would yield the convergence of {T1 (Vn )} to T1 (V ) and even the convergence of the geodesics, but the metrics do not seem sufficiently strong to do that. To appreciate this point, we first establish an inequality that really follows from the triangle inequality for dG . Associate with W a w such that T1 (w) = T1 (W ), with V a v such that T1 (v) = T1 (V ), and let u be such that T1 (u) = T1 (W )◦T1 (V )−1 . Then the concatenation w ∗ v− is such that T1 (v ∗ u) = T1 (W ). From this we get the inequalities inf T1 (w)=T1 (W )

wL1 ≤ v ∗ uL1 = vL1 + uL1

and by taking infima we get the inequality inf T1 (w)=T1 (W )

wL1 ≤

inf

T1 (u)=T1 (W )◦T1 (V )−1

uL1 +

inf T1 (v)=T1 (V )

vL1

inf ⇒ inf wL1 − inf vL1 ≤ uL1 . T1 (w)=T1 (W ) T1 (v)=T1 (V ) T1 (u)=T1 (W )◦T1 (V )−1 Pick any Cauchy sequence {T1 (Vn )}. For any ε > 0, there exists N such that for all n, m > N 1 1 w − inf v uL1 < ε. inf inf L L ≤ −1 T1 (w)=T1 (Vm )

T1 (v)=T1 (Vn )

T1 (u)=T1 (Vm )◦T1 (Vn )

Therefore, there exists a sequence {vn } such that T1 (vn ) = T1 (Vn ) and 0 ≤ vn L1 −

inf T1 (v)=T1 (Vn )

vL1 < ε

⇒ |vm L1 − vn L1 | < 3ε. The sequence {vn L1 } is bounded in L1 (0, 1; Θ), but this does not seem sufficient to get the convergence of the vn ’s in L1 (0, 1; Θ).

2. Metrics on Transformations Generated by Velocities

169

One could think of many ways to strengthen the metric. For instance, one could introduce the minimization of the whole W 1,1 (0, 1; Θ)-norm of the path t → Tt (v) with respect to v from which the minimizing velocity v(t) = ∂Tt /∂t ◦ Tt−1 could be recovered, but the triangle inequality would be lost.

2.3

Constructions of Azencott and Trouv´ e

In 1994 R. Azencott [1] defined the following program. Given a smooth manifold M , consider the time continuous curve T = (Tt )0≤t≤1 in Aut(M ) solutions of ∂Tt = V (t) ◦ Tt , ∂t

T0 = I,

(2.23)

for a continuous time family V (t) of vector fields in a Hilbert space H ⊂ Aut(M ) and define  1 def l(φ) with l(φ) = |V (t)|I dt. (2.24) d(I, φ) = inf V ∈C([0,1];H) T1 (V )=φ

0

A fully rigorous construction of this distance, in the context of infinite-dimensional ´ [2] in 1995 and [3] in 1998. He shows Lie groups, was performed by A. Trouve that the subset of φ ∈ Aut(M ) for which d(I, φ) can be defined is a subgroup A of invertible mappings. He extends this distance between two arbitrary mappings def φ and ψ by right-invariance: d(φ, ψ) = d(I, ψ ◦ φ−1 ). Hence, in this extended framework, the new problem should be to find in A  1 1 φˆ = argmin |f˜ − f ◦ φ|2 dx + d(Id, φ)2 , 2 M 2 where M is a smooth manifold, the function f : M → R is a high dimensional representation template, and f˜ is a new observed image belonging to this family (cf. ´ [3]). This approach was further developed by the group of Azencott A. Trouve (cf., for instance, L. Younes [1]). ´ [3] in 1998: Quoting from the introduction of A. Trouve . . . Hence, for a large deformation φ, one can consider φ as the concatenation of small deformations φui . More precisely, if φ = Φn , where the Φk are recursively defined by Φ0 = Id and φk+1 = φuk+1 ◦ Φk , the family Φ = (Φk )0≤k≤n defines a polygonal ) line in Aut(M ) whose length l(Φ) can be approximated by l(φ) = nk=1 |uk |e . At a naive level, one could define the distance d(Id, φ) as the infimum of l(Φn ) for all family Φ such that Φn = φ. For a more rigorous setting, one should consider the time continuous curve Φ = (Φt )0≤t≤1 in Aut(M ). . . . In fact what he describes as a naive approach is the very precise construction of A. M. Micheletti [1] in 1972. He is certainly not the first author who has overlooked that paper in Italian, where the central completeness result that is of interest to us was just a lemma in her analysis of the continuity of the first eigenvalue of the Laplacian. We brought it up only in the 2001 edition of this book.

170

Chapter 4. Transformations Generated by Velocities

By introducing a Hilbert space H ⊂ Θ = C 0,1 (RN , RN ) and minimizing over L (0, 1; H) rather than L1 (0, 1; Θ), the bounded sequence {vn } lives in L2 (0, 1; H), where there exist a v ∈ L2 (0, 1; H) and a weakly converging subsequence to v. By using the complete metric of Micheletti, we had the existence of a F ∈ F(Θ) such that T1 (vn ) → F , but now we have a candidate v for which F = T1 (v). This is very much analogous to the Hilbert uniqueness method of J.-L. Lions [1] in the context of the controllability of partial differential equations (cf. also R. Glowinski and J.-L. Lions [1, 2]). 2

Remark 2.3. It is important to understand that finding a complete metric of Courant type for the various spaces of diffeomorphisms is a much less restrictive problem than constructing a metric from geodesics. The existence of a geodesic is not required for the Courant metrics.

3 3.1

Semiderivatives via Transformations Generated by Velocities Shape Function

All spaces F(Θ) and F(Θ)/G(Ω0 ) in Chapter 3 associated with the family of sets X (Ω0 ) = {F (Ω0 ) : ∀F ∈ F(Θ)}

(3.1)

are nonlinear and nonconvex and the elements of F(Θ)/G(Ω0 ) are equivalence classes of transformations. Defining a differential with respect to such spaces is similar to defining a differential on an infinite-dimensional manifold. Fortunately, the tangent space is invariant and equal to Θ in every point of F(Θ). This considerably simplifies the analysis. Definition 3.1. Given a nonempty subset D of RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D will be referred to as the underlying holdall or universe. A shape functional is a map J :A→E (3.2) from some admissible family A of sets in P(D) into a topological space E. For instance, A could be the set X (Ω). D can represent some physical or mechanical constraint, a submanifold of RN , or some mathematical constraint. In most cases, it can be chosen large and as smooth as necessary for the analysis. In the unconstrained case, D is equal to RN .

3.2

Gateaux and Hadamard Semiderivatives

Consider the solution (flow) of the differential equation dx (t, X) = V (t, x(t, X)), t ≥ 0, dt

x(t, X) = X,

(3.3)

3. Semiderivatives via Transformations Generated by Velocities

171

V x(t)

X Ωt = Tt (Ω)

Ω

Figure 4.1. Transport of Ω by the velocity field V . def

for velocity fields V (t)(x) = V (t, x) (cf. Figure 4.1). It generates the family of transformations {Tt : RN → RN : t ≥ 0} defined as follows: def

X → Tt (X) = x(t; X) : RN → RN .

(3.4)

Given an initial set Ω ⊂ RN , associate with each t > 0 the new set def

Ωt = Tt (Ω) = {Tt (X) : ∀X ∈ Ω} .

(3.5)

This perturbation of the initial set is the basis of the velocity (speed ) method . The choice of the terminology “velocity” to describe this method is accurate but may become ambiguous in problems where the variables involved are themselves “physical velocities”: this situation is commonly encountered in continuum mechanics. In such cases it may be useful to distinguish between the “artificial velocity” and the “physical velocity.” This is at the origin of the terminology speed method, which has often been used in the literature. The latter terminology is convenient, but is not as accurate as the velocity method. We shall keep both terminologies and use the one that is most suitable in the context of the problem at hand. It is important to work with the weakest notion of semiderivative that preserves the basic elements of the classical differential calculus such as the chain rule for the differentiation of the composition of functions. It should be able to handle the norm of a function or functions defined as an upper or lower envelope of a family of differentiable functions. Since the spaces of geometries are generally nonlinear and nonconvex, we also need a notion of semiderivative that yields differentials in the tangent space as for manifolds. We are looking for a very general notion of semidifferential but not more! In that context the most suitable candidate is the Hadamard semiderivative3 that will be introduced later. Important nondifferentiable functions are Hadamard semidifferentiable and the chain rule is verified. 3 In his 1937 paper, M. Fre ´chet [1] extends to function spaces the Hadamard derivative and promotes it as more general than his own Fr´echet derivative since it does not require the space to have a metric in infinite-dimensional spaces. In the same paper, he also introduces a relaxation of the Hadamard derivative that is almost a semiderivative. A precise definition of the Hadamard semiderivative explicitly appears in J.-P. Penot [1] in 1978 and in A. Bastiani [1] in 1964 as a well-known notion.

172

Chapter 4. Transformations Generated by Velocities

We proceed step by step. A first candidate for the shape semiderivative of J at Ω in the direction V would be def

dJ(Ω; V ) = lim

t0

J(Ωt (V )) − J(Ω) t

(when the limit exists).

(3.6)

This very weak notion depends on the history of V for t > 0 and will be too weak for most of our purposes. For instance, its composition with another function would not verify the chain rule. To make it compatible with the usual notion of semiderivative in a direction θ of the tangent space Θ to F(Θ) in F (Ω), it must be strengthened as follows. Definition 3.2. Given θ ∈ Θ def

dJ(Ω; θ) =

J(Ωt (V )) − J(Ω) , t ([0,τ ];Θ)

lim

0

V ∈C V (0)=θ t0

(3.7)

where the limit depends only on θ and is independent of the choice of V for t > 0. Equivalently, the limit dJ(Ω; θ) is independent of the way we approach Ω for a fixed direction θ. This is precisely the adaptation of the Hadamard semiderivative.4 Definitions based on a perturbation of the identity def

Tt (X) = x(t) = X + t θ(X),

t ≥ 0,

(3.8)

can be recast in the velocity framework by choosing the velocity field V (t) = θ◦Tt−1 , that is, ∀x ∈ RN , ∀t ≥ 0,

V (t, x) = θ(Tt−1 (x)) = θ ◦ (I + tθ)−1 (x), def

(3.9)

which requires the existence of the inverse for sufficiently small t. With that choice for each X, x(t) = Tt (X) is the solution of the differential equation   dx (t) = V t, x(t) , dt

x(0) = X.

(3.10)

Under appropriate continuity and differentiability assumptions, ∀x ∈ RN , ∀x ∈ RN ,

def

V (0)(x) = V (0, x) = θ(x), def ∂V ˙ (t, x) V (0)(x) = = − [Dθ(x)] θ(x), ∂t t=0

(3.11) (3.12)

where Dθ(x) is the Jacobian matrix of θ at the point x. In compact notation, V (0) = θ

and V˙ (0) = − [Dθ] θ.

(3.13)

This last computation shows that at time 0, the points of the domain Ω are simultaneously affected by the velocity field V (0) = θ and the acceleration field 4 Cf. footnote 2 in section 2 of Chapter 9 for the definitions and properties and the comparison of the various notions of derivatives and semiderivatives in Banach spaces.

3. Semiderivatives via Transformations Generated by Velocities

173

V˙ (0) = −[Dθ]θ. The same result will be obtained without acceleration term by replacing I + tθ by the transformation Tt generated by the solutions of the differential equation dx (t) = θ(x(t)), dt

x(0) = X,

def

Tt (X) = x(t),

for which V (0) = θ and V˙ (0) = 0. Under suitable assumptions the two methods will produce the same first-order semiderivative. The definition of a shape semiderivative def

dJ(Ω; θ) = lim

t0

J([I + tθ]Ω) − J(Ω) t

(3.14)

by perturbation of the identity is similar to a Gateaux semiderivative that will not verify the chain rule. Moreover, second-order semiderivatives will differ by an acceleration term that will appear in the expression obtained by the method of perturbation of the identity.

3.3

Examples of Families of Transformations of Domains

In section 3.2 we have formally introduced a notion of semiderivative of a shape functional via transformations generated by flows of velocity fields and perturbations of the identity operator as special cases of the velocity method. Before proceeding with a more abstract treatment, we present several examples of definitions of shape semiderivatives that can be found in the literature. We consider special classes of domains (C ∞ , C k , Lipschitzian), Cartesian graphs, polar coordinates, and level sets that provide classical examples of parametrized and/or constrained deformations. In each case we construct the associated underlying family (not necessarily unique) of transformations {Tt : 0 ≤ t ≤ τ }. 3.3.1

C ∞ -Domains

Let Ω be an open domain of class C ∞ in RN . Recall from section 5 of Chapter 2 that in any point x ∈ Γ the unit outward normal is given by def

∀y ∈ Γx = U(x) ∩ Γ,

n(y) =

mx (y) , |mx (y)|

(3.15)

where mx (y) = − ∗(Dhx )−1 (h−1 x (y)) eN in U(x) ⇒ n=−

∗

(3.16)

−1

(Dhx ) eN ◦ h−1 x . | ∗(Dhx )−1 eN |

(3.17)

When Γ is compact, it is possible to find a finite sequence of points {xj : 1 ≤ j ≤ J} in Γ such that def

Γ⊂U =

J * j=1

Uj ,

def

Uj = U(xj ).

174

Chapter 4. Transformations Generated by Velocities

As in the definition of the boundary integral, associate with {Uj } a partition of unity {rj }: rj ∈ D(Uj ),

0 ≤ rj ≤ 1,

J

rj = 1 in U0

j=1

for some neighborhood U0 of Γ such that Γ ⊂ U0 ⊂ U0 ⊂ U. For the C ∞ -domain Ω the normal satisfies n=

J

rj n =

j=1

since

∀j,

J j=1

rj

mj ∈ C ∞ (Γ; RN ) |mj |

mj ◦ hj ∈ C ∞ (B; RN ). |mj |

Given any ρ ∈ C ∞ (Γ) and t ≥ 0, consider the following perturbation Γt of Γ along the normal field n: def

Γt = {x ∈ RN : x = X + tρ(X)n(X), ∀X ∈ Γ}.

(3.18)

We claim that for τ sufficiently small and all t, 0 ≤ t ≤ τ , the set Γt is the boundary of a C ∞ -domain Ωt by constructing a transformation Tt of RN which maps Ω onto Ωt and Γ onto Γt . First construct an extension N ∈ D(RN , RN ) of the normal field n on Γ. Define J def rj mj ∈ D(U, RN ). (3.19) m = j=1

By construction

m ∈ C ∞ (Γ; RN ),

(3.20)

m = 0 on Γ, and there exists a neighborhood U1 of Γ contained in U0 where m = 0 since m is at least C 1 . Now construct a function r0 in D(U1 ), 0 ≤ r0 (x) ≤ 1, and a neighborhood V of Γ such that r0 = 1 in V and Γ ⊂ V ⊂ V ⊂ U1 . Define the vector field ∀x ∈ RN ,

def

N (x) = r0 (x)

m(x) . |m(x)|

(3.21)

Hence N belongs to D(RN , RN ) since supp N ⊂ V is compact. Moreover m N= in V ⇒ N (x) = n(x) on Γ = Γ ∩ V. |m| For each j, ρ ◦ hj ∈ C ∞ (B0 ) and the extensions ρhj (ζ) = ρhj (ζ  , ζN ) = ρ(hj (ζ  , 0)), def

∀ζ ∈ B,

def

ρ˜j = ρhj ◦ gj ,

3. Semiderivatives via Transformations Generated by Velocities

175

belong, respectively, to C ∞ (B) and C ∞ (Uj ). Then def

ρ˜ =

J

rj ρ˜j ∈ D(RN )

j=1

is an extension of ρ from Γ to RN with compact support since supp rj ⊂ Uj ⊂ U. Define the following transformation of RN : def

Tt (X) = X + t ρ˜(X)N (X),

t ≥ 0.

(3.22)

By construction, ρ˜ N is uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists 0 < τ such that Tt is bijective and bicontinuous from RN onto itself. As a result from J. Dugundji [1] for 0 ≤ t ≤ τ Ωt = Tt (Ω) = [I + t˜ ρ N ](Ω), ∂Ωt = Tt (∂Ω) = [I + t˜ ρ N ](∂Ω) = [I + tρ n](∂Ω) = Γt . Since the domain Ωt is specified by its boundary Γt , it depends only on ρ and not on its extension ρ˜. The special transformation Tt introduced here is of class C ∞ , that is, Tt ∈ C ∞ (RN , RN ), and 1t [Tt − I] is proportional to the normal field n on Γ, but it is not proportional to the normal nt on Γt for t > 0. In other words, at t = 0 the deformation is along n, but at t > 0 the deformation is generally not along nt . If J(Ω) is a real-valued shape functional defined on C ∞ -domains in D, the semiderivative (if it exists) is defined as follows: for all ρ ∈ C ∞ (Γ)   J (I + t˜ ρN )Ω − J(Ω) def dn J(Ω; ρ˜) = lim . (3.23) t0 t It turns out that this limit depends only on ρ and not on its extension ρ˜. 3.3.2

C k -Domains

When Ω is a domain of class C k with boundary Γ, the normal field n belongs to C k−1 (Γ, RN ). Therefore, choosing deformations along the normal would yield transformations {Tt } mapping C k -domains Ω onto C k−1 -domains Ωt = Tt (Ω). The obvious way to deal with C k -domains is to relax the constraint that the perturbation ρ˜N be carried by the normal. Choose vector fields θ in D k (RN , RN ) and consider the family of transformations Tt = I + tθ,

Ωt = Tt (Ω),

t ≥ 0.

(3.24)

This is a generalization of the family of transformations (3.22) in section 3.3.1 from ρ˜N to θ. For k ≥ 1, θ is again uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists τ > 0 such that Tt is bijective and bicontinuous from RN onto itself. Thus for 0 ≤ t ≤ τ , Ωt = Tt (Ω) = [I + tθ](Ω)

and ∂Ωt = Tt (∂Ω) = [I + tθ](∂Ω).

176

Chapter 4. Transformations Generated by Velocities

A more restrictive approach to get around the lack of sufficient smoothness of the normal n to Γ would be to introduce a transverse field p on Γ such that p ∈ C k (Γ; RN ),

∀x ∈ Γ,

p(x) · n(x) > 0.

Given p and ρ ∈ C k (Γ) define for t ≥ 0  def

Γt = x ∈ RN : x = X + tρ(X)p(X), ∀X ∈ Γ .

(3.25)

(3.26)

Choosing C k -extensions ρ˜ and p˜ of ρ and p we can go back to the case where θ = ρ˜p˜ ∈ Dk (RN , RN ).

(3.27)

For any θ ∈ Dk (RN , RN ) the semiderivative is defined as   J (I + tθ)(Ω) − J(Ω) def dk J(Ω; θ) = lim . t0 t 3.3.3

(3.28)

Cartesian Graphs

In many applications it is convenient to work with domains Ω which are the hypograph of some positive function γ in Cartesian coordinates. Such domains are typically of the form   def (3.29) Ω = (x , xN ) ∈ RN : x ∈ U ⊂ RN −1 and 0 < xN < γ(x ) , where U is a connected open set in RN −1 and γ ∈ C 0 (U ; R+ ) is a positive function. Many free boundary and contact problems are formulated over such domains. Usually the domains Ω (and hence the functions γ) will be constrained. The function γ can be specified on U \U or not. In some examples the derivative of γ could also be specified, that is, ∂γ/∂ν = g on ∂U , where U is smooth and ν is the outward unit normal field along ∂U . When γ (resp., ∂γ/∂ν) is specified along ∂U , the directions of deformation µ are chosen in C 0 (U ; R+ ) such that µ = 0 (resp., ∂µ/∂ν = 0) on ∂U.

(3.30)

For small t ≥ 0 and each such µ, define the perturbed domain

 Ωt = (x , xN ) ∈ RN : 0 < xN < γ(x ) + tµ(x ) and the family of transformations (x , xN ) → Tt (x , xN ) =



x , xN +

(3.31)



xN tµ(x ) γ(x )

: Ω → Ωt ,

(3.32)

which can be extended to a neighborhood D of Ω containing the perturbed domains Ωt , 0 ≤ t ≤ t1 , for some small t1 > 0. In general, D will be such that D = U × [0, L] for some L > 0. This construction is also appropriate for domains that are Lipschitzian or of class C k , 1 ≤ k ≤ ∞, that is, when γ is a Lipschitzian or a C k -function. Again the

3. Semiderivatives via Transformations Generated by Velocities transformation Tt is equal to I + tθ, where   µ(x ) def . θ(x , xN ) = 0, xN γ(x )

177

(3.33)

But of course this θ is not the only choice for that Ωt = Tt (Ω). For instance, let λ : R+ → R+ be any smooth increasing function such that λ(0) = 0 and λ(1) = 1. Then we could consider the transformations     xN def  ) . (3.34) µ(x Tt (x , xN ) = x , xN + tλ γ(x ) This example illustrates the general principle that the transformation Tt of D such that Ωt = Tt (Ω) is not unique. This means that, at least for smooth domains, only the trace Tt |Γt on Γt is important, while the displacement of the inner points does not contribute to the definition of Ωt . Nevertheless this statement is to be interpreted with caution. Here we implicitly assume that the objective function J(Ω) and the associated constraints are only a function of the shape of Ω. However, in some problems involving singularities at inner points of Ω (e.g., when the solution y(Ω) of the state equation has a singularity or when some constraints on the domain are active), the situation might require a finer analysis. One such example is the internal displacement of the interior nodes of a triangularization τh when the solution y(Ω) of a partial differential equation is approximated by a piecewise polynomial solution over the triangularized domain Ωh in the finite element method. Such a displacement does not change the shape of Ωh , but it does change the solution yh of the problem. When the displacement of the interior nodes is a priori parametrized by the boundary nodes the solution yh will depend only on the position of the boundary nodes, but the interior nodes will contribute through the choice of the specified parametrization. 3.3.4

Polar Coordinates and Star-Shaped Domains

In some examples domains are star-shaped with respect to a point. Since a domain can always be translated, there is no loss of generality in assuming that this point is the origin. Then such domains Ω can be parametrized as follows:  def

(3.35) Ω = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) , where SN −1 is the unit sphere in RN ,  def

SN −1 = x ∈ RN : |x| = 1 ,

(3.36)

and f : SN −1 → R+ is a positive continuous mapping from SN −1 such that def

m = min {f (ζ) : ζ ∈ SN −1 } > 0.

(3.37)

Given any g ∈ C 0 (SN −1 ) and a sufficiently small t ≥ 0 the perturbed domains are defined as  def

Ωt = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) + tg(ζ) . (3.38)

178

Chapter 4. Transformations Generated by Velocities

For example, choose t, 0 ≤ t ≤ t1 , for some m t1 = >0 gC 0 (SN −1 )

(3.39)

and define the transformation Tt as Tt (X) = .0,

/

Tt (X) = ρ +

ρ t f (ζ) g(ζ)

if X = 0, ζ, if X = ρζ = 0.

(3.40)

As in the previous example, Tt is not unique and for any continuous increasing function λ : R+ → R+ such that λ(0) = 0 and λ(t) = 1 the transformation Tt (X) = .0,



Tt (X) = ρ + tλ

ρ f (ζ)



/

if X = 0,

g(ζ) ζ, if X = ρζ = 0,

(3.41)

yields the same domain Ωt . 3.3.5

Level Sets

In sections 3.3.1 to 3.3.4, the perturbed domain Ωt always appears in the form Ωt = Tt (Ω), where Tt is a bijective transformation of RN and Tt is of the form I +tθ. In some free boundary problems (e.g., plasma physics, propagation of fronts) the free boundary Γ is a level curve of a smooth function u defined over an open domain D. Assume that D is bounded open with smooth boundary ∂D. Let u ∈ C 2 (D) be a positive function on D such that u ≥ 0 in D,

u = 0 on ∂D,

∃ a unique xu ∈ D such that ∀x ∈ D − {xu } , |∇u(x)| > 0.

 If m = max u(x) : x ∈ D , then for each t in [0, m[ the level set Γt = u−1 (t) 2

(3.42)

(3.43)

N

is a C -submanifold of R in D, which is the boundary of the open set Ωt = {x ∈ D : u(x) > t} .

(3.44)

By definition Ω0 = D, for all t1 > t2 , Ωt1 ⊂ Ωt2 , and the domains Ωt converge in the Hausdorff complementary topology5 to the point xu . The outward unit normal field on Γt is given by x ∈ Γt ,

nt (x) = −|∇u(x)|−1 ∇u(x).

(3.45)

This suggests introducing the velocity field ∀x ∈ D − {xu } ,

V (x) = |∇u(x)|−2 ∇u(x), def

(3.46)

which is continuous everywhere but at x = xu . If V were continuous everywhere, then for each X, the path x(t; X) generated by the differential equation   dx (t) = V x(t) , x(0) = X (3.47) dt 5 Cf.

section 2 of Chapter 6.

3. Semiderivatives via Transformations Generated by Velocities would have the property that

  u x(t) = u(X) + t

179

(3.48)

since formally

   dx d  u x(t) = ∇u x(t) · (t) = 1. (3.49) dt dt This means that the map X → Tt (X) = x(t; X) constructed from (3.47) would map the level set

 (3.50) Γ0 = X ∈ D : u(X) = 0

onto the level set

 Γt = x ∈ D : u(x) = t

(3.51)

and eventually Ω0 onto Ωt . Unfortunately, it is easy to see that this last property fails on the function u(x) = 1 − x2 defined on the unit disk. To get around this difficulty, introduce for some arbitrarily small ε, 0 < ε < m/2, an infinitely differentiable function ρε : RN → [0, 1] such that ρε (x) =

0, 1,

if |∇u(x)| < ε, if |∇u(x)| > 2ε,

(3.52)

x ∈ D.

(3.53)

and the velocity Vε (x) = ρε (x)V (x), As above, define the transformation X → Ttε (X) = x(t; X),

(3.54)

where x(t; X) is the solution of the differential equation   dx (t) = Vε x(t) , t ≥ 0, x(0) = X ∈ D. (3.55) dt For 0 ≤ t < m − 2ε, Tt maps Γ0 onto Γt ; for 0 ≤ s < m − ε such that s + t < m − ε, Tt maps Γs onto Γt+s . However, for s > m − ε, Tt is the identity operator. As a result, for 0 ≤ t < m − 2ε, Tt (Ω0 ) = Ωt

and Tt (Γ0 ) = Γt .

(3.56)

Of course ε > 0 is arbitrary and we can make the construction for t’s arbitrary close to m. This is an example that can be handled by the velocity (speed) method and not by a perturbation of the identity. Here the domains Ωt are implicitly constrained to stay within the larger domain D. We shall see in section 5 how to introduce and characterize such a constraint. Another example of description by level sets is provided by the oriented distance function bΩ for some open domain Ω of class C 2 with compact boundary Γ (cf. Chapter 7). We shall see that there exists h > 0 and a neighborhood

 Uh (Γ) = x ∈ RN : |bΩ (x)| < h such that bΩ ∈ C 2 (Uh (Γ)). Then for 0 ≤ t < h the flow corresponding to the velocity field V = ∇bΩ maps Ω and its boundary Γ onto



 Tt (Ω) = Ωt = x ∈ RN : bΩ (x)| < t , Tt (Γ) = Γt = x ∈ RN : bΩ (x)| = t .

180

4

Chapter 4. Transformations Generated by Velocities

Unconstrained Families of Domains

´sio [12, Here we study equivalences between the velocity method (cf. J.-P. Zole 8]) and methods using a family of transformations. In section 4.1, we give some general conditions to construct a family of transformations of RN from a velocity field. Conversely, we show how to construct a velocity field from a family of transformations of RN . In section 4.2, this construction is applied to Lipschitzian perturbations of the identity. In section 4.3, the various equivalences of section 4.1 are specialized to velocities in C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0.

4.1

Equivalence between Velocities and Transformations

Let the real number τ > 0 and the map V : [0, τ ] × RN → RN be given. If t is interpreted as an artificial time, the map V can be viewed as the (time-dependent) velocity field {V (t) : 0 ≤ t ≤ τ } defined on RN : def

x → V (t)(x) = V (t, x) : RN → RN . Assume that there exists τ = τ (V ) > 0 such that   ∀x ∈ RN , V (·, x) ∈ C [0, τ ]; RN , (V) ∃c > 0, ∀x, y ∈ RN , V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

(4.1)

(4.2)

where V (·, x) is the function t → V (t, x). Note that V is continuous on [0, τ ] × RN . Hence it is uniformly continuous on [0, τ ] × D for any bounded open subset D of RN and V (·) ∈ C([0, τ ]; C(D; RN )).

(4.3)

Associate with V the solution x(t; V ) of the vector ordinary differential equation   dx (t) = V t, x(t) , dt

t ∈ [0, τ ],

x(0) = X ∈ RN

(4.4)

and define the transformations def

X → Tt (V )(X) = xV (t; X) : RN → RN

(4.5)

and the maps (whenever the inverse of Tt exists) def

(t, X) → TV (t, X) = Tt (V )(X) : [0, τ ] × RN → RN ,

(4.6)

(t, x) → TV−1 (t, x) = Tt−1 (V )(x) : [0, τ ] × RN → RN .

(4.7)

def

Notation 4.1. In what follows we shall drop the V in TV (t, X), TV−1 (t, x), and Tt (V ) whenever no confusion arises.

4. Unconstrained Families of Domains

181

Theorem 4.1. (i) Under assumptions (V) the map T specified by (4.4)–(4.6) has the following properties:   ∀X ∈ RN , T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1) ∀X, Y ∈ RN , T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2) (T3)

∀t ∈ [0, τ ], X → Tt (X) = T (t, X) : RN → RN is bijective,   ∀x ∈ RN , T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0, ∀x, y ∈ RN ,

(4.8)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|.

(ii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) to (T3), the map def

(t, x) → V (t, x) =

 ∂T  t, Tt−1 (x) : [0, τ ] × RN → RN ∂t

(4.9)

satisfies conditions (V), where Tt−1 is the inverse of X → Tt (X) = T (t, X). If, in addition, T (0, ·) = I, then T (·, X) is the solution of (4.4) for that V . (iii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) and (T2) and T (0, ·) = I, then there exists τ  > 0 such that the conclusions of part (ii) hold on [0, τ  ]. A more general version of this theorem for constrained domains (Theorem 5.1) will be given and proved in section 5.1. Proof. (i) Conditions (T1) follow by standard arguments. Condition (T2). Associate with X in RN the function y(s) = Tt−s (X), Then

dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t.

0 ≤ s ≤ t,

y(0) = Tt (X).

(4.10)

y(0) = x ∈ RN

(4.11)

For each x ∈ RN , the differential equation dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t,

has a unique solution in C 1 ([0, t]; RN ). The solutions of (4.11) define the map def

x → St (x) = y(t) : RN → RN such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ RN ,

|St (y) − St (x)| ≤ c|y − x|.

(4.12)

182

Chapter 4. Transformations Generated by Velocities

In view of (4.10) and (4.11) St (Tt (X)) = y(t) = Tt−t (X) = X

⇒ St ◦ Tt = I on RN .

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (4.11) for some arbitrary x in RN . By definition dz (r) = V (r, z(r)), dr

z(0) = y(t, x),

and necessarily x = y(0; x) = z(t) = Tt (y(t; x)) = Tt (St (x)) ⇒ Tt ◦ St = I on RN

⇒ St = Tt−1 : RN → RN .

Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12), and we need only show that ∀x ∈ RN ,

T −1 (·, x) ∈ C([0, τ ]; RN ).

Given t in [0, τ ] pick an arbitrary sequence {tn }, tn → t. Then for each x ∈ RN there exists X ∈ RN such that Tt (X) = x and Ttn (X) → Tt (X) = x from the first condition (T1). But (x) − Tt−1 (x) = Tt−1 (Tt (X)) − Tt−1 (Tt (X)) Tt−1 n n (Tt (X)) − Tt−1 (Ttn (X)) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1 |Tt−1 (x) − Tt−1 (x)| = |Tt−1 (Tt (X)) − Tt−1 (Ttn (X)) | ≤ c|Tt (X) − Ttn (X)|, n n n and the last term converges to zero as tn goes to t. (ii) The first part of conditions (V) is satisfied since, for each x ∈ RN and t, s in [0, τ ], |V (t, x) − V (s, x)| ∂T ∂T ∂T ∂T (t, Tt−1 (x)) − (t, Ts−1 (x)) + (t, Ts−1 (x)) − (s, Ts−1 (x)) ≤ ∂t ∂t ∂t ∂t ∂T ∂T −1 −1 −1 −1 (t, Ts (x)) − (s, Ts (x)) . ≤ c|Tt (x) − Ts (x)| + ∂t ∂t

4. Unconstrained Families of Domains

183

Thus from (T3) and (T1) t → V (t, x) is continuous at s = t, and hence for all x in RN , V (., x) ∈ C([0, τ ]; RN ). The Lipschitzian property follows directly from the Lipschitzian properties (T1) and (T3): for all x and y in RN , ∂T ∂T −1 −1 |V (t, y) − V (t, x)| = (t, Tt (y)) − (t, Tt (x)) ∂t ∂t −1 −1 ≤ c Tt (y) − Tt (x) ≤ cc |y − x|. This proves that V satisfies condition (V). (iii) From (T1) and (T2) for f (t) = Tt − I and t ≥ s,  t ∂T ∂T (r, Tr (y)) − (r, Tr (x)) dr, f (t)(y) − f (t)(x) = ∂t ∂t 0  t c |Tr (y) − Tr (x)| dr ≤ c2 t |y − x|. |f (t)(y) − f (t)(x)| ≤ 0 

For τ = min{τ, 1/(2c )} and 0 ≤ t ≤ τ  , c(f (t)) ≤ 1/2, 2

g(t) = Tt−1 − I = (I − Tt ) ◦ Tt−1 = −f (t) ◦ [I + g(t)], (1 − c(f (t))) c(g(t)) ≤ c(f (t))

⇒ c(g(t)) ≤ 1 and c(Tt−1 ) ≤ 2,

and the second condition (T3) is satisfied on [0, τ  ]. The first one follows by the same argument as in part (i). Therefore the conclusions of part (ii) are true on [0, τ  ]. This equivalence theorem says that we can start either from a family of velocity fields {V (t)} on RN or a family of transformations {Tt } of RN provided that the map V , V (t, x) = V (t)(x), satisfies (V) or the map T , T (t, X) = Tt (X), satisfies (T1) to (T3). Starting from V , the family of homeomorphisms {Tt (V )} generates the family def

Ωt = Tt (V )(Ω) = {Tt (V )(X) : X ∈ Ω}

(4.13)

of perturbations of the initial domain Ω. Interior (resp., boundary) points of Ω are mapped onto interior (resp., boundary) points of Ωt . This is the basis of the velocity method which will be used to define shape derivatives.

4.2

Perturbations of the Identity

In examples it is usually possible to show that the transformation T satisfies assumptions (T1) to (T3) and construct the corresponding velocity field V defined in (4.9). For instance, consider perturbations of the identity to the first (A = 0) or second order: for t ≥ 0 and X ∈ RN , def

Tt (X) = X + tU (X) +

t2 A(X), 2

(4.14)

where U and A are transformations of RN . It turns out that for Lipschitzian transformations U and A, assumptions (T1) to (T3) are satisfied in some interval [0, τ ].

184

Chapter 4. Transformations Generated by Velocities

Theorem 4.2. Let U and A be two uniform Lipschitzian transformations of RN : there exists c > 0 such that for all X, Y ∈ RN , |U (Y ) − U (X)| ≤ c|Y − X|

and

|A(Y ) − A(X)| ≤ c|Y − X|.

There exists τ > 0 such that the map T given by (4.14) satisfies conditions (T1) to (T3) on [0, τ ]. The associated velocity V given by     (t, x) → V (t, x) = U Tt−1 (x) + tA Tt−1 (x) : [0, τ ] × RN → RN (4.15) satisfies conditions (V) on [0, τ ]. Remark 4.1. Observe that from (4.14) and (4.15) V (0) = U,

∂V V˙ (0)(x) = (t, x) t=0 = A − [DU ]U, ∂t

(4.16)

where DU is the Jacobian matrix of U . The term V˙ (0) is an acceleration at t = 0 which will always be present even when A = 0, but it can be eliminated by choosing A = [DU ]U . Proof. (i) By the definition of T in (4.14), t → T (t, X) and t → ∂T (t, X) = U (X) + ∂t tA(X) are continuous on [0, ∞[. Moreover, for all X and Y ,   t2 |T (t, Y ) − T (t, X)| ≤ 1 + tc + c |Y − X| 2 and

∂T ∂T ≤ [c + tc] |Y − X|. (t, Y ) − (t, X) ∂t ∂t

Thus conditions (T1) are satisfied for any finite τ > 0. To check condition (T2), def

consider for any Y ∈ RN the mapping h(X) = Y −[Tt (X)−X]. For any X1 and X2 |h(X2 ) − h(X1 )| ≤ t|U (X2 ) − U (X1 )| +

t2 |A(X2 ) − A(X1 )| 2

t2 ≤ t c |X2 − X1 | + c |X2 − X1 | = t c [1 + t/2] |X2 − X1 |. 2

(4.17)

For τ = min{1, 1/(4c)}, and any t, 0 ≤ t ≤ τ , t c [1 + t/2] < 1/2 and h is a contraction. So for all 0 ≤ t ≤ τ and Y ∈ RN , there exists a unique X ∈ RN such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y and Tt is bijective. Therefore (T2) is satisfied in [0, τ  ]. The last part of the proof is the uniform Lipschitzian property of Tt−1 . In view of (4.17), for all t, 0 ≤ t ≤ τ ,

4. Unconstrained Families of Domains

185

t c [1 + t/2] < 1/2 and |Tt (X2 ) − X2 − (Tt (X1 ) − X1 )| = |h(X2 ) − h(X1 )| ≤

1 |X2 − X1 | 2

1 |X2 − X1 | 2 ⇒ |X2 − X1 | ≤ 2 |Tt (X2 ) − Tt (X1 )|.

⇒ |X2 − X1 | − |Tt (X2 ) − Tt (X1 )| ≤

In view of condition (T2) for all x and y, |Tt−1 (y) − Tt−1 (x)| ≤ 2 |Tt (Tt−1 (y)) − Tt (Tt−1 (x))| = 2 |y − x|.

(4.18)

To complete our argument we prove the continuity with respect to t for each x. Let X = Tt−1 (x). For any s in [0, τ ] Ts−1 (x) − Tt−1 (x) = Ts−1 (Tt (X)) − Tt−1 (Tt (X)) = Ts−1 (Tt (X)) − Ts−1 (Ts (X)), and in view of (4.18) |Ts−1 (x) − Tt−1 (x)| ≤ 2|Tt (X) − Ts (X)|. The continuity of Ts−1 (x) at s = t now follows from the continuity of Ts (X) at s = t. Thus conditions (T3) are satisfied.

4.3

Equivalence for Special Families of Velocities

In this section we specialize Theorem 4.1 to velocities in C k,1 (RN , RN ), C0k+1 (RN , RN ), and C k+1 (RN , RN ), k ≥ 0. The following notation will be convenient: def

f (t) = Tt − I,

f  (t) =

dTt , dt

g(t) = Tt−1 − I, def

whenever Tt−1 exists and the identities g(t) = −f (t) ◦ Tt−1 = −f (t) ◦ [I + g(t)], dTt ◦ Tt−1 = f  (t) ◦ Tt−1 = f  (t) ◦ [I + g(t)]. V (t) = dt Recall also for a function F : RN → RN the notation def

|F (y) − F (x)| and ∀k ≥ 1, |y − x| y=x

def

c(F ) = sup

ck (F ) =



c(∂ α F ).

|α|=k

Theorem 4.3. Let k ≥ 0 be an integer. (i) Given τ > 0 and a velocity field V such that V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(4.19)

186

Chapter 4. Transformations Generated by Velocities for some constant c > 0 independent of t, the map T given by (4.4)–(4.6) satisfies conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )) ∩ C([0, τ ]; C k,1 (RN , RN )),

ck (f  (t)) ≤ c, (4.20)

for some constant c > 0 independent of t. Moreover, conditions (T3) are satisfied and there exists τ  > 0 such that g ∈ C([0, τ  ]; C k (RN , RN )),

ck (g(t)) ≤ ct,

(4.21)

for some constant c independent of t. (ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.20) and T (0, ·) = I, there exists τ  > 0 such that the velocity field V (t) = f  (t) ◦ Tt−1 satisfies conditions (V) and (4.19) in [0, τ  ]. Proof. We prove the theorem for k = 0. The general case is obtained by induction over k. (i) By assumption on V , the conditions (V) given by (4.2) are satisfied and by Theorem 4.1 the corresponding family T satisfies conditions (T1) to (T3). Conditions (4.20) on f . For any x and s ≤ t  t Tt (x) − Ts (x) = V (r) ◦ Tr (x) dr, s  t |Tt (x) − Ts (x)| ≤ c|Tr (x) − Ts (x)| + |V (r) ◦ Ts (x)| dr, s  t c|f (r)(x) − f (s)(x)| + V (r)C dr. |f (t)(x) − f (s)(x)| ≤ s

By assumption on V and Gronwall’s inequality, ∀t, s ∈ [0, τ ],

f (t) − f (s)C ≤ c |t − s|

for another constant c independent of t. Moreover, |(f (t) − f (s))(y) − (f (t) − f (s))(x)| = |(Tt − Ts )(y) − (Tt − Ts )(x)|  t ≤ |V (r) ◦ Tr (y) − V (r) ◦ Tr (x)| dr s  t c|(Tr − Ts )(y) − (Tr − Ts )(x)| + c|Ts (y) − Ts (x)| dr ≤ s  t ≤ c|(f (r) − f (s))(y) − (f (r) − f (s))(x)| + cc |y − x| dr s

for some other constant c by the second condition (T1). Again by Gronwall’s inequality there exists another constant c such that |(f (t) − f (s))(y) − (f (t) − f (s))(x)| ≤ c|t − s| |y − x| ⇒ c(f (t) − f (s)) ≤ c |t − s| ⇒ f ∈ C([0, τ ]; C 0,1 (RN , RN )) and f (t) − f (s)C0,1 ≤ c |t − s|.

(4.22)

4. Unconstrained Families of Domains

187

Moreover, f  (t) = V (t) ◦ Tt and |f  (t)(x) − f  (s)(x)| ≤ |V (t)(Tt (x)) − V (s)(Tt (x))| + |V (s)(Tt (x)) − V (s)(Ts (x))| ≤ V (t) − V (s)C + c(V (s)) Tt − Ts C ≤ V (t) − V (s)C + c f (t) − f (s)C . Finally, |f  (t)(y) − f  (t)(x)| ≤ |V (t)(Tt (y)) − V (t)(Tt (x))| ≤ c(V (t)) |Tt (y) − Ts (x)| ≤ c c(Tt ) |y − x| and c(f  (t)) ≤ c for some new constant c independent of t. Therefore, f ∈ C 1 ([0, τ ]; C(RN , RN )) and c(f  (t)) ≤ c. Conditions (4.21) on g. Since conditions (T1) and (T2) are satisfied there exists τ  > 0 such that conditions (T3) are satisfied by Theorem 4.1 (iii). Moreover, from conditions (4.20) |g(t)(y) − g(t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t))|Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) (|g(t)(y) − g(t)(x)| + |y − x|) ⇒ (1 − c(f (t))) |g(t)(y) − g(t)(x)| ≤ c(f (t)) |y − x| ≤ ct |y − x|. Choose a new τ  = min{τ  , 1/(2c)}. Then for 0 ≤ t ≤ τ  , c(g(t)) ≤ 2ct. Now g(t) − g(s) = −f (t) ◦ [I + g(t)] + f (s) ◦ [I + g(s)], g(t) − g(s)C ≤f (t) ◦ [I + g(t)] − f (t) ◦ [I + g(s)]C + f (t) ◦ [I + g(s)] − f (s) ◦ [I + g(s)]C ≤c(f (t))g(t) − g(s)C + f (t) − f (s)C ≤ct g(t) − g(s)C + f (t) − f (s)C . For t in [0, τ  ], ct ≤ 1/2, and g(t) − g(s)C ≤ 2f (t) − f (s)C ⇒ g ∈ C([0, τ  ]; C(RN , RN )) and c(g(t)) ≤ 2ct. The conditions (4.20) on f are satisfied for k = 0. For k = 1 we start from the equation 

t

DTt − DTs =

DV (r) ◦ Tr DTr dr s

188

Chapter 4. Transformations Generated by Velocities

and use the fact that DTt−1 = [DTt ]−1 ◦ Tt−1 in connection with the identity Dg(t) = −Df (t) ◦ Tt−1 DTt−1 = −(Df (t)[DTt ]−1 ) ◦ Tt−1 . (ii) From conditions (4.20) on f , the transformation T satisfies conditions (T1). To check condition (T2) we consider two cases: k ≥ 1 and k = 0. For k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ  > 0 such that Tt is invertible for all t in [0, τ  ] and (T2) is satisfied in [0, τ  ]. In the case k = 0 consider for any Y the map h(X) = Y − f (t)(X). For any X1 and X2 , |h(X2 ) − h(X1 )| ≤ c(f (t)) |X2 − X1 |. But by assumption f ∈ C([0, τ ]; C 0,1 (RN )) and c(f (0)) = 0 since f (0) = 0. Hence there exists τ  > 0 such that c(f (t)) ≤ 1/2 for all t in [0, τ  ] and h is a contraction. So for all Y in RN there exists a unique X such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y, Tt is bijective, and condition (T2) is satisfied in [0, τ  ]. By Theorem 4.1 (iii) from (T1) and (T2), there exists another τ  > 0 for which conditions (T3) on g and (V) on V (t) = f  (t) ◦ Tt−1 are also satisfied. Moreover, we have seen in the proof of part (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions (4.20) and (4.21), |V (t)(y) − V (t)(x)| ≤ |f  (t)(Tt−1 (y)) − f  (t)(Tt−1 (x))| ≤ c(f  (t)) |Tt−1 (y) − Tt−1 (x)| ≤ c(f  (t)) [1 + c(g(t))] |y − x| ≤ c |y − x| and c(V (t)) ≤ c . Also |V (t)(x) − V (s)(x)| = |f  (t)(Tt−1 (x)) − f  (s)(Ts−1 (x))| ≤ |f  (t)(Tt−1 (x)) − f  (t)(Ts−1 (x))| + |f  (t)(Ts−1 (x)) − f  (s)(Ts−1 (x))| ≤ c(f  (t)) |Tt−1 (x) − Ts−1 (x)| + f  (t) − f  (s)C ≤ c g(t) − g(s)C + f  (t) − f  (s)C . Therefore, since both g and f  are continuous, V ∈ C([0, τ  ]; C(RN , RN )) and c(V (t)) ≤ c for some constant c independent of t. This proves the result for k = 0. As in part (i), for k = 1 we use the identity DV (t) = Df  (t) ◦ Tt−1 DTt−1 = (Df  (t)[DTt ]−1 ) ◦ Tt−1 and proceed in the same way. The general case is obtained by induction over k.

4. Unconstrained Families of Domains

189

We now turn to the case of velocities in C0k (RN , RN ). As in Chapter 2, it will be convenient to use the notation C0k for the space C0k (RN , RN ), C k (RN ) for the space C k (RN , RN ), and C k,1 (RN ) for the space C k,1 (RN , RN ). Theorem 4.4. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity field V such that V ∈ C([0, τ ]; C0k (RN , RN )),

(4.23)

the map T given by (4.4)–(4.6) satisfies conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C0k (RN , RN )).

(4.24)

Moreover, conditions (T3) are satisfied and there exists τ  > 0 such that g ∈ C([0, τ  ]; C0k (RN , RN )).

(4.25)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.24) and T (0, ·) = I, there exists τ  > 0 such that the velocity field V (t) = f  (t) ◦ Tt−1 satisfies conditions (V) and (4.23) on [0, τ  ]. Proof. As in the proof of Theorem 4.3, we prove only the theorem for k = 1. The general case is obtained by induction on k, the various identities on f , g, f  , and V , and the techniques of Lemma 2.4, Theorems 2.11 and 2.12, and Lemmas 2.5 and 2.6 of section 2.5 in Chapter 3. (i) By the embedding C01 (RN , RN ) ⊂ C 1 (RN , RN ) ⊂ C 0,1 (RN , RN ), it follows from (4.23) that V ∈ C([0, τ ]; C 0,1 (RN , RN )) and condition (4.19) of Theorem 4.3 are satisfied. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satisfied in some interval [0, τ  ], τ  > 0. Conditions (4.24) on f . It remains to show that f (t) and f  (t) belong to the subspace C0 (RN , RN ) of C(RN , RN ) and to prove the appropriate properties for Df (t) and Df  (t). Recall from the proof of the previous theorems that there exists c > 0 such that  t  t |f (t)(x)| ≤ c |V (r)(x)| dr ≤ c |(V (r) − V (0))(x)| dr + c t |V (0)(x)|. 0

0

By assumption on V (0), for ε > 0 there exists a compact set K such that ∀x ∈ K,

|V (0)(x)| ≤ ε/(2c)

and there exists δ, 0 < δ < 1, such that ∀t, 0 ≤ t ≤ δ,

V (r) − V (0)C ≤ ε/(2c)

⇒ ∀t, 0 ≤ t ≤ δ, ∀x ∈ K,

|f (t)(x)| ≤ ε

⇒ f (t) ∈ C0 .

Proceeding in this fashion from the interval [0, δ] to the next interval [δ, 2δ] using the inequality  t |(f (t) − f (s))(x)| ≤ c |(V (r) − V (δ))(x)| dr + c |t − δ| |V (δ)(x)|, s

190

Chapter 4. Transformations Generated by Velocities

the uniform continuity of V ∀t, s,

|t − s| < δ,

V (t) − V (s)C ≤ ε/(2c),

and the fact that V (δ) ∈ C0 , that is, that there exists a compact set K(δ) such that ∀x ∈ K(δ),

|V (δ)(x)| ≤ ε/(2c),

we get f (t) ∈ C0 , δ ≤ t ≤ 2δ, and hence f ∈ C([0, τ ]; C0 ). For f  (t) we make use of the identity f  (t) = V (t) ◦ Tt . Again by assumption for any ε > 0 there exists a compact set K(t) such that |V (t)(x)| ≤ ε on K(t). Thus by choosing the compact set Kt = Tt−1 (K(t)), |f  (t)(x)| ≤ ε on Kt , and f  ∈ C([0, τ ]; C0 ). In order to complete the proof, it remains to establish the same properties for Df (t) and Df  (t). The matrix Df (t) is solution of the equations d Df (t) = DV (t) ◦ Tt DTt , dt

Df (0) = 0

⇒ Df  (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt .

(4.26)

From the proofs of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, for each t the elements of the matrix def

A(t) = DV (t) ◦ Tt = DV (t) ◦ [I + f (t)] belong to C0 since DV (t) and f (t) do. By assumption, V ∈ C([0, τ ]; C0k ) and V and all its derivatives ∂ α V are uniformly continuous in [0, τ ] × RN . Therefore, for each ε > 0 there exists δ > 0 such that ∀ |t − s| < δ, ∀ |y  − x | < δ,

|DV (t)(y  ) − DV (s)(x )| < ε.

Pick 0 < δ  < δ such that ∀ |t − s| < δ  ,

Tt − Ts C = f (t) − f (s)C < δ ⇒ ∀x, ∀ |t − s| < δ  , |Tt (x) − Ts (x)| < δ ⇒ |DV (t)(Tt (x)) − DV (s)(Ts (x))| < ε

⇒ A(t) − A(s)C < ε

⇒ A ∈ C([0, τ ]; (C0 )N ).

For each x, Df (t)(x) is the unique solution of the linear matrix equation (4.26). To show that Df (t) ∈ (C0 )N we first show that Df (t)(x) is uniformly continuous for x in RN . For any x and y  t |V (r, Tr (y)) − V (r, Tr (x))| dr |Df (t)(y) − Df (t)(x)| ≤ 0  t ≤ c|Tr (y) − Tr (x)| dr 0  t |f (r)(y) − f (r)(x)| + |y − x| dr. ≤c 0

4. Unconstrained Families of Domains

191

But f ∈ C([0, τ ]; C0 ) is uniformly continuous in (t, x): for each ε > 0 there exists δ, 0 < δ < ε/(2cτ ), such that ∀ |t − s| < δ, ∀ |y − x| < δ,

|f (t)(y) − f (s)(x)| < ε/(2cτ ).

Substituting in the previous inequality for each ε > 0, there exists δ > 0 such that ∀ t, ∀ |y − x| < δ,

|Df (t)(y) − Df (t)(x)| < ε.

Hence Df (t) is uniformly continuous in RN . Furthermore, from (4.26) we have the following inequality:  t |Df (t)(x)| ≤ |DV (r)(Tr (x))| |Df (r)(x)| + |DV (r)(Tr (x))| dr  t 0 (4.27) (|Df (r)(x)| + 1) dr ≤c 0

since V ∈ C([0, τ ]; C01 ). By Gronwall’s inequality |Df (t)(x)| ≤ c t for some other constant c independent of t. Hence Df (t) ∈ C(RN , RN )N . Finally to show that Df (t) vanishes at infinity we start from the integral form of (4.26):  t DV (r)(Tr (x)) DTr (x) dr, Df (t)(x) =  t 0 |DV (r)(Tr (x)) − DV (r)(x)| + |DV (r)(x)| dr |Df (t)(x)| ≤ c  t 0  ≤c |f (r)(x)| + |DV (r)(x)| dr. 0

By the same technique as for f (t), it follows that the elements of Df (t) belong to C0 since both f (s) and DV (r) do. Finally for the continuity with respect to t  t A(r)Df (r) + A(r) dr, Df (t) − Df (s) = s  t Df (t) − Df (s)C ≤ A(r)C Df (r) − Df (s)C s + A(r)C (1 + Df (r)C ) dr. Again, by Gronwall’s inequality, there exists another constant c such that Df (t) − Df (s)C ≤ c |t − s|. Therefore, Df ∈ C([0, τ ]; (C0 )N ) and f ∈ C([0, τ ]; C01 ). For Df  we repeat the proof for f  using the identity Df  (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt to get Df  ∈ C([0, τ ]; (C0 )N )

⇒ f  ∈ C([0, τ ]; C01 ).

192

Chapter 4. Transformations Generated by Velocities

Conditions (4.25) on g. From the remark at the beginning of part (i) of the proof, the conclusions of Theorem 4.3 are true for g, and it remains to check the remaining properties for g and Dg using the identities g(t) = −f (t) ◦ [I + g(t)],

Dg(t) = −Df (t) ◦ [I + g(t)] (I + Dg(t)).

From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, g(t) ∈ C0 since Df (t) and g(t) belong to C0 . Therefore, g(t) ∈ C01 . The continuity follows by the same argument as for f  and g ∈ C([0, τ ]; C01 ). (ii) By assumption from conditions (4.24), conditions (T1) are satisfied. For (T2) observe that for k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ  > 0 such that Tt is invertible for all t in [0, τ  ] and (T2) is satisfied in [0, τ  ]. Furthermore, from the proof of part (i) conditions (T3) and (4.25) on g are also satisfied in some interval [0, τ  ], τ  > 0. Therefore, the velocity field V (t) = f  (t) ◦ Tt−1 = f  (t) ◦ [I + g(t)] satisfies the conditions (V) specified by (4.2) in [0, τ  ]. From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, V (t) ∈ C0k since f  (t) and g(t) belong to C0k . By assumption, f ∈ C 1 ([0, τ ]; C0k ). Hence f  and all its derivatives ∂ α f  , |α| ≤ k, are uniformly continuous on [0, τ ] × RN ; that is, given ε > 0, there exists δ > 0 such that ∀t, s, |t − s| < δ, ∀y  , x , |y  − x | < δ,

|∂ α f  (t)(y ) − ∂ α f  (s)(x )| < ε.

Similarly g ∈ C([0, τ  ]; C0k ) and there exists 0 < δ  ≤ δ such that ∀t, s, |t − s| < δ  , ∀y, x, |y − x| < δ  ,

|∂ α g(t)(y) − ∂ α g(s)(x)| < δ.

Therefore, for |t − s| < δ  Tt−1 − Ts−1 C = g(t) − g(s)C < δ, and since δ  < δ ∀x, |f  (t)(Tt−1 (x)) − f  (t)(Ts−1 (x))| < ε ⇒ V (t) − V (s)C < ε ⇒ V ∈ C([0, τ  ]; C0 ). We then proceed to the first derivative of V , DV (t) = Df  (t) ◦ Tt−1 DTt−1 = Df  (t) ◦ [I + g(t)] [I + Dg(t)], and by uniform continuity of the right-hand side V ∈ C([0, τ  ]; C01 ). By induction on k, we finally get V ∈ C([0, τ  ]; C0k ). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at infinity is not an essential element of the proof. Therefore, the theorem is valid with C k (RN , RN ) in place of C0k (RN , RN ).

5. Constrained Families of Domains

193

Theorem 4.5. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity field V such that V ∈ C([0, τ ]; C k (RN , RN )),

(4.28)

the map T given by (4.4)–(4.6) satisfies conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )).

(4.29)

Moreover, conditions (T3) are satisfied and there exists τ  > 0 such that g ∈ C([0, τ  ]; C k (RN , RN )).

(4.30)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.29) and T (0, ·) = I, there exists τ  > 0 such that the velocity field V (t) = f  (t) ◦ Tt−1 satisfies conditions (V) and (4.28) on [0, τ  ].

5

Constrained Families of Domains

We now turn to the case where the family of admissible domains Ω is constrained to lie in a fixed larger subset D of RN or its closure. For instance, D can be an open set or a closed submanifold of RN .

5.1

Equivalence between Velocities and Transformations

Given a nonempty subset D of RN , consider a family of transformations T : [0, τ ] × D → RN

(5.1)

for some τ = τ (T ) > 0 with the following properties:   ∀X ∈ D, T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1D ) ∀X, Y ∈ D, T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2D ) (T3D )

X → Tt (X) = T (t, X) : D → D is bijective,   ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0,

∀t ∈ [0, τ ], ∀x, y ∈ D,

(5.2)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|,

where under assumption (T2D ) T −1 is defined from the inverse of Tt as (t, x) → T −1 (t, x) = Tt−1 (x) : [0, τ ] × D → RN . def

(5.3)

These three properties are the analogue for D of the same three properties obtained for RN . In fact, Theorem 4.1 extends from RN to D by adding one assumption to (V). Specifically, consider a velocity field V : [0, τ ] × D → RN

(5.4)

194

Chapter 4. Transformations Generated by Velocities

for which there exists τ = τ (V ) > 0 such that (V1D ) (V2D )

∀x ∈ D, ∀x, y ∈ D,

  V (·, x) ∈ C [0, τ ]; RN ,

∃c > 0,

V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

∀x ∈ D, ∀t ∈ [0, τ ],

(5.5)

±V (t, x) ∈ TD (x),

where TD (x) is Bouligand’s contingent cone6 to D at the point x in D  % % * 1 def TD (X) = (D − X) + εB h ε>0 α>0

(5.6)

0
and B is the unit disk in RN (cf. J.-P. Aubin and A. Cellina [1, p. 176]). This definition is equivalent to     D−X 1 = v lim inf dD (X + tv) = 0 (5.7) TD (X) = lim sup t0 t t t0 (cf. J.-P. Aubin and H. Frankowska [1, pp. 121–122, 17, 21]). Note that when D is bounded in RN , V (·) ∈ C([0, τ ]; C(D; RN ) ∩ Lip (D; RN )) = C([0, τ ]; C 0,1 (D; RN )). When D is equal to RN , TD (x) = RN for all x and condition (V2D ) can be dropped. When D is equal to the boundary ∂A of a set A of class C 1,1 in RN , ∂A is a C 1,1 submanifold of RN and ∀x ∈ ∂A,

±V (t, x) ∈ T∂A (x) ⇐⇒ ∀x ∈ ∂A,

V (t, x) · ∇bA (x) = 0;

that is, at each point of ∂A, the velocity field is tangent to ∂A: it belongs to the tangent linear space of ∂A. The next theorem is a generalization of Theorem 4.1 from RN to an arbitrary set D which shows the equivalence between velocity and transformation viewpoints. Theorem 5.1. (i) Let τ > 0 and let V be a family of velocity fields satisfying conditions (V1D ) and (V2D ) and consider the family of transformations (t, X) → T (t, X) = x(t; X) : [0, τ ] × D → RN ,

(5.8)

where x(·, X) is the solution of   dx (t) = V t, x(t) , dt

0 ≤ t ≤ τ, x(0) = X.

(5.9)

Then the family of transformations T satisfies conditions (T1D ) to (T3D ). 6 This

is an equivalent characterization of Bouligand’s contingent cone of Definition 2.4 in section 2.4 of Chapter 2.

5. Constrained Families of Domains

195

(ii) Conversely, given a family of transformations T satisfying conditions (T1D ) to (T3D ), the family of velocity fields (t, x) → V (t, x) =

 ∂T  t, Tt−1 (x) : [0, τ ] × D → RN ∂t

(5.10)

satisfies conditions (V1D ) and (V2D ). If, in addition, T (0, ·) = I, then T (·, X) is the solution of (5.9) for that V . (iii) Given a real number τ > 0 and a map T : [0, τ ] × D → D satisfying assumptions (T1D ) and (T2D ) and T (0, ·) = I, then there exists τ  > 0 such that the conclusions of part (ii) hold on [0, τ  ]. Remark 5.1. Assumption (V 2D ) is a double viability condition. M. Nagumo [1]’s usual viability condition (5.11) V (t, x) ∈ TD (x), ∀t ∈ [0, τ ], ∀x ∈ D is a necessary and sufficient condition for a viable solution to (5.9):   ∀t ∈ [0, τ ], ∀X ∈ D, x(t; X) ∈ D or Tt (D) ⊂ D

(5.12)

(cf. J.-P. Aubin and A. Cellina [1, p. 174 and p. 180]). Condition (V 2D ), ∀t ∈ [0, τ ], ∀x ∈ D,

±V (t, x) ∈ TD (x),

(5.13)

is a strict viability condition which says that Tt maps D into D and ∀t ∈ [0, τ ],

Tt : D → D

is a homeomorphism.

(5.14)

In particular it maps interior points onto interior points and boundary points onto boundary points (cf. J. Dugundji [1, pp. 87–88]). Remark 5.2. Condition (V 2D ) is a generalization to an arbitrary set D of the following condition ´sio [12] in 1979: for all x in ∂D, used by J.-P. Zole V (t, x) · n(x) = 0, 0,

if the outward normal n(x) exists, otherwise.

Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Apply M. Nagumo [1]’s theorem to the augmented system on [0, τ ]    dx   (t) = V t, x(t) , x(0) = X ∈ D, dt (5.15)   dx0 (t) = 1, x0 (0) = 0, dt that is,

   x  dˆ (t) = Vˆ x ˆ(t) , dt x ˆ def ˆ(0) = (0, X) ∈ D = R+ ×D,

(5.16)

196

Chapter 4. Transformations Generated by Velocities

    x) = 1, V˜ (ˆ x) , and where x ˆ(t) = x0 (t), x(t) ∈ RN +1 , Vˆ (ˆ   V (x0 , x), 0 ≤ x0 ≤ τ ˜ V (x0 , x) = , x ∈ D. V (τ, x), τ < x0

(5.17)

It is easy that systems (5.15) and (5.16) are equivalent on [0, τ ] and that  to check  x ˆ(t) = t, x(t) . The new velocity field on Vˆ ⊂ RN +1 is continuous at each point ˆ by the first assumption (V 1D ) since x ˆ∈D   Vˆ (ˆ y ) − Vˆ (ˆ x) = 0, V˜ (y0 , y) − V˜ (x0 , x) , and for 0 ≤ x0 , y0 ≤ τ |V (y0 , y) − V (x0 , x)| ≤ |V (y0 , y) − V (y0 , x)| + |V (y0 , x) − V (x0 , x)| ≤ n|y − x| + |V (y0 , x) − V (x0 , x)|. In addition, TDˆ (ˆ x) = TR+ (x0 ) × TD (x)   ⇒ Vˆ (ˆ x) = 1, V (ˆ x) ∈ TR+ (x0 ) × TD (x). ˆ is bounded and RN +1 is finite-dimensional. By using the version Moreover Vˆ (D) of M. Nagumo[1]’s theorem given in J.-P. Aubin and A. Cellina [1, Thm. 3, part b), pp. 182–183], there exists a viable solution xˆ to (5.16) for all t ≥ 0. In particular, ˆ = R+ ×D, ∀t ∈ [0, τ ], x ˆ(t) ∈ D which is necessarily of the form x ˆ(t) = (t, x(t)). Hence there exists a viable solution x, x(t) ∈ D on [0, τ ], to (5.9). The uniqueness now follows from the Lipschitz condition (V 1D ). The Lipschitzian continuity (T 1D ) can be established by a standard argument. Condition (T2D ). Associate with X in D the function y(s) = Tt−s (X),

0 ≤ s ≤ t.

Then

  dy (s) = −V t − s, y(s) , 0 ≤ s ≤ t, ds For each x ∈ D, the differential equation   dy (s) = −V t − s, y(s) , ds

0 ≤ s ≤ t,

y(0) = Tt (X).

(5.18)

y(0) = x ∈ D

(5.19)

has a unique viable solution in C 1 ([0, t]; RN ): ∀s ∈ [0, t],

y(s) ∈ D,

since by assumption (V 2D ) ∀t ∈ [0, τ ], ∀x ∈ D,

−V (t, x) ∈ TD (x).

(5.20)

5. Constrained Families of Domains

197

The proof is the same as above. The solutions of (5.19) define a Lipschitzian mapping x → St (x) = y(t) : D → D such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ D,

|St (y) − St (x)| ≤ c|y − x|.

Now in view of (5.18) and (5.19)   St Tt (X) = y(t) = Tt−t (X) = X

⇒

(5.21)

St ◦ Tt = I on D.

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (5.19). By definition,   dz (r) = V r, z(r) , dr

z(0) = y(t, x)

and necessarily     x = y(0; x) = z(t) = Tt y(t; x) = Tt St (x) ⇒ Tt ◦ St = I on D ⇒ St = Tt−1 : D → D. Condition (T3D ). The uniform Lipschitz continuity in (T3D ) follows from (5.21) and (T2D ), and we need only show that   ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN . Given t in [0, τ ] choose an arbitrary sequence {tn }, tn → t. Then for each x ∈ D there exists X ∈ D such that Tt (X) = x and Ttn (X) → Tt (X) = x from (T1D ). But     Tt−1 Tt (X) − Tt−1 Tt (X) (x) − Tt−1 (x) = Tt−1 n n     Tt (X) − Tt−1 Ttn (X) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1     Tt (X) − Tt−1 Ttn (X) | |Tt−1 (x) − Tt−1 (x)| = |Tt−1 n n n ≤ c|Tt (X) − Ttn (X)|, and the last term converges to zero as tn goes to t.

198

Chapter 4. Transformations Generated by Velocities ¯ and t, s in [0, τ ] (ii) The first condition (V1D ) is satisfied since for each x ∈ D ∂T   ∂T   |V (t, x) − V (s, x)| ≤ t, Tt−1 (x) − t, Ts−1 (x) ∂t ∂t ∂T    ∂T  t, Ts−1 (x) − s, Ts−1 (x) + ∂t ∂t ≤ c|Tt−1 (x) − Ts−1 (x)| ∂T   ∂T   t, Ts−1 (x) − s, Ts−1 (x) . + ∂t ∂t

The second condition (V1D ) follows from (T1D ) and (T3D ) and the following inequality: for all x and y in D ∂T   ∂T   −1 −1 |V (t, y) − V (t, x)| = t, Tt (y) − t, Tt (x) ∂t ∂t ≤ c Tt−1 (y) − Tt−1 (x) ≤ cc |y − x|. To check condition (V2D ), recall definition (5.6) of the Bouligand contingent cone:  % % * 1 TD (X) = (D − X) + εB , h ε>0 α>0 0
where B is the unit disk in RN . We first show that  ∂T  ∀x ∈ D, V (t, x) = t, Tt−1 (x) ∈ TD (x). ∂t By (T2D ) Tt is bijective. So it is equivalent to show that ∀X ∈ D,

  ∂T (t, X) ∈ TD Tt (X) . ∂t

For simplicity we use the notation x(t) = Tt (X) = T (t, X)

and x (t) =

∂T (t, X). ∂t

By the definition of TD (x(t)), we must prove that ∀ε > 0, ∀α > 0, ∃h ∈ ]0, α[ , ∃u such that x (t) ∈ u + εB and x(t) + hu ∈ D. Choose δ, 0 < δ < α, such that ∀s,

|s − t| < δ

⇒

|x (s) − s (t)| < ε.

Then fix t . For all t such that 0 < t − t < δ,  t 

 x(t ) − x(t) = x (s) − x (t) ds + (t − t)x (t) t  x(t ) − x(t)  ⇒ − x (t) < ε.  t −t

(5.22)

5. Constrained Families of Domains

199

Therefore choose u = [x(t ) − x(t)]/(t − t). Now x(t) + hu = x(t) +

t

 h  x(t ) − x(t) −t

and choose h = t − t since 0 < t − t < δ < α:

 T (t, X) + (t − t)u = T (t, X) + T (t , X) − T (t, X) = T (t , X) ∈ D by assumption on T . This proves (5.22). The second part of (V2D ) is ∀x ∈ D,

−V (t, x) = −

 ∂T  t, Tt−1 (x) ∈ TD (x), ∂t

which is equivalent to proving that ∀X ∈ D,

−

  ∂T (t, X) ∈ TD Tt (X) , ∂t

or with the simplified notation   −x (t) ∈ TD x(t) .

(5.23)

We proceed exactly as in the proof of (5.22) except that we choose t such that 0 < t − t < δ, h = t − t , and u = −[x(t) − x(t )]/(t − t ). Then |u + x (t)| < ε    x(t) − x(t ) and x(t) + hu = x(t) + (t − t ) − = x(t ) ∈ D t − t and we get (5.23). (iii) From (T1D ) and (T2D ) 

t

f (t)(y) − f (t)(x) = 

0

∂T ∂T (r, Tr (y)) − (r, Tr (x)) dr, ∂t ∂t

t

|f (t)(y) − f (t)(x)| ≤

c |Tr (y) − Tr (x)| dr ≤ c2 t |y − x|. 0

For τ  = min{τ, 1/(2c2 )} and 0 ≤ t ≤ τ  , c(f (t)) ≤ 1/2, g(t) = Tt−1 − I = (I − Tt ) ◦ Tt−1 = −f (t) ◦ [I + g(t)], (1 − c(f (t))) c(g(t)) ≤ c(f (t))

⇒ c(g(t)) ≤ 1 and c(Tt−1 ) ≤ 2,

and the second condition (T3) is satisfied on [0, τ  ]. The first one follows by the same argument as in part (i). Therefore, the conclusions of part (ii) are true on [0, τ  ]. This completes the proof of the theorem.

200

Chapter 4. Transformations Generated by Velocities

5.2

Transformation of Condition (V2D ) into a Linear Constraint

Condition (V2D ) is equivalent to ∀t ∈ [0, τ ], ∀x ∈ D,

V (t, x) ∈ {−TD (x)} ∩ TD (x)

(5.24)

since TD (x) = TD (x). If TD (x) were convex, then the above intersection would be a closed linear subspace of RN . This is true when D is convex. In that case TD (x) = CD (x), where CD (x) is the Clarke tangent cone and LD (x) = {−CD (x)} ∩ CD (x)

(5.25)

is a closed linear subspace of RN . This means that (V2D ) reduces to ∀t ∈ [0, τ ], ∀x ∈ D,

V (t, x) ∈ LD (x).

(5.26)

It turns out that for continuous vector fields V (t, ·), the equivalence of (V2D ) and (5.26) extends to arbitrary domains D. This equivalence generally fails for discontinuous vector fields. Other equivalences might be possible between TD and some intermediary convex cone between CD and TD , but there is no evidence so far of that fact. For smooth bounded open domains Ω, the two cones coincide and the condition reduces to V · n = 0, n the normal to ∂Ω, and V (t, x) belongs to the tangent space to ∂Ω in each point of ∂Ω. Theorem 5.2. (i) Given a velocity field V satisfying (V1D ), condition (V2D ) is equivalent to (V2C )

∀t ∈ [0, τ ], ∀x ∈ D,

V (t, x) ∈ LD (x) = {−CD (x)} ∩ CD (x),

where CD (x) is the (closed convex ) Clarke tangent cone to D at x,   dD (y + tv)   lim = 0 CD (x) = v ∈ RN : t0 , t y−→x  

(5.27)

D

and −→ denotes the convergence in D. D

(ii) LD (x) is a closed linear subspace of RN . Proof. (i) The equivalence of (V2D ) and (V2C ) is a direct consequence of the following lemma. Lemma 5.1. Given a vector field W ∈ C(D; RN ), the following two conditions are equivalent: ∀x ∈ D,

W (x) ∈ TD (x),

(5.28)

∀x ∈ D,

W (x) ∈ CD (x).

(5.29)

5. Constrained Families of Domains

201

(ii) The set LD (x) is closed as the intersection of two closed sets. To show that it is linear, we show that for all α ∈ R and V ∈ LD (x), αV ∈ LD (x), and for all V and W in LD (x), V + W ∈ LD (x). Since ±CD (x) are cones, ∀α ∈ R, ∀V ∈ LD (x),

±|α|V ∈ CD (x) ⇒ ±αV ∈ CD (x) ⇒ αV ∈ LD (x).

By convexity of ±CD (x) ∀V, W ∈ LD (x),

±(V + W ) ∈ CD (x) ⇒ V + W ∈ LD (x).

This completes the proof of the theorem. Proof of Lemma 5.1. Assume that (5.29) is verified. By definition CD (x) ⊂ TD (x) and (5.29) ⇒ (5.28). Conversely, either x is an isolated point and TD (x) = {0} = CD (x) or there are points x = y ∈ D such that y −→ x. In the latter case we D

know that lim inf TD (y) = CD (x) y−→x D

(cf., for instance, J.-P. Aubin and H. Frankowska [1, Thm. 4.1.10, sect. 4.1.5, p. 130]). Since W is continuous in D and (5.28) is satisfied, then for each x ∈ D W (x) = lim W (y) ∈ lim inf TD (y) = CD (x) and (5.28) implies (5.29).

y−→x

y−→x

D

D

Remark 5.3. Lemma 5.1 essentially says that for continuous vector fields we can relax the condition of M. Nagumo [1]’s theorem from (V2D ) involving the Bouligand contingent cone to (V2C ) involving the smaller Clarke convex tangent cone. In dimension N = 3, LD (x) is {0}, a line, a plane, or the whole space. Notation 5.1. In what follows, it will be convenient to introduce the spaces and subspaces 

(5.30) L = V : [0, τ ] × RN → RN : V satisfies (V) on RN , and for an arbitrary domain D in RN

 LD = {V : [0, τ ] × D → RN : V satisfies (V1D ) and (V2C ) on D .

(5.31)

For any integers k ≥ 0 and m ≥ 0 and any compact subset K of RN define the following subspaces of L:    m,k VK (5.32) = C m [0, τ ], D k K, RN ∩ L, where Dk (K, RN ) is the space of k-times continuously differentiable transformations m,k ⊂ LK . As usual D∞ (K, RN ) of RN with compact support in K. In all cases VK N will be written D(K, R ).

202

6

Chapter 4. Transformations Generated by Velocities

Continuity of Shape Functions along Velocity Flows

Throughout this section, assume that Ω is a closed subset or an open crack-free7 subset of RN and that B is a Banach space. Since the construction of the Courant metric is relatively abstract, can the continuity of a shape functional be characterized in a more direct way? The answer is yes thanks to the sharp theorems giving the equivalence between transformations and velocities in the previous section without restricting the analysis to the subgroup GΘ defined by (2.7) in section 2. This is due to the fact that continuity and semiderivatives are local properties and that the velocity character naturally pops out as a sequence of transformations goes to identity. In this section, we give a characterization via velocities of the continuity of a shape functional of the form Ω → J(Ω ) : X (Ω) → B,

X (Ω) = {F (Ω) : ∀F ∈ F(Θ)}

(6.1)

with respect to the Courant metric topology of the quotient F(Θ)/G(Ω) using the equivalence Theorems 4.3, 4.4, and 4.5. Checking the continuity along flows of a velocity is easier and more natural. As in Definition 3.1, assume that J verifies the compatibility condition of Definition 3.1: ∀H, F ∈ F(Θ) such that H(F (Ω)) = F (Ω),

J(H(F (Ω))) = J(F (Ω)).

We specifically consider the continuity of shape functionals with respect to the Courant metric associated with Θ equal to C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0, but similar equivalences are true for the other spaces of Chapter 3. Remark 6.1. An obvious consequence of Theorems 6.1, 6.2, and 6.3 is that when a shape functional is semidifferentiable in the sense of Definition 3.2 (Hadamard), it is continuous at that point for the Courant metric topology in complete analogy with the classical Euclidean calculus. We begin with the space C0k (RN ) = C0k (RN , RN ) of A. M. Micheletti [1]. Theorem 6.1. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonempty open subset of RN . Consider a shape functional J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F(C0k (RN , RN ))/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

t0

(6.2)

for all families of velocity fields {V (t) : 0 ≤ t ≤ τ } satisfying the condition V ∈ C([0, τ ]; C0k (RN , RN )). 7 Cf.

Definition 7.1 (ii) of Chapter 8.

(6.3)

6. Continuity of Shape Functions along Velocity Flows

203

Proof. It is sufficient to prove the theorem for a real-valued function J. The Banach space case is readily obtained by considering the new real-valued function j(T ) = |J(T (Ω)) − J(Ω)|. (i) If J is dG -continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which implies conditions (4.24) and (4.25): f ∈ C 1 ([0, τ ]; C0k (RN , RN ))

and g ∈ C([0, τ ]; C0k (RN , RN ))

⇒ Tt − ICk (RN ) → 0 and Tt−1 − ICk (RN ) → 0 as t → 0. But by definition of the metric dG d([Tt ], [I]) ≤ Tt−1 − IC k + Tt − ICk → 0 as t → 0, and we get the convergence (6.2) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.3). (ii) Conversely, it is sufficient to prove that for any sequence {[Tn ]} such that dG ([Tn ], [I]) goes to zero, there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞. Indeed let  = lim inf J(Tn (Ω)) n→∞

and L = lim sup J(Tn (Ω)). n→∞

By definition of the liminf, there is a subsequence, still indexed by n, such that  = lim inf n→∞ J(Tn (Ω)). But since there exists a subsequence {Tnk } of {Tn } such that J(Tnk (Ω)) → J(Ω), then necessarily  = J(Ω). The same reasoning applies to the limsup, and hence the whole sequence J(Tn (Ω)) converges to J(Ω) and we have the continuity of J at Ω. We prove that we can construct a velocity V associated with a subsequence of {Tn } verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k + gn Ck = Tn−1 − IC k + Tn − IC k ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Define the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ],  def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

 (Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1).

204

Chapter 4. Transformations Generated by Velocities

Conditions on f . By definition, for all t, 0 ≤ t ≤ 1/2, f (t) = Tt − I ∈ C0k (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T (tn , X) = 0 = (tn+1 , X), ∂t   ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn

Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X),

f  (t) = ∂T /∂t(t, ·) ∈ C0k (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By definition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and   tn+1 − t (fn+1 − fn )C k f (t) − f (0)Ck = f (t)C k = fn + p tn+1 − tn ≤ 2fn Ck + fn+1 C k ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Define at t = 0, f  (t) = 0. By the same technique, there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f  (t) − f  (0)Ck = f  (t)C k 3 3 3 ∂T 3 3 ≤ c Tn+1 − Tn Ck = c fn+1 − fn C k =3 (t, ·) 3 ∂t 3 k |tn+1 − tn | 2−(n+1) C

≤ c 2 2−2(n+2) /2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t ⇒ f  (t)C k ≤ ct. So for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C0k (RN )) and the condition (4.24) of Theorem 4.4 is satisfied. Hence the corresponding velocity V satisfies conditions (4.23). Finally V satisfies conditions (6.3) and by (6.2) for all ε > 0 there exists δ > 0 such that ∀t, 0 ≤ t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ, and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and this proves the d-continuity for the subsequence {Tn }. The case of the Courant metric associated with the space C k (RN ) = C k (RN , RN ) is a corollary to Theorem 6.1. Theorem 6.2. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonempty open subset of RN . Consider a shape functional J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F(C k (RN , RN ))/G(Ω).

6. Continuity of Shape Functions along Velocity Flows

205

Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.4)

t0

for all families of velocity fields {V (t) : 0 ≤ t ≤ τ } satisfying the condition V ∈ C([0, τ ]; C k (RN , RN )).

(6.5)

The proof of the theorem for the Courant metric topology associated with the space C k,1 (RN ) = C k,1 (RN , RN ) is similar to the proof of Theorem 6.1 with obvious changes. Theorem 6.3. Let k ≥ 0 be an integer, Ω be a nonempty open subset of RN , and B be a Banach space. Consider a shape functional J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F(C k,1 (RN , RN ))/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.6)

t0

for all families {V (t) : 0 ≤ t ≤ τ } of velocity fields in C k,1 (RN , RN ) satisfying the conditions V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(6.7)

for some constant c independent of t. Proof. As in the proof of Theorem 6.1, it is sufficient to prove the theorem for a real-valued function J. (i) If J is dG -continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Under condition (6.7) from Theorem 4.3 f = T − I ∈ C([0, τ ]; C k,1 (RN )) and Tt − IC k,1 → 0 as t → 0, g(t) = Tt−1 − I ∈ C k,1 (RN ) and Tt−1 − IC k,1 ≤ ct → 0 as t → 0. But by definition of the metric dG dG ([Tt ], [I]) ≤ Tt−1 − IC k,1 + Tt − IC k,1 → 0 as t → 0, and we get the convergence (6.6) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.7). (ii) Conversely, as in the proof of Theorem 6.1, it is sufficient to prove that given any sequence {[Tn ]} such that δ([Tn ], [I]) → 0 there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞.

206

Chapter 4. Transformations Generated by Velocities

By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k,1 + gn Ck,1 = Tn−1 − IC k,1 + Tn − ICk,1 ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Define the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ],  def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

 (Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1). Conditions on f . By definition for all t, 0 ≤ t ≤ 1/2, f (t) = Tt −I ∈ C k,1 (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X), (tn , X) = 0 = (tn+1 , X), ∂t  ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn f  (t) = ∂T /∂t ∈ C k,1 (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By definition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and 3 3   3 3 tn+1 − t 3 (f f (t) − f (0)C k,1 = f (t)C k,1 = 3 f + p − f ) n+1 n 3 3 n tn+1 − tn C k,1 ≤ 2fn C k,1 + fn+1 C k,1 ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Define at t = 0, f  (t) = 0. By the same technique there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f  (t) − f  (0)C k,1 = f  (t)C k,1 3 3 3 3 ∂T fn+1 − fn Ck,1 Tn+1 − Tn C k,1 3 (t, ·)3 =c =3 ≤c 3 ∂t |tn+1 − tn | 2−(n+1) k,1 C −2(n+2)

≤ c22

/2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t

⇒ f  (t)C k,1 ≤ ct and ck (f  (t)) ≤ ct and for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C k (RN )). Moreover, it can be shown that ck (f  (t)) ≤ ct

⇒ ∀t, s ∈ [0, τ ],

ck (f (t) − f (s)) ≤ c |t − s|

for some c > 0. The result is straightforward for k = 0 and then the general case follows by induction on k. As a result f ∈ C([0, 1/2]; C k,1 (RN )) and the condition

6. Continuity of Shape Functions along Velocity Flows

207

(4.20) of Theorem 4.3 is satisfied. Hence the corresponding velocity V satisfies conditions (4.19). Finally the velocity field V satisfies conditions (6.7), and by (6.6) for all ε > 0 there exists δ > 0 such that ∀t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and we have the dG -continuity for the subsequence {Tn }. Remark 6.2. The conclusions of Theorems 6.1, 6.2, and 6.3 are generic. They also have their counterpart in the constrained case. The difficulty lies in the second part of the theorem, which requires a special construction to make sure that the family of transformations {Tt : 0 ≤ t ≤ τ } constructed from the sequence {Tn } are homeomorphisms of D.

Chapter 5

Metrics via Characteristic Functions 1

Introduction

The constructions of the metric topologies of Chapter 3 are limited to families of sets which are the image of a fixed set by a family of homeomorphisms or diffeomorphisms. If it is connected, bounded, or C k , k ≥ 1, then the images will have the same properties under C k -transformations of RN . In this chapter we considerably enlarge the family of available sets by relaxing the smoothness assumption to the mere Lebesgue measurability and even just measurability to include Hausdorff measures. This is done by identifying Ω ⊂ RN with its characteristic function & 1, if x ∈ Ω def def χΩ (x) = if Ω = ∅ and χ∅ = 0. (1.1) 0, if x ∈ /Ω We first introduce an Abelian group structure1 on characteristic functions of measurable 2 subsets of a fixed holdall D in section 2 and then construct complete metric spaces of equivalence classes of measurable characteristic functions via the Lp norms, 1 ≤ p < ∞, that turn out to be algebraically and topologically equivalent. Starting with section 3, we specialize to the Lebesgue measure in RN . The complete topology generated by the metric defined as the Lp -norm of the difference of two characteristic functions is called the strong topology in section 3.1. In section 3.2 we consider the weak Lp -topology on the family of characteristic functions. Weak limits of sequences of characteristic functions are functions with values in [0, 1] which belong to the closed convex hull of the equivalence classes of measurable characteristic functions. This occurs in optimization problems where the function to be optimized depends on the solution of a partial differential equation on the variable domain, as was shown, for instance, by F. Murat [1] in 1971. It usually corresponds to the appearance of a microstructure or a composite material in mechanics. 1 Cf.

the group structure on subsets of a holdall D in section 2.2 of Chapter 2. in the sense of L. C. Evans and R. F. Gariepy [1]. It simultaneously covers Hausdorff and Lebesgue measures. 2 Measurable

209

210

Chapter 5. Metrics via Characteristic Functions

One important example in that category was the analysis of the optimal thickness of a circular plate by K.-T. Cheng and N. Olhoff [2, 1] in 1981. The reader is referred to the work of F. Murat and L. Tartar [3, 1], initially published in 1985, for a comprehensive treatment of the calculus of variations and homogenization. Section 3.3 deals with the question of finding a nice representative in the equivalence class of sets. Can it be chosen open? We introduce the measure theoretic representative and characterize its interior, exterior, and boundary. It will be used later in section 6.4 to prove the compactness of the family of Lipschitzian domains verifying a uniform cone property. Section 3.4 shows that the family of convex subsets of a fixed bounded holdall is closed in the strong topology. The use of the Lp -topologies is illustrated in section 4 by revisiting the optimal ´a and K. Malanowski [1] in 1970. By relaxing design problem studied by J. Ce the family of characteristic functions to functions with values in the interval [0, 1] a saddle point formulation is obtained. Functions with values in the interval [0, 1] can also be found in the modeling of a dam by H. W. Alt [1] and H. W. Alt and G. Gilardi [1], where the “liquid saturation” in the soil is a function with values in [0, 1]. Another problem amenable to that formulation is the buckling of columns as will be illustrated in section 5 using the work of S. J. Cox and M. L. Overton [1]. It is one of the very early optimal design problems, formulated by Lagrange in 1770. The Caccioppoli or finite perimeter sets of the celebrated Plateau problem are revisited in section 6. Their characteristic function is a function of bounded variation. They provide the first example of compact families of characteristic functions in Lp (D)-strong, 1 ≤ p < ∞. This was developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem of minimal surfaces. Section 6.3 exploits the embedding3 of BV(D) ∩ L∞ (D) into the Sobolev spaces W ε,p (D), 0 < ε < 1/p, p ≥ 1, to introduce a cascade of complete metrics between Lp (D) and BV(D) on characteristic functions. In section 6.4 we show that the family of Lipschitzian domains in a fixed bounded holdall verifying the uniform cone property of section 6.4.1 in Chapter 2 is also compact. This condition naturally yields a uniform bound on the perimeter of the sets in the family and hence can be viewed as a special case of the first compactness theorem. Section 7 gives an example of the use of the perimeter in the Bernoulli free boundary problem and in particular for the water wave. There the energy associated with the surface tension of the water is proportional to the perimeter, that is, the surface area of the free boundary.

2

Abelian Group Structure on Measurable Characteristic Functions

2.1

Group Structure on Xµ (RN )

From section 2.2 in Chapter 2, the Abelian group structure on the subsets of RN extends to the corresponding family of their characteristic functions. 3 BV(D)

is the space of functions of bounded variations in D.

2. Abelian Group Structure on Measurable Characteristic Functions

211

Theorem 2.1. Let ∅ = D ⊂ RN . The space of characteristic functions X(D) endowed with the multiplication
and the neutral multiplicative element χ∅ = 0 def

χA
χB = |χA − χB | = χ(A∩
B)∪(B∩
A) = χAB

(2.1)

is an Abelian group, that is, χ A
χB = χ B
χA ,

χA
χ∅ = χA ,

χA
χA = χ∅ ,

χ−1 A = χA .

(2.2)

Proof. Note that the underlying group structure on subsets of D is the one of section 2.2 of Chapter 2. The product is well-defined in X(D) since it is the characteristic function of a subset of A ∪ B in D. The first two identities (2.2) are obvious. As for the inverse χ A
χB = χ ∅

⇔

|χA − χB | = 0

⇔

B = A,

and there is a unique multiplicative inverse χ−1 A = χA or χA
χA = χ∅ . Remark 2.1. We chose the product terminology
, but we could have chosen to call that algebraic operation an addition ⊕.

2.2

Measure Spaces

Definition 2.1. Let X be a set and P(X) be the collection of subsets of X. (i) A mapping µ : P(X) → [0, ∞] is called a measure 4 on X if (a) µ(∅) = 0, and ) ∞ (b) µ(A) ≤ ∞ k=1 µ(Ak ), whenever A ⊂ ∪k=1 Ak . (ii) A subset A ⊂ X is σ-finite with respect to µ if it can be written A = ∪∞ k=1 Bk , where Bk is µ-measurable and µ(Bk ) < ∞ for k = 1, 2, . . . . (iii) A measure µ on RN is a Radon measure if µ is Borel regular and µ(K) < ∞ for each compact set K ⊂ RN . The Lebesgue measure mN on RN is a Radon measure since RN is σ-finite with respect to mN . The s-dimensional Hausdorff measure Hs , 0 ≤ s < N , on RN (cf. section 3.2.3 in Chapter 2) is Borel regular but not necessarily a Radon measure since bounded Hs -measurable subsets of RN are not necessarily σ-finite with respect to Hs . Yet, a Borel regular measure µ can be made σ-finite by restricting it to a µ-measurable set A ⊂ RN such that µ(A) < ∞ or, more generally, to a σ-finite µ-measurable set A ⊂ RN . For smooth submanifolds of dimension s, s an integer, Hs gives the same area as the integral of the canonical density in section 3.2.2 of Chapter 2. 4 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

212

Chapter 5. Metrics via Characteristic Functions

Given a µ-measurable subset D ⊂ RN , let Lp (D, µ) denote the Banach space of equivalence classes of µ-measurable functions such that |f |p dµ < ∞, 1 ≤ p < ∞, and let L∞ (D, µ) be the space of equivalence classes of µ-measurable functions such that ess supD |f | < ∞. Denote by [A]µ the equivalence class of µ-measurable subsets of D that are equal almost everywhere and def

Xµ (D) = {χA : A ⊂ D is µ-measurable} . Since A
B is µ-measurable for A and B µ-measurable, {[A]µ : A ⊂ D is µ-measurable} is a subgroup of P(D) and Xµ (D) ∩ L1 (D, µ) = {χA : A ⊂ D is µ-measurable and µ(A) < ∞}

(2.3)

is a group in Lp (D, µ) with respect to
. When µ = mN , we drop the subscript µ.

2.3

Complete Metric for Characteristic Functions in Lp -Topologies

Now that we have revealed the underlying group structure of Xµ (D) ∩ L1 (D, µ), we construct a metric on that group. We have a structure similar to the Courant metric of A. M. Micheletti [1] in Chapter 3 on groups of transformations of RN . We identify equivalence classes of µ-measurable subsets of D and characteristic functions via the bijection [A]µ ↔ χA . Theorem 2.2. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) Xµ (D) ∩ L1 (D, µ) is closed in Lp (D, µ). The function def

ρD ([A2 ]µ , [A1 ]µ ) = χA2 − χA1 Lp (D,µ) defines a complete metric structure on the Abelian group Xµ (D)∩L1 (D, µ) that makes it a topological group. If µ(D) < ∞, then Xµ (D) ∩ L1 (D, µ) = Xµ (D). (ii) If, in addition, D is σ-finite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then Xµ (D) is closed in Lploc (D, µ) and def

ρ([A2 ]µ , [A1 ]µ ) =

∞ χA2 − χA1 Lp (Dk ,µ) 1 2n 1 + χA2 − χA1 Lp (Dk ,µ) k=1

defines a complete metric structure on the Abelian group Xµ (D) that makes it a topological group. When µ is a Radon measure on RN , the assumption that D is σ-finite with respect to µ can be dropped. Remark 2.2. For the Lebesgue measure mN and D measurable (resp., measurable and bounded), X(D) is a topological Abelian group in Lploc (D, mN ) (resp., Lp (D, mN )). For Hausdorff measures Hs , 0 ≤ s < N , the theorem does not say more.

2. Abelian Group Structure on Measurable Characteristic Functions

213

Proof. (i) Let {χAn } be a Cauchy sequence of µ-measurable characteristic functions in Xµ (D) ∩ L1 (D, µ) ⊂ Lp (D, µ) converging to some f ∈ Lp (D, µ). There exists a subsequence, still denoted by {χAn }, such that χAn (x) → f (x) in D except for a subset Z of zero µ-measure. Hence 0 = χAn (x) (1 − χAn (x)) → f (x) (1 − f (x)). Define the set def

A = {x ∈ D\Z : f (x) = 1}. Clearly A is µ-measurable and χA = f on D\Z since f (x) (1 − f (x)) = 0 on D\Z. Hence f = χA almost everywhere on D, χAn → χA in Lp (D, µ), and χA ∈ Xµ (D) ∩ L1 (D, µ). Finally, the metric is compatible with the group structure −1 ρD ([Ω2 ]µ , [Ω1 ]µ ) = χΩ2 − χΩ1 Lp = χΩ2
χ−1 Ω1 Lp = χΩ1
χΩ2 Lp def

as in Chapter 3 and Xµ (D) ∩ L1 (D, µ) is a topological group. (ii) From part (i) using the Fr´echet topology associated with the family of seminorms qk (f ) = f Lp (Dk ) , k ≥ 1. For p = 1 and µ-measurable subsets A1 and A2 in D, the metric ρD ([A2 ]µ , [A1 ]µ ) is the µ-measure of the symmetric difference A1 ∆A2 = (A1 ∩ A2 ) ∪ (A2 ∩ A1 ) and for 1 ≤ p < ∞ all the topologies are equivalent on Xµ (D) ∩ L1 (D, µ). Theorem 2.3. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) The topologies induced by Lp (D, µ) on Xµ (D) ∩ L1 (D, µ) are all equivalent for 1 ≤ p < ∞. (ii) If, in addition, D is σ-finite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then the topologies induced by Lploc (D, µ) on Xµ (D) are all equivalent for 1 ≤ p < ∞. Proof. For D µ-measurable and 1 ≤ p < ∞, Xµ (D) ∩ Lp (D, µ) = Xµ (D) ∩ L1 (D, µ), and for any χ and χ in Xµ (D) ∩ L1 (D, µ)   χ − χpLp (D,µ) = |χ − χ(x)|p dµ = |χ − χ(x)| dµ = χ − χL1 (D,µ) , D

D

since χ(x) and χ(x) are either 0 or 1 almost everywhere in D. Therefore the linear (identity) mapping between Xµ (D)∩Lp (D, µ) and Xµ (D)∩L1 (D, µ) is bicontinuous and the topologies are equivalent. Remark 2.3. At this stage, it is not clear what the tangent space to this topological Abelian group is. If µ is a Radon measure, then for all ϕ ∈ D(RN )  

 1 1 χA
χBε (x)
χ−1 ϕ dµ = χB (x) ϕ dµ A µ(Bε (x)) RN µ(Bε (x)) RN ε

214

Chapter 5. Metrics via Characteristic Functions

and by the Lebesgue–Besicovitch differentiation theorem 

 1 N χA
χBε (x)
χ−1 A ϕ dµ → ϕ(x), µ a.e. in R . µ(Bε (x)) RN

3

Lebesgue Measurable Characteristic Functions

In this section we specialize to the Lebesgue measure and the corresponding family of characteristic functions  def

X(RN ) = χΩ : ∀Ω Lebesgue measurable in RN . (3.1) Clearly X(RN ) ⊂ L∞ (RN ), and for all p ≥ 1, X(RN ) ⊂ Lploc (RN ). Also associate with ∅ = D ⊂ RN the set def

X(D) = {χΩ : ∀Ω Lebesgue measurable in D} .

(3.2)

The above definitions are special cases of the definitions of section 2.3, and Theorems 2.2 and 2.3 apply to mN as a Radon measure.

3.1

Strong Topologies and C ∞ -Approximations

In view of the equivalence Theorem 2.3, we introduce the notion of strong convergence. Definition 3.1. (i) If D is a bounded measurable subset of RN , a sequence {χn } in X(D) is said to be strongly convergent in D if it converges in Lp (D)-strong for some p, 1 ≤ p < ∞. (ii) A sequence {χn } in X(RN ) is said to be locally strongly convergent if it converges in Lploc (RN )-strong for some p, 1 ≤ p < ∞. The following approximation theorem will also be useful Theorem 3.1. Let Ω be an arbitrary Lebesgue measurable subset of RN . There exists a sequence {Ωn } of open C ∞ -domains in RN such that χΩn → χΩ

in L1loc (RN ).

Proof. The construction of the family of C ∞ -domains {Ωn } can be found in many places (cf., for instance, E. Giusti [1, sect. 1.14, p. 10 and Lem. 1.25, p. 23]). Associate with χ = χΩ the sequence of convolutions fn = χ ∗ ρn for a sequence {ρn } of symmetric mollifiers. By construction, 0 ≤ fn ≤ 1. For t, 0 < t < 1, define  def

Fnt = x ∈ RN : fn (x) > t . By definition, fn − χ > t in Fnt \Ω, χ − fn > 1 − t in Ω\Fnt , and 1 |χ − fn | a.e. in RN . |χ − χFnt | ≤ min{t, 1 − t}

3. Lebesgue Measurable Characteristic Functions

215

By Sard’s Theorem 4.3 of Chapter 2, the set Fnt is a C ∞ -domain for almost all t in (0, 1). Fix α, 0 < α < 1/2, and choose a sequence {tn } such that, for all n, def

α ≤ tn ≤ 1 − α, and define Ωn = Fntn . Therefore |χΩ − χΩn | ≤

1 1 |χΩ − fn | ≤ |χΩ − fn | a.e. in RN min{tn , 1 − tn } α

and for any bounded measurable subset D of RN χΩ − χΩn L1 (D) ≤ α−1 χΩ − fn L1 (D) , which goes to zero as n goes to infinity.

3.2

Weak Topologies and Microstructures

Some shape optimization problems lead to apparent paradoxes. Their solution is no longer a geometric domain associated with a characteristic function, but a fuzzy domain associated with the relaxation of a characteristic function to a function with values ranging in [0, 1]. The intuitive notion of a geometric domain is relaxed to the notion of a probability distribution of the presence of points of the set. When the underlying problem involves two different materials characterized by two constants k1 = k2 , the occurrence of such a solution can be interpreted as the mixing or homogenization of the two materials at the microscale. This is also referred to as a composite material or a microstructure. Somehow this is related to the fact that the strong convergence needs to be relaxed to the weak Lp -convergence and the space X(D) needs to be suitably enlarged. Even if X(D) is strongly closed and bounded in Lp (D), it is not strongly compact. However, for 1 < p < ∞ its closed convex hull co X(D) is weakly compact in the reflexive Banach space Lp (D). In fact, co X(D) = {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D} .

(3.3)

Indeed, by definition co X(D) ⊂ {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D}. Conversely any χ that belongs to the right-hand side of (3.3) can be approximated by a sequence of convex combinations of elements of X(D). Choose χn =

n 1 χBnm , n m=1

 m Bnm = x : χ(x) ≥ , n

for which |χn (x) − χ(x)| < 1/n. The elements of co X(D) are not necessarily characteristic functions of a domain; that is, the identity χ(x) (1 − χ(x)) = 0 a.e. in D is not necessarily satisfied. We first give a few basic results and then consider a classical example from the theory of homogenization of differential equations.

216

Chapter 5. Metrics via Characteristic Functions

Lemma 3.1. Let D be a bounded open subset of RN , let K be a bounded subset of R, and let def

K = {k : D → R : k is measurable and k(x) ∈ K a.e. in D} . (i) For any p, 1 ≤ p < ∞, and any sequence {kn } ⊂ K the following statements are equivalent: (a) {kn } converges in L∞ (D)-weak; (b) {kn } converges in Lp (D)-weak; (c) {kn } converges in D(D) , where D(D) is the space of scalar distributions on D. (ii) If K is bounded, closed, and convex, then K is convex and compact in L∞ (D)weak, Lp (D)-weak, and D(D) . The above results remain true in the vectorial case when K is the set of mappings k : D → K for some bounded subset K ⊂ Rp and a finite integer p ≥ 1. Proof. (i) It is clear that (a) ⇒ (b) ⇒ (c). To prove that (c) ⇒ (a) recall that since K is bounded, there exists a constant c > 0 such that K ⊂ cB, where B denotes the unit ball in RN . By density of D(D) in L1 (D), any ϕ in L1 (D) can be approximated by a sequence {ϕm } ⊂ D(D) such that ϕm → ϕ in L1 (D). So for each ε > 0 there exists M > 0 such that ∀m ≥ M,

ϕm − ϕL1 (D) ≤

ε . 4c

Moreover, there exists N > 0 such that  ε ∀n ≥ N, ∀ ≥ N, ϕM (kn − k
) dx ≤ . 2 D Hence for each ε > 0, there exists N such that for all n ≥ N and  ≥ N    ϕ(kn − k
) dx ≤ ϕM (kn − k
) dx + (ϕ − ϕM )(kn − k
) dx D D D ε ≤ + 2cϕ − ϕM L1 (D) ≤ ε. 2 (ii) When K is bounded, closed, and convex, K is also bounded, closed, and convex. Since L2 (D) is a Hilbert space, K is weakly compact. In view of the equivalences of part (i), K is also sequentially compact in all the other weak topologies.5 5 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E
, any closed ball is a compact metrizable space for the weak topology. Since K is a bounded subset of the normed reflexive separable Banach space Lp (D), 1 ≤ p < ∞, the weak compactness of K coincides ´ [1, Vol. 2, Chap. XII, sect. 12.15.9, p. 75]). with its weak sequential compactness (cf. J. Dieudonne

3. Lebesgue Measurable Characteristic Functions

217

In view of this equivalence we adopt the following terminology. Definition 3.2. Let ∅ = D ⊂ RN be bounded open. A sequence {χn } in X(D) is said to be weakly convergent if it converges for some topology between L∞ (D)-weak and D(D) . It is interesting to observe that working with the weak convergence makes sense only when the limit element is not a characteristic function. Theorem 3.2. Let the assumptions of the theorem be satisfied. Let {χn } and χ be elements of X(RN ) (resp., X(D)) such that χn  χ weakly in L2 (D) (resp., L2loc (RN )). Then for all p, 1 ≤ p < ∞,   χn → χ strongly in Lp (D) resp., Lploc (RN ) . Proof. It is sufficient to prove the result for D. The strong L2 (D)-convergence follows from the property     |χn |2 dx = χn 1 dx → χ1 dx = |χ|2 dx D D D D    2 2 2 |χn − χ| dx = |χn | − 2 χχn + |χ| dx → |χ|2 − 2 χχ+ |χ|2 dx = 0. ⇒ D

D

D

The convergence for any p ≥ 1 now follows from Theorem 2.2. Working with the weak convergence creates new phenomena and difficulties. For instance when a characteristic function is present in the coefficient of the higherorder term of a differential equation the weak convergence of a sequence of characteristic functions {χn } to some element χ of co X(D), χn  χ in L2 (D)-weak

⇒ χn  χ in L∞ (D)-weak,

does not imply the weak convergence in H 1 (D) of the sequence {y(χn )} of solutions to the solution of the differential equation corresponding to y(χ), y(χn )  y(χ) in H 1 (D)-weak. By compactness6 of the injection of H 1 (D) into L2 (D) this would have implied convergence in L2 (D)-strong: y(χn ) → y(χ) in L2 (D)-strong. This fact was pointed out in 1971 by F. Murat [1] in the following example, which will be rewritten to emphasize the role of the characteristic function. 6 This is true since D is a bounded Lipschitzian domain. Examples of domains between two spirals can be constructed where the injection is not compact.

218

Chapter 5. Metrics via Characteristic Functions

Example 3.1. Consider the following boundary value problem:  d  dy  − k + ky = 0 in D = ]0, 1[ , dx dx  y(0) = 1 and y(1) = 2,

(3.4)

where K = {k = k1 (x)χ + k2 (x)(1 − χ) : χ ∈ X(D)} with

4 k1 (x) = 1 −

1 x2 − , 2 6

4 k2 (x) = 1 +

1 x2 − . 2 6

(3.5)

(3.6)

This is equivalent to K = {k ∈ L∞ (0, 1) : k(x) ∈ {k1 (x), k2 (x)} Associate with the integer p ≥ 1 the function 4   m 2m + 1 1 x2    1 − 2 − 6 , p < x ≤ 2p , k p (x) = 4   1 x2 2m + 1 m+1  1 + − , <x≤ , 2 6 2p p and the characteristic function  m 2m + 1   <x≤ ,  1, p 2p χp (x) = m+1 2m + 1    0, <x≤ , 2p p

a.e. in [0, 1]} .

0 ≤ m ≤ p − 1,

0 ≤ m ≤ p − 1.

(3.7)

(3.8)

(3.9)

It is readily seen that kp = k1 χp + k2 (1 − χp ). He shows that for each p, k p ∈ K (resp., χp ∈ X(D)) and that   1 k p  k∞ = 1 in L∞ (0, 1)-weak, resp., χp  (3.10) 2   1 1 1 1 1 = in L∞ (0, 1)-weak  .  + (3.11) p k 2 k1 k2 1/2 + x2 /6 Moreover, yp  y in H 1 (0, 1)-weak,

(3.12)

where yp denotes the solution of (3.4) corresponding to k = k p and y the solution of the boundary value problem      1 x2 dy − d + + y = 0 in ]0, 1[ , dx 2 6 dx (3.13)   y(0) = 1, y(1) = 2.

3. Lebesgue Measurable Characteristic Functions

219

Define the function kH (x) = which corresponds to

1 x2 + , 2 6

  4 1 1 x2 χH (x) = 1+ − ∈ co X(D). 2 2 6

(3.14)

(3.15)

Notice that kH appears in the second-order term and k∞ in the zeroth-order term in (3.13):   dy d (3.16) kH + k∞ y = 0 in ]0, 1[ . − dx dx It is easy to check that y(x) = 1 + x2 in [0, 1],

(3.17)

which is not equal to the solution y∞ of (3.4) for the weak limit k∞ = 1: y∞ (x) =

2(ex − e−x ) + e1−x − e−(1−x) . e − e−1

(3.18)

To our knowledge this was the beginning of the theory of homogenization in France. This is only one part of F. Murat [1]’s example. He also constructs an objective function for which the lower bound is not achieved by an element of K. This is a nonexistence result. The above example uses space varying coefficients k1 (x) and k2 (x). However, it is still valid for two positive constants k1 > 0 and k2 > 0. It is easy to show that   1 k1 + k2 1 1 1 1 k2 , χH = k∞ = , χ∞ = , and = + , 2 2 kH 2 k1 k2 k1 + k2 and the solution y of the boundary value problem (3.16) is given by k1 + k2 2 sinh cx + sinh c(1 − x) ≥ 1, , where c = √ sinh c 2 k1 k2 and the solution y∞ by 2 sinh x + sinh(1 − x) . y∞ (x) = sinh 1 Thus for k1 = k2 , or equivalently c > 1, y = y∞ . In fact, using the same sequence of χp ’s, the sequence yp = y(χp ) of solutions of (4.5) weakly converges to yH = y(χH ), which is different from the solution y∞ = y(χ∞ ) for k1 = k2 . This is readily seen by noticing that since χ∞ and χH are constant y(x) =

−∆(kH yH ) = χ∞ f = −∆(k∞ y∞ ) ⇒ kH yH = k∞ y∞

⇒ yH =

(k1 + k2 )2 y∞ = y∞ , for k1 = k2 . 4k1 k2

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Chapter 5. Metrics via Characteristic Functions

Despite this example, we shall see in the next section that in some cases we obtain the existence (and uniqueness) of a maximizer in co X(D) which belongs to X(D).

3.3

Nice or Measure Theoretic Representative

Since the strong and weak Lp -topologies are defined on equivalence classes [Ω] of (Lebesgue) measurable subsets Ω of RN , it is natural to ask if there is a nice representative that is generic of the class [Ω]. For instance we have seen in Chapter 2 that within the equivalence class of a set of class C k there is a unique open and a unique closed representative and that all elements of the class have the same interior, boundary, and exterior. The same question will again arise for finite perimeter sets. As an illustration of what is meant by a nice representative, consider the smiling and the expressionless suns in Figure 5.1. The expressionless sun is obtained by adding missing points and lines “inside” Ω and removing the rays “outside” Ω. This “restoring/cleaning” operation can be formalized as follows.

Ω


Ω


Figure 5.1. Smiling sun Ω and expressionless sun Ω. Definition 3.3. Associate with a Lebesgue measurable set Ω in RN the sets   def

Ω0 = x ∈ RN : ∃ρ > 0 such that m(Ω ∩ B(x, ρ) = 0 ,     def

Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) ,     def

Ω• = x ∈ RN : ∀ρ > 0 such that 0 < m Ω ∩ B(x, ρ) < m B(x, ρ) and the measure theoretic exterior O, interior I, and boundary ∂∗ Ω   m(B(x, r) ∩ Ω) def O = x ∈ RN : lim =0 , r0 m(B(x, r))   m(B(x, r) ∩ Ω) def I = x ∈ RN : lim =1 , r0 m(B(x, r)) def

∂∗ Ω =

x∈R

N

& m(B(x, r) ∩ Ω) m(B(x, r) ∩ Ω) : 0 ≤ lim inf < lim sup ≤1 . r0 m(B(x, r)) m(B(x, r)) r0

We shall say that I is the nice or measure theoretic representative of Ω.

3. Lebesgue Measurable Characteristic Functions

221

The six sets Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω are invariant for all sets in the equivalence class [Ω] of Ω. They are two different partitions of RN Ω0 ∪ Ω 1 ∪ Ω • = R N

and

O ∪ I ∪ ∂∗ Ω = RN

(3.19)

like int Ω ∪ int Ω ∪ ∂Ω = RN . The next theorem links the six invariant sets and describes some of the interesting properties of the measure theoretic representative. Theorem 3.3. Let Ω be a Lebesgue measurable set in RN and let Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω be the sets constructed from Ω in Definition 3.3. (i) The sets O, I, and ∂∗ Ω are invariants of the equivalence class [Ω]. Moreover, they are Borel measurable and χI = χΩ ,

χO = χ
Ω ,

and

χ∂∗ Ω = 0

a.e. in RN .

(ii) The sets Ω0 , Ω1 , and Ω• are invariants of the equivalence class [Ω]. The sets Ω0 and Ω1 are open, Ω• is closed, and int Ω ⊂ Ω1 = int I, int Ω ⊂ Ω0 = int O, ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω, (3.20) ∂∗ Ω ∪ ∂O ∪ ∂I ⊂ Ω• ⊂ ∂Ω. (3.21) (iii) The condition m(∂Ω) = 0 implies m(∂I) = 0. When m(∂I) = 0, then int I and I can be chosen as the respective open and closed representatives of Ω. In particular, this is true for Lipschitzian sets and hence for convex sets. Proof. (i) All the sets involved depend only on χΩ almost everywhere. They are invariants of the equivalence class [Ω]. From L. C. Evans and R. F. Gariepy [1, Lem. 2, sect. 5.11, p. 222, Cor. 3, sect. 1.7.1, p. 45], I, O, and ∂∗ Ω are Borel measurable and m(I\Ω ∪ Ω\I) = 0, and lim

r0

m(B(x, r) ∩ Ω) = 1 a.e. in Ω, m(B(x, r))

lim

r0

m(B(x, r) ∩ Ω) = 0 a.e. in Ω. m(B(x, r))

Therefore, m(∂∗ Ω) = 0, χI = χΩ , and χO = χ
Ω almost everywhere in RN . (ii) We use the proof given by E. Giusti [1, Prop. 4.1, pp. 42–43] for Ω0 and Ω1 . To show that Ω0 is open, pick any   point x in Ω0 . By construction, there exists ρ > 0 such that m Ω ∩ B(x, ρ) = 0. For any y ∈ B(x, ρ) (that is, |y − x| < ρ), ρ0 = ρ − |x − y| > 0. Then B(y, ρ0 ) ⊂ B(x, ρ) and  choose  m B(y, ρ0 ) ∩ Ω ≤ m B(x, ρ) ∩ Ω = 0. So B(x, ρ) ⊂ Ω0 and Ω0 is open. Similarly, by repeating the argument for Ω, we obtain that Ω1 is open. By complementarity, Ω• is closed. By definition, int Ω ⊂ Ω1 ⊂ I and, since Ω1 is open, Ω1 ⊂ int I. For all x ∈ int I, there exists r > 0 such that B(x, r) ⊂ I and       m B(x, r) = m B(x, r) ∩ I = m B(x, r) ∩ Ω . Hence int I ⊂ Ω1 and necessarily int I = Ω1 . Similarly, int Ω ⊂ Ω0 = int O. By taking the union term by term of the two chains int Ω ⊂ int I ⊂ Ω1 = int I ⊂ I

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Chapter 5. Metrics via Characteristic Functions

and int Ω ⊂ Ω0 = int O ⊂ O and then the complement, we get ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω. From the first two relations (3.20), Ω1 = int I implies Ω1 = I ⊂ O and Ω0 = int O implies Ω0 = O ⊂ I. Therefore, Ω• = Ω1 ∩ Ω0 ⊃ I ∩ I = ∂I and similarly Ω• = Ω1 ∩ Ω0 ⊃ O ∩ O = ∂O. (iii) From part (ii). The inclusions (3.20) indicate that the measure theoretic interior and exterior are enlarged and that the measure theoretic boundary is reduced. In general those operations do not commute with the set theoretic operations. For instance, they do not commute with the closure, as can be seen from the following simple example. Example 3.2. Consider the set Ω of all rational numbers in [0, 1]. Then O = Ω0 = R I = Ω1 = ∅ ⇒ I(Ω) = ∅, ∂∗ Ω = Ω• = ∅ O(Ω) = (Ω)0 = R \[0, 1] I(Ω) = (Ω)1 = ]0, 1[ ⇒ I(Ω) = ]0, 1[ . ∂ (Ω) = (Ω) = {0, 1} ∗

•

However, the complement operation commutes with the other set of operations and determines the same sets (Ω)0 = Ω1 , (Ω)1 = Ω0 , I(Ω) = O(Ω), I(Ω) = O(Ω),

(Ω)• = Ω• , ∂∗ (Ω) = ∂∗ (Ω).

(3.22) (3.23)

The next theorem is a companion to Lemma 5.1 in Chapter 2. Theorem 3.4. The following conditions are equivalent: (i) int Ω = Ω (ii) int Ω = Ω1 ,

and

int Ω = Ω.

int Ω = Ω0 ,

∂Ω = ∂Ω0 = ∂Ω1 (= Ω• ).

If either of the above two conditions is satisfied, then int I = int Ω,

int O = int Ω,

I ∩ O = ∂Ω = ∂Ω = ∂Ω.

Proof. (i) ⇒ (ii). First note that Ω1 = (Ω0 ∪ Ω• ) = Ω0 ∩ Ω• ⊂ Ω0 . Similarly, Ω0 ⊂ Ω1 . From the first two relations (3.20), int Ω ⊂ Ω1 and int Ω ⊂ Ω0 . Combining them with the two identities in (i), Ω = int Ω ⊂ Ω1 ⊂ Ω0 and Ω = int Ω ⊂ Ω0 ⊂ Ω1 . By taking the intersection of the two chains, ∂Ω ⊂ Ω0 ∩ Ω1 = (Ω0 ∪ Ω1 ) = Ω• . But int Ω ∪ int Ω ∪ ∂Ω and Ω1 ∪ Ω0 ∪ Ω• are two disjoint partitions of RN such that int Ω ⊂ Ω1 , int Ω ⊂ Ω0 , and ∂Ω ⊂ Ω• . We conclude that int Ω = Ω1 , int Ω = Ω0 , ∂Ω = Ω• , and ∂Ω = Ω• = ∂Ω0 = ∂Ω1 since,

3. Lebesgue Measurable Characteristic Functions

223

by Lemma 5.1 in Chapter 2, int Ω = Ω and int Ω = Ω imply ∂ int Ω = ∂Ω = ∂ int Ω. (ii) ⇒ (i). By assumption, Ω = int Ω ∪ ∂Ω = Ω1 ∪ ∂Ω1 = Ω1 = int Ω, Ω = int Ω ∪ ∂Ω = Ω0 ∪ ∂Ω0 = Ω0 = int Ω ⇒ ∂Ω = Ω ∩ Ω = int Ω ∩ Ω = Ω ∩ Ω = ∂ Ω ⇒ ∂Ω = Ω ∩ Ω = Ω ∩ int Ω = Ω ∩ Ω = ∂ Ω.

3.4

The Family of Convex Sets

Convex sets will play a special role here and in the subsequent chapters. We shall say that an equivalence class [Ω] of Lebesgue measurable subsets of D is convex if there exists a convex Lebesgue measurable subset Ω∗ of D such that [Ω] = [Ω∗ ]. We also introduce the notation def

C(D) = {χΩ : Ω convex subset of D} .

(3.24)

Theorem 3.5. Let Ω = ∅ be a convex subset of RN . (i) Ω and int Ω are convex.7 (ii) (int Ω) = Ω = Ω = int Ω. If int Ω = ∅, then int Ω = Ω. If int Ω = ∅ and Ω = RN , then ∂Ω = ∅ and int Ω = ∅. (iii) Given a measurable (resp., bounded measurable) subset D in RN ∀1 ≤ p < ∞,

C(D) is closed in Lploc (D) (resp., Lp (D)).

Proof. (i) For all x and y in Ω, there exist {xn } and {yn } in Ω such that xn → x and yn → y. Then for all λ ∈ [0, 1] Ω  λ xn + (1 − λ) yn → λ x + (1 − λ) y ∈ Ω. If int Ω = ∅, then for all x and y in int Ω, there exist rx > 0 and ry > 0 such that B(x, rx ) ⊂ Ω and B(y, ry ) ⊂ Ω. So for all λ ∈ [0, 1] def

xλ = λx + (1 − λ)y ∈B(xλ , λrx + (1 − λ)ry ) ⊂ λB(x, rx ) + (1 − λ)B(y, ry ) ⊂ Ω and xλ ∈ int Ω. (ii) By definition, int Ω = Ω and int Ω = Ω. If Ω = RN , Ω ⊂ Ω = ∅. This implies int Ω = Ω = Ω = ∅ and int Ω = Ω. If Ω = RN and int Ω = ∅, 7 By

convention ∅ is convex.

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Chapter 5. Metrics via Characteristic Functions

from Theorem 5.6 in Chapter 2, Ω is locally Lipschitzian, and from Theorem 5.4 in Chapter 2, ∂Ω = ∅, int Ω = ∅, int Ω = Ω, and Ω = Ω. Finally, if int Ω = ∅, then Ω = RN and Ω belongs to a linear subspace L of RN of dimension strictly less than N . Therefore Ω ⊃ Ω ⊃ L, RN = Ω ⊃ Ω ⊃ L = RN , and Ω = Ω. (iii) It is sufficient to prove the result for D bounded. For any Cauchy sequence {χn } in C(D), there exist a sequence of convex sets {Ωn } and χΩ in X(D) such that χΩn → χΩ in Lp (D). In particular, there exists a subsequence χk = χΩnk such that χk (x) → χΩ (x) almost everywhere in D. Define Ω∗ = {x ∈ D : χk (x) → 1 as k → ∞}. def

To show that Ω∗ is convex consider x and y in Ω∗ . There exists K ≥ 1 such that ∀k ≥ K,

|χk (x) − 1| < 1/2,

|χk (y) − 1| < 1/2,

and, since χk is either 0 or 1, for all k ≥ K, χk (x) = 1 = χk (y). By convexity for all λ ∈ [0, 1], xλ = λx + (1 − λ)y ∈ Ωnk and ∀k ≥ K,

χk (λx + (1 − λ)y) = 1 → 1

⇒ x λ ∈ Ω∗ .

But χk (x) → χΩ (x) almost everywhere and lim χk (x) ∈ {0, 1} a.e.

k→∞

So for x ∈ D\Ω∗ , χk (x) → 0 = χΩ∗ (x) almost everywhere in D. By the Lebesguedominated convergence theorem, χk → χΩ∗ in Lp (D) and necessarily χΩ = χΩ∗ . This means that [Ω] is a convex equivalence class. We shall show in section 6.1 (Corollary 1) that for bounded domains D, C(D) is also compact for the Lp (D) topology, 1 ≤ p < ∞.

3.5

Sobolev Spaces for Measurable Domains

The lack of a priori smoothness on Ω may introduce technical difficulties in the formulation of some boundary value problems. However, it is possible to relax such boundary value problems from smooth bounded open connected domains Ω to ´sio [7]). For instance consider the homogeneous measurable domains (cf. J.-P. Zole Dirichlet boundary value problem −y = f in Ω,

y = 0 on Γ

(3.25)

over a bounded open connected domain Ω with a boundary Γ of class C 1 and associate with its solution y = y(Ω) the objective function and volume constraint   1 2 J(Ω) = |y − g| dx, dx = π. (3.26) 2 Ω Ω There is a priori no reason to assume that an optimal (minimizing) domain Ω∗ is of class C 1 or is connected. So the problem must be suitably relaxed to a large

3. Lebesgue Measurable Characteristic Functions

225

enough class of domains, which preserves the meaning of the underlying function spaces, the well-posedness of the original problem, and the volume. To extend problem (3.25)–(3.26) to Lebesgue measurable sets, we first have to make sense of the Sobolev space for measurable subsets Ω of D. Theorem 3.6. Let D be an open domain in RN . For any Lebesgue measurable subset Ω of RN , the spaces  def

H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D , (3.27)

 def H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D (3.28) are closed subspaces of H01 (D) and hence Hilbert spaces.8 Similarly, for any χ ∈ co X(D),  def

H•1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)∇ϕ = 0 a.e. in D , (3.29)

 def H1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)ϕ = 0 a.e. in D (3.30) are also closed subspaces of H01 (D) and hence Hilbert spaces. Furthermore9 H1 (Ω; D) ⊂ H•1 (Ω; D),

H1 (χ; D) ⊂ H•1 (χ; D).

Proof. We give only the proof for H•1 (Ω; D). Let {ϕn } in H•1 (Ω; D) be a Cauchy sequence. It converges to an element ϕ in the H01 (D)-topology. Hence {∇ϕn } converges to ∇ϕ in L2 (D). But for all n (1 − χΩ )∇ϕn = 0 in L2 (D). By Schwarz’s inequality the map ϕ → (1 − χΩ )∇ϕ is continuous and (1 − χΩ )∇ϕ = 0 in L2 (D). So finally ϕ ∈ H•1 (Ω; D) and H•1 (Ω; D) is a closed subspace of H01 (D). Assuming that D is bounded, the variational problems  1  find y = y(Ω) ∈ H• (Ω; D) such that    ∀ϕ ∈ H•1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx, D D  1  find y = y(Ω) ∈ H (Ω; D) such that    ∀ϕ ∈ H1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx D

(3.31)

(3.32)

D

8 Observe that for two open domains in the same equivalence class, the spaces defined by (3.27)– (3.28) coincide. Therefore their functions do not see cracks in the underlying domain. The case of cracks will be handled in Chapter 8 by capacity methods. 9 From L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 130], ∇ϕ = 0 a.e. on {ϕ = 0}.

226

Chapter 5. Metrics via Characteristic Functions

now make sense and have unique solutions for measurable subsets Ω of D (or even χ ∈ co X(D)), and the associated objective function   J(Ω) = h χΩ , y(Ω) ,

def

h(χ, ϕ) =

1 2

 χ|ϕ − g|2 dx

(3.33)

D

can be minimized over all measurable subsets Ω of D (or all χ ∈ co X(D)) with fixed measure m(Ω) = π. The above problems are now well-posed, and their restriction to smooth bounded open connected domains coincides with the initial problem (3.25)–(3.26). Indeed, if Ω is a connected open domain with a boundary Γ of class C 1 , then Γ has a zero Lebesgue measure and the definition (3.28) of H•1 (Ω; D) coincides with H1 (Ω; D). Therefore, the problem specified by (3.31)–(3.33) is a well-defined extension of problem (3.25)–(3.26). In general, when Ω contains holes, the elements of H•1 (Ω; D) are not necessarily equal to zero on the boundary of each hole and can be equal to different constants from hole to hole, as in physical problems involving a potential. Example 3.3. Let D = ]−2, 2[ , Ω = ]−2, −1[ ∪ ]1, 2[ , and f = 1 on D. The solution of (3.31) is given by  d2 y   − 2 =1 dx   y(−2) = 0,

on ]−2, −1[ ,

 d2 y   − 2 = 1 on ]1, 2[ , dx   y(2) = 0, dy (1) = 0, dx

dy (−1) = 0, dx 1 y= on [−1, 1],  2 x  −(x + 2) 2 in ]−2, −1[ , ⇒ y(x) = 12 in [−1, 1],   −(x − 2) x2 in ]1, 2[ .

Example 3.4. Let D = B(0, 2), the open ball of radius 2 centered in 0 in R2 , H = B(0, 1), Ω = D H, and f = 1. In polar coordinates the solution of (3.31) is given by    1 d dy   r = 1 in ]1, 2[ , − 1 1 3 r dr dr andy(r) = ln +  2 2 4   y(2) = 0, dy (1) = 0, dr or explicitly y(r) =

1 2 1 2

ln 2r + 1 − ln 12 + 34

r2 4

in ]1, 2[ , in [0, 1].

in [0, 1],

3. Lebesgue Measurable Characteristic Functions

227

The last example is a special case that retains the characteristics of the onedimensional example. In higher dimensions the normal derivative is not necessarily zero on the “internal boundary” of Ω. Example 3.5. Let D = B(0, 1) in R2 , let H be a bounded open connected hole in D such that H ⊂ D, and let Ω = D H. Then it can be checked that the solution of (3.31) is of the form −∆y = f in Ω,

y = 0 on ∂D,

where the constant c on H is determined by the condition  ∂y ∀ϕ ∈ H•1 (Ω; D), ϕ dγ = 0, ∂Ω ∂n or equivalently

 ∂H

∂y dγ = 0. ∂n

Example 3.6. Let D = ]−2, 2[ × ]−2, 2[ and let Ω1 and Ω2 be two open squares in G = ]−1, 1[ × ]−1, 1[ such that Ω1 ⊂ G, Ω2 ⊂ G, and Ω1 ∩ Ω2 = ∅. Define the domain as Ω = Ω0 ∪ Ω1 ∪ Ω2 ,

Ω0 = D\G

(cf. Figure 5.2).

Then the solution of (3.31) is characterized by −∆y = f in Ω,

y = 0 on ∂D,



y = c on G\ Ω1 ∪ Ω2 ,

where the constant c is determined by the condition    ∂y ∂y ∂y dγ + dγ + dγ = 0 int ∂n ∂n ∂n ∂Ω0 ∂Ω1 ∂Ω2 and ∂Ωint 0 = ∂G, the interior boundary of Ω0 .

Ω2

Ω0 = D\G

Ω1

Figure 5.2. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 .

228

4

Chapter 5. Metrics via Characteristic Functions

Some Compliance Problems with Two Materials

As a first example consider the optimal compliance problem where the optimization variable is the distribution of two materials with different physical characteristics within a fixed domain D. It cannot a priori be assumed that the two regions are separated by a smooth boundary and that each region is connected. The optimal solution may lead to a nonsmooth interface and even to the mixing of the two materials. This type of solution occurs in control or optimization problems over a bounded nonconvex subset of a function space. In general, their relaxed solution lies in the closed convex hull of that subset. In control theory, the phenomenon is known as chattering control. To illustrate this approach it is best to consider a generic example. In section 4.1, we consider a variation of the optimal design problem ´a and K. Malanowski [1] in 1970 and then discuss its original studied by J. Ce version in section 4.2. The variation of the problem is constructed in such a way that the form of the associated variational equation is similar to the one of Example 3.1, which provided a counterexample to the weak continuity of the solution. Yet the maximization of the minimum energy yields as solution a characteristic function in both cases. However, this is no longer true for the minimization of the same minimum energy as was shown by F. Murat and L. Tartar [1, 3].

4.1

Transmission Problem and Compliance

Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth boundary ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 5.3. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem   − k1 y = f in Ω1 , −k2 y = 0 in Ω2 , (4.1) ∂y ∂y  y = 0 on ∂D, k1 + k2 = 0 on ∂Ω1 ∩ ∂Ω2 , ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Our objective is to maximize the equivalent of the compliance  J(Ω1 ) = − f y dx (4.2) Ω1

Ω1

Ω2

Figure 5.3. Fixed domain D and its partition into Ω1 and Ω2 .

4. Some Compliance Problems with Two Materials

229

over all domains Ω1 in D. In mechanics the compliance is associated with the total work of the body forces f . Denote by Ω the domain Ω1 . By complementarity Ω2 = D Ω and let χ = χΩ . Problem (4.1) can be rewritten in the following variational form: find y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D),   [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = χf ϕ dx. D

(4.3)

D

For k1 > 0 and k2 > 0 and all χ in X(D) def

k(x) = k1 χ(x) + k2 (1 − χ(x)), 0 < min{k1 , k2 } ≤ k(x) ≤ max{k1 , k2 } a.e. in D.

(4.4)

By the Lax–Milgram theorem, the variational equation (4.3) still makes sense and has a unique solution y = y(χ) in H01 (D) that coincides with the solution of the boundary value problem −div (k ∇y) = χf in D,

y = 0 on ∂D.

(4.5)

Notice that the high-order term has the same form as the term in Example 3.1. As for the objective function, it can be rewritten as  J(χ) = − χf y(χ) dx. (4.6) D

Thus the initial boundary value problem (4.1) has been transformed into the variational problem (4.3), and the initial objective function (4.2) into (4.6). Both make sense for χ in X(D) and even in co X(D). The family of characteristic functions X(D) and its closed convex hull co X(D) have been defined and characterized in (3.2) and (3.3). The objective function (4.6) can be further rewritten as a minimum J(χ) = for the energy function def

min

ϕ∈H01 (D)

E(χ, ϕ)

(4.7)

 (k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx

E(χ, ϕ) =

(4.8)

D

associated with the variational problem (4.3). The initial optimal design problem becomes a max-min problem max J(χ) = max χ∈X(D)

min

χ∈X(D) ϕ∈H01 (D)

E(χ, ϕ),

where X(D) can be considered as a subset of Lp (D) for some p, 1 ≤ p < ∞.

(4.9)

230

Chapter 5. Metrics via Characteristic Functions This problem can also be relaxed to functions χ with value in [0, 1] max

J(χ) =

χ∈co X(D)

max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.10)

We shall refer to this problem as the relaxed problem. ´a and K. Malanowski These formulations have been introduced by J. Ce [1], who used the variable k(x) and the following equality constraint on its integral:   γ − k2 m(D) k(x) dx = γ or χ(x) dx = k1 − k 2 D D for some appropriate γ > 0. Moreover the force f in (4.1)–(4.3) was exerted everywhere in D and not only in Ω. So the objective function (4.2)–(4.6) was the integral of f y over all of D. We shall see in section 4.2 that the fact that the support of f is Ω or all of D does not affect the nature of the results. In both maximization problems (4.1)–(4.2) and (4.3)–(4.6) it is necessary to introduce a volume constraint in order to avoid a trivial solution. Notice that by using (4.3) with ϕ = y(χ) the objective function (4.6) becomes  (k1 χ + k2 (1 − χ)) |∇y(χ)|2 dx ≤ 0. J(χ) = − D

So maximizing J(χ) is equivalent to minimizing the integral of k(x) |∇y|2 . Therefore, χ = 0 (only material k2 ) is a maximizer since the corresponding solution of (4.3) is y = 0. In order to eliminate this situation we introduce the following constraint on the volume of material k1 :  χ dx ≥ α > 0 (4.11) D

for some α, 0 < α ≤ m(D). The case α = m(D) yields the unique solution χ = 1 (only material k1 ). So we can further assume that α < m(D). For 0 < α < m(D), the optimal design problem becomes max J(χ) = max χ∈X(D) χ∈X(D) D

χ dx≥α

D

min

E(χ, ϕ)

(4.12)

min

E(χ, ϕ).

(4.13)

ϕ∈H01 (D)

χ dx≥α

and its relaxed version max

χ∈co X(D)  χ dx≥α D

J(χ) =

max

χ∈co X(D) ϕ∈H01 (D)  χ dx≥α D

We shall now show that problem (4.13) has a unique solution χ∗ in co X(D) and that χ∗ is a characteristic function, χ∗ ∈ X(D), for which the inequality constraint is saturated:  ∗ χ∗ dx = α. (4.14) χ ∈ X(D) and D

4. Some Compliance Problems with Two Materials

231

At this juncture it is advantageous to incorporate the volume inequality constraint into the problem formulation by introducing a Lagrange multiplier λ ≥ 0. The formulation of the relaxed problem becomes   max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ),

def

χ dx − α

G(χ, ϕ, λ) = E(χ, ϕ) + λ

(4.15)

D

and E(χ, ϕ) is the energy function given in (4.7). We first establish the existence of saddle point solutions to problem (4.15). We use a general result in I. Ekeland and R. Temam [1, Prop. 2.4, p. 164]. The set co X(D) is a nonempty bounded closed convex subset of L2 (D) and the set H01 (D)×[R+ ∪{0}] is trivially closed and convex. The function G is concave-convex with the following properties:   ∀χ ∈ co X(D), (ϕ, λ) → G(χ, ϕ, λ) is convex, continuous, and (4.16) G(χ0 , ϕ, λ) = +∞;  ∃χ0 ∈ co X(D) such that ϕ 1lim +|λ|→∞ H

∀ϕ ∈

H01 (D),

∀λ ≥ 0,

0

the map χ → G(χ, ϕ, λ) is affine and

2

continuous for L (D)-strong.

(4.17)

For the first condition recall that 0 < α < m(D) and pick χ0 = 1 on D. To check the second condition pick any sequence {χn } in co X(D) which converges to some χ in L2 (D)-strong. Then the sequence also converges in L2 (D)-weak and by Lemma 3.1 in L∞ (D)-weak∗ . Hence G(χn , ϕ, λ) converges to G(χ, ϕ, λ). ˆ is of the form X × Y ⊂ co X(D) × {H 1 (D) × The set of saddle points (χ, ˆ y, λ) 0 + [R ∪ {0}]} and is completely characterized by the following variational equation and inequalities (cf. I. Ekeland and R. Temam [1, Prop. 1.6, p. 157]):  1 [k2 + (k1 − k2 )χ] ˆ ∇y · ∇ϕ − χf ˆ ϕ dx = 0, (4.18) ∀ϕ ∈ H0 (D),  D. / ˆ (χ − χ) (k1 − k2 )|∇y|2 − 2f y + λ ˆ dx ≤ 0, (4.19) ∀χ ∈ co X(D), D    ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≥ 0, λ (4.20) D

D

But for each χ ∈ co X(D) there exists a unique y(χ) solution of (4.18) and then

 ∀χ ˆ ∈ X, {χ} ˆ × Y ⊂ {χ} ˆ × {y(χ)} ˆ × [R+ ∪ {0}] . Therefore, y(χ) ˆ is independent of χ ˆ ∈ X, that is, Y = {y} × Λ and ˆ ∈ Λ, ∀χ ˆ ∈ X, λ ˆ inequality (4.19) is For each λ, terization of the maximizer χ ˆ ∈ X:   if 1, χ(x) ˆ = ∈ [0, 1], if   0, if

y(χ) ˆ = y.

completely equivalent to the following characˆ > 0, (k1 − k2 )|∇y|2 − 2f y + λ 2 ˆ = 0, (k1 − k2 )|∇y| − 2f y + λ 2 ˆ < 0. (k1 − k2 )|∇y| − 2f y + λ

(4.21)

232

Chapter 5. Metrics via Characteristic Functions

Associate with an arbitrary λ ≥ 0 the sets D+ (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ > 0}, D0 (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ = 0}.

(4.22)

In order to complete the characterization of the optimal triplets, we need the following general result. Lemma 4.1. Consider a function G:A×B →R

(4.23)

for some sets A and B. Define  def

g = inf sup G(x, y), x∈A y∈B

def

A0 =

 x ∈ A : sup G(x, y) = g ,



def

h = sup inf G(x, y), y∈B x∈A

def

B0 =

(4.24)

y∈B

 y ∈ B : inf G(x, y) = h . x∈A

(4.25)

When g = h the set of saddle points (possibly empty) will be denoted by def

S = {(x, y) ∈ A × B : g = G(x, y) = h} .

(4.26)

Then the following hold. (i) In general h ≤ g and ∀(x0 , y0 ) ∈ A0 × B0 ,

h ≤ G(x0 , y0 ) ≤ g.

(4.27)

(ii) If h = g, then S = A0 × B0 . Proof. (i) If A0 × B0 = ∅, there is nothing to prove. If there exist x0 ∈ A0 and y0 ∈ B0 , then by definition h = inf G(x, y0 ) ≤ G(x0 , y0 ) ≤ sup G(x0 , y) = g. x∈A

(4.28)

y∈B

(ii) If h = g, then in view of (4.26)–(4.27), A0 × B0 ⊂ S. Conversely if there exists (x0 , y0 ) ∈ S, then h = G(x0 , y0 ) = g and by the definitions of A0 and B0 , (x0 , y0 ) ∈ A0 × B0 . ˆ the characteristic function Associate with an arbitrary solution (χ, ˆ y, λ), χλˆ =

χD+ (λ) ˆ , 0,

ˆ = ∅, if D+ (λ) ˆ = ∅. if D+ (λ)

(4.29)

Then from (4.21)     ˆ = χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ ˆ a.e. in D χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ λ

4. Some Compliance Problems with Two Materials and

233 

 ˆ [χk ˆ 1 + (1 − χ)k ˆ 2 ] |∇y| − 2χf ˆ y dx + λ

ˆ = G(χ, ˆ y, λ) D

.

 χ ˆ dx − α

2

/

D

ˆ dx − λα ˆ k2 |∇y|2 + χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ

= D = D

. / ˆ dx − λα ˆ k2 |∇y|2 + χλˆ (k1 − k2 )|∇y|2 − 2f y + λ

ˆ = G(χλˆ , y, λ). ˆ is also a saddle point. So there exists a maximizer From Lemma 4.1, (χλˆ , y, λ) χλˆ ∈ X(D) that is a characteristic function and necessarily max

min

χ∈X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ) =

max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

ˆ ∈ Λ such that λ ˆ > 0, then by But we can show more than that. If there exists λ construction χ ˆ ≥ χλˆ and by (4.20)   α= χ ˆ dx ≥ χλˆ dx = α. D

D

ˆ then χ ˆ = χλˆ . But, always by Since χ ˆ = χλˆ = 1 almost everywhere in D+ (λ), construction, χλˆ is independent of χ. ˆ Thus the maximizer is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} is a degenerate one. Set ϕ = y in (4.18) and regroup the terms as follows:  (k2 + (k1 − k2 )χ) ˆ |∇y|2 − f y χ ˆ dx 0= D       k1 − k2 k1 − k2 2 2 χ ˆ |∇y| dx + χ ˆ |∇y| − f y χ ˆ dx. = k2 + 2 2 D D The integrand of the first integral is positive. From the characterization of χ, ˆ the integrand of the second one is also positive. Hence they are both zero almost everywhere in D. As a result m(D+ (0)) = 0

and ∇y = 0.

ˆ = Therefore y = 0 in D. Hence the saddle points are of the general form (χ, ˆ y, λ) (χ, ˆ 0, 0), χ ˆ ∈ X. In particular, D+ (0) = ∅ and from our previous  considerations χ∅ ; that is, χ ˆ = 0 is a solution. But this is impossible since D χ ˆ dx ≥ α > 0. ˆ = 0 cannot occur. Therefore λ In conclusion, there exists a (unique for α > 0) maximizer χ∗ in co X(D) that is in fact a characteristic function, and necessarily max χ∈X(D) D

χ dx≥α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D)  χ dx≥α D

E(χ, ϕ).

(4.30)

234

Chapter 5. Metrics via Characteristic Functions

Moreover, for 0 < α ≤ m(D),



χ∗ dx = α

(4.31)

D

and χ∗ is the unique solution of the problem with an equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx=α

max

min

χ∈co X(D) ϕ∈H01 (D)  χ dx≥α

E(χ, ϕ).

(4.32)

D

As a numerical illustration of the theory consider (4.1) over the diamond-shaped domain D = {(x, y) : |x| + |y| < 1}

(4.33)

with the function f of Figure 5.4. This function has a sharp peak in (0, 0) which has been scaled down in the picture. The variational form (4.3) of the boundary value problem was approximated by continuous piecewise linear finite elements on each triangle, and the function χ by a piecewise constant function on each triangle. The constant on each triangle was constrained to lie between 0 and 1 together with the global constraint on its integral over the whole domain D. Figure 5.5 shows the optimal partition (the domain has been rotated by 45 degrees to save space). The ˆ m ), where χ grey triangles correspond to the region D0 (λ ˆ ∈ [0, 1]. The presence of this grey zone in the approximated problem is due to the fact that equality for the total area where χ ˆ = 1 could not be exactly achieved with the chosen triangulation of the domain. Thus the problem had to adjust the value of χ ˆ between 0 and 1 in

60 50 40 30 20

1 0.8

10

0.6 0.4

0 1

0.2 0.8

0 0.6

0.4

−0.2 0.2

−0.4 0

−0.2

−0.6 −0.4

−0.6

−0.8 −0.8

−1

−1

Figure 5.4. The function f (x, y) = 56 (1 − |x| − |y|)6 .

4. Some Compliance Problems with Two Materials

235

Figure 5.5. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of section 4.1. a few triangles in order to achieve equality for the integral of χ. ˆ For this example the Lagrange multiplier associated with the problem is strictly positive.

4.2

The Original Problem of C´ ea and Malanowski

We now put the force f everywhere in the fixed domain D. The same technique and the same conclusion can be drawn: there exists at least one maximizer of the compliance, which is a characteristic function. For completeness we give the main elements below. Fix the bounded open Lipschitzian domain D in RN and let Ω be a smooth subset of D. Let y be the solution of the transmission problem   − k1 y = f in Ω, −k2 y = f in D\Ω,    y = 0 on ∂D, (4.34)  ∂y ∂y    k1 + k2 = 0 on ∂Ω ∩ D, ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω (resp., D Ω) and f is a given function in L2 (D). Again problem (4.34) can be reformulated in terms of the characteristic function χ = χΩ : find y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D),   [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = f ϕ dx, D

(4.35)

D

with the objective function

 J(χ) = −

f y(χ) dx D

(4.36)

236

Chapter 5. Metrics via Characteristic Functions

to be maximized over all χ ∈ X(D). As in the previous case, the function J(χ) can be rewritten as the minimum of the energy function  def [k1 χ + k2 (1 − χ)] |∇ϕ|2 − 2f ϕ dx (4.37) E(χ, ϕ) = D

over H01 (D), J(χ) =

min

ϕ∈H01 (D)

E(χ, ϕ),

(4.38)

and we have the relaxed max-min problem max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.39)

Without constraint on the integral of χ, the problem is trivial and χ = 1 (resp., χ = 0) if k1 > k2 (resp., k2 > k1 ). In other words it is optimal to use only the strong material. In order to make the problem nontrivial assume that the strong material is k1 , that is, k1 > k2 , and put an upper bound on the volume of material k1 which occupies the part Ω of D:  χ dx ≤ α, 0 < α < m(D). (4.40) D

The case α = 0 trivially yields χ = 0. Under assumption (4.40) the case χ = 1 with only the strong material k1 is no longer admissible. Thus we consider for 0 < α < m(D) the problem max

min

χ∈co X(D) ϕ∈H01 (D)  χ dx≤α

E(χ, ϕ).

(4.41)

D

We shall now show that problem (4.41) has a unique solution χ∗ in co X(D) and that in fact χ∗ is a characteristic function for which the inequality constraint is saturated:  χ∗ dx = α. (4.42) D

This solves the original problem of C´ea and Malanowski with the equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ).

(4.43)

χ dx=α

As in section 4.1 it is convenient to reformulate the problem with a Lagrange multiplier λ ≥ 0 for the constraint inequality (4.40):   max min G(χ, ϕ, λ), G(χ, ϕ, λ) = E(χ, ϕ) − λ χ dx − α . (4.44) 1 χ∈X(D) ϕ∈H0 (D) λ≥0

D

4. Some Compliance Problems with Two Materials

237

We then relax the problem to co X(D): max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

(4.45)

By the same arguments as the ones used in section 4.1 we have the existence of ˆ which are completely characterized by saddle points (χ, ˆ y, λ)  ∀ϕ ∈ H01 (D), [k2 + (k1 − k2 )χ]∇y ˆ · ∇ϕ − f ϕ dx = 0, (4.46) D  . / ˆ (χ − χ) (k1 − k2 )|∇y|2 − λ ˆ dx ≤ 0, (4.47) ∀χ ∈ co X(D), D   ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≤ 0, λ (4.48) D

D

As before y is unique and the set of saddle points of G is the closed convex set



 X × {y} × Λ ⊂ X(D) × {y} × {λ : λ ≥ 0} . ˆ m ≥ 0. The closed convex set Λ has a minimal element λ ˆ each maximizer χ For each λ, ˆ ∈ X is necessarily of the form  ˆ > 0,  if (k1 − k2 )|∇y|2 − λ 1, 2 ˆ = 0, χ(x) ˆ = ∈ [0, 1], if (k1 − k2 )|∇y| − λ   2 ˆ < 0. 0, if (k1 − k2 )|∇y| − λ Associate with each λ > 0 the sets

 D+ (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ > 0 ,

 D0 (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ = 0 ,

 D− (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ < 0 .

(4.49)

(4.50) (4.51) (4.52)

Define the characteristic function χm = χD+ (λm ) . By construction all χ ˆ ∈ X are of the form  ˆ χ ˆ dx ≤ α. (4.53) χm ≤ χ, D

ˆ m ) is also a saddle point of G. Therefore, we Again it is easy to show that (χm , y, λ have a maximizer χm ∈ X(D) over co X(D) which is a characteristic function and   χm dx ≤ χ ˆ dx ≤ α. D

D

ˆ > 0 in Λ, then for all χ If there exists λ ˆ∈X   χm dx = α = χ ˆ dx. D

D

238

Chapter 5. Metrics via Characteristic Functions

As a result the maximizer χm is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} cannot occur since the triplet (1, y1 , 0) would be a saddle point, where y1 is the solution of the variational equation (4.34) for χ = 1. To see this, first observe that for k1 > k2 ∀χ, ϕ,

G(χ, ϕ, 0) = E((χ, ϕ) ≤ E((1, ϕ) = G(1, ϕ, 0).

As a result inf G(χ, ϕ, λ) ≤ inf G(χ, ϕ, 0) ≤ inf G(1, ϕ, 0), ϕ

ϕ,λ

ϕ

inf G(1, ϕ, 0) = G(1, y1 , 0) ≤ sup inf G(χ, ϕ, 0) ϕ

χ

ϕ

⇒ sup inf G(χ, ϕ, λ) ≤ sup inf G(χ, ϕ, 0) = G(1, y1 , 0). χ ϕ,λ

χ

ϕ

But we know that there exists a unique y ∈ H01 (D) such that sup inf G(χ, ϕ, 0) ≤ sup G(χ, y, 0) = inf sup G(χ, ϕ, λ). χ

ϕ

ϕ,λ χ

χ

So from the above two inequalities, sup inf G(χ, ϕ, λ) = G(1, y1 , 0) = inf sup G(χ, ϕ, λ), χ ϕ,λ

ϕ,λ χ

(1, y1 , 0) is a saddle point of G, and  α≥ χ dx = m(D). D

This contradicts the fact that α < m(D). In conclusion in all cases there exists a (unique when 0 ≤ α < m(D)) maximizer χ∗ in co X(D), which is in fact a characteristic function, and max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx≥α

max

min

χ∈co X(D) ϕ∈H01 (D)  χ dx≥α

E(χ, ϕ).

D

Moreover, for 0 ≤ α < m(D)



χ∗ dx = α. D

This is precisely the solution of the original problem of C´ea and Malanowski and max χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D)  χ dx≥α

E(χ, ϕ).

D

Again, as an illustration of the theoretical results, consider (4.34) over the diamondshaped domain D defined in (4.33) with the function f of Figure 5.4 in section 4.1.

4. Some Compliance Problems with Two Materials

239

The variational form (4.35) of the boundary value problem was approximated in the same way as the variational form (4.3) of (4.1) in section 4.1. Figure 5.6 shows the optimal partition of the domain (rotated 45 degrees). The grey triangles ˆ m ), where χ ˆ ∈ [0, 1]. The black region corresponds correspond to the region D0 (λ ˆ m /(k1 − k2 ) and the white region to the ones where to the points where |∇y|2 > λ ˆ m /(k1 − k2 ), as can be readily seen in Figure 5.6. For this example the |∇y|2 < λ Lagrange multiplier associated with the problem is strictly positive. It is interesting to compare this computation with the one of Figure 5.5 in section 4.1, where the support of the force was restricted to Ω.

Figure 5.6. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of C´ea and Malanowski. Remark 4.1. The formulation and the results remain true if the generic variational equation is replaced by a variational inequality (unilateral problem) max χ∈X(D) D

4.3

min

ϕ∈{ψ∈H01 (D):ψ≥0 in D}

E(χ, ϕ).

χ dx=α

Relaxation and Homogenization

In sections 4.1 and 4.2 the possible homogenization phenomenon predicted in Example 3.1 of section 3.2 did not take place, and in both examples the solution was a characteristic function. Yet, it occurs when the maximization is changed to a ´a and K. Malanowski [1] of section 4.2: minimization in the problem of J. Ce inf χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ).

240

Chapter 5. Metrics via Characteristic Functions

In 1985 F. Murat and L. Tartar [1] gave a general framework to study this class of problems by relaxation. It is based on the use of L. C. Young [2]’s generalized functions (measures). They present a fairly complete analysis of the homogenization theory of second-order elliptic problems of the form  ∂  ∂u aij (x) = f. − ∂xi ∂xj ij

They give as examples the maximization and minimization versions of the problem of section 4.2. This material, which would have deserved a whole chapter in this book, is fortunately available in English (cf. F. Murat and L. Tartar [3]). More results on composite materials can be found in the book edited by A. Cherkaev and R. Kohn [1], which gathers a selection of translations of key papers originally written in French and Russian.

5

Buckling of Columns

One of the very early optimal design problem was formulated by Lagrange in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by T. Clausen in 1849. It consists in finding the best profile of a vertical column to prevent buckling. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. Consider a normalized column of unit height and unit volume. Denote by  the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies  1  1  2 EI |u | dx −  |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For sufficiently small load  the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest  for which this minimum is zero. This is equivalent to finding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (5.1) 1 0=u∈V |u |2 dx 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

5. Buckling of Columns

241

cross-sectional area A(x),



I(x) = c A(x)

1

⇒

A(x) dx = 1. 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we finally get

1

A |u |2 dx , 1 0=u∈V A∈A |u |2 dx 0    1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(5.2) (5.3)

0

This problem can also be reformulated by rewriting  1 def 1 − A0 A(x) = A0 + χ(x) (A1 − A0 ), χ(x) dx = α = A1 − A0 0 for some χ ∈ co X([0, 1]). Clearly the problem makes sense only for 0 < A0 ≤ 1. Then 1 [A0 + (A1 − A0 ) χ] |u |2 dx def 0 ˜ ˜ λ(χ), λ(χ) = inf . (5.4) sup 1 0=u∈V |u |2 dx χ∈co X([0,1]) 0 1

0

χ(x) dx=α

Rayleigh’s quotient is not a nice convex concave function with respect to (v, A), and its analysis necessitates tools different from the ones of section 4. One of the original elements of the paper of S. J. Cox and M. L. Overton [1] was to replace Rayleigh’s quotient by G. Auchmuty [1]’s dual variational principle for the eigenvalue problem (5.1). We first recall the existence of solution to the minimization of Rayleigh’s quotient. In what follows we shall use the norm u L2 for the space H01 (0, 1) and u L2 for the space H02 (0, 1). Theorem 5.1. There exists at least one nonzero solution u ∈ V to the minimization problem 1 A |u |2 dx def 0 λ(A) = inf . (5.5) 1 0=u∈V |u |2 dx 0 Then λ(A0 ) > 0 and for all A ∈ A, λ(A) ≥ λ(A0 ), and  1  1 ∀v ∈ V, |v  |2 dx ≤ λ(A0 )−1 A |v  |2 dx. 0

(5.6)

0

The solutions are completely characterized by the variational equation:  1  1 A u v  dx = λ(A) u v  dx. ∃u ∈ V, ∀v ∈ V, 0

0

(5.7)

242

Chapter 5. Metrics via Characteristic Functions

Proof. The infimum is bounded below by 0 and is necessarily finite. Let {un } be a minimizing sequence such that un L2 = 1. Then the sequence un L2 (D) is bounded. Hence {un } is a bounded sequence in H 2 (0, 1) and there exist u ∈ H 2 (0, 1) and a subsequence, still indexed by n, such that un  u in H02 (0, 1)-weak. Therefore the subsequence strongly converges in H01 (0, 1) and  1  1  1  1  2  2  2 |un | dx → |u | dx and A |u | dx ≤ lim inf A |un |2 dx 1= 0

1 ⇒

0

0

A |u |2 dx

1 0

n→∞

0

|u |2

dx

1

A |un |2 dx

≤ lim inf 0 1 n→∞

0

|un |2 dx

0

= λ(A),

and, by the definition of λ(A), u ∈ H02 (0, 1) is a minimizing element. Notice that & 1  2 A |u | dx 0 0 ∀A ∈ A, λ(A0 ) = inf : ∀u ∈ H02 (0, 1), u = 0 1  |2 dx |u 0 & 1 A |u |2 dx 2 0 : ∀u ∈ H0 (0, 1), u = 0 = λ(A) ≤ inf 1 |u |2 dx 0 and similarly λ(A) ≤ λ(A1 ). If λ(A0 ) = 0, we repeat the above construction and end up with an element u ∈ H02 (0, 1) such that  1  1  1 |u |2 dx = 1 and A0 |u |2 dx = A0 |u |2 dx = 0, 0

0

0

which is impossible in H02 (0, 1) since A0 = 0 and u = 0 implies u = 0. If u = 0, then Rayleigh’s quotient is differentiable and its directional semiderivative in the direction v is given by 1   1  1 A u v  dx u v dx  2 0 −2 A |u | dx 0  4 . 2 u 2L2 u L2 0 A nonzero solution u of the minimization problem is necessarily a stationary point, and for all v ∈ H02 (0, 1) 1  1   A0 |u |2 dx 1     A u v dx = 0 1 u v dx = λ(A) u v  dx.  |2 dx |u 0 0 D 0 Conversely any nonzero solution of (5.7) is necessarily a minimizer of Rayleigh’s quotient. The dual variational principle of G. Auchmuty [1] for the eigenvalue problem (5.5) can be chosen as

 def µ(A) = inf L(A, v) : v ∈ H02 (0, 1) , (5.8)  1 1/2  1 def 1 L(A, v) = A |v  |2 dx − |v  |2 dx . (5.9) 2 0 0

5. Buckling of Columns

243

Theorem 5.2. For each A ∈ A, there exists at least one minimizer of L(A, v), µ(A) = −

1 , 2λ(A)

(5.10)

and the set of minimizers of (5.8) is given by   u is solution of (5.7) and     def 1/2 E(A) = u ∈ H02 (0, 1) :   1 .  |u |2 dx = 1/λ(A)  

(5.11)

0

Proof. The existence of the solution follows from the fact that v → L(A, v) is weakly lower semicontinuous and coercive. From Theorem 5.1 the set E(A) is not empty, and for any u ∈ E(A) µ(A) ≤ L(A, u) = −

1 < 0. 2λ(A)

(5.12)

Therefore, the minimizers of (5.7) are different from the zero functions. For u = 0, the function L(A, u) is differentiable and its directional semiderivative is given by 

1

dL(A, u; v) = 0

1 A u v dx −  u L2 





1

u v  dx,

(5.13)

0

and any minimizer u of L(A, v) is a stationary point of dL(A, u; v), that is, 

1

∀v ∈ H02 (0, 1),

A u v dx −

0

1 u L2



1

u v  dx = 0.

(5.14)

0

Therefore, u is a solution of the eigenvalue problem with λ =

1 u 

L2

⇒ −

1 1 = µ(A) ≤ −  2λ 2λ(A)

from inequality (5.12). By the minimality of λ(A), we necessarily have λ = λ(A), and this concludes the proof of the theorem. Theorem 5.3. (i) The set A is compact in the L2 (0, 1)-weak topology. (ii) The function A → µ(A) is concave and upper semicontinuous with respect to the L2 (0, 1)-weak topology. (iii) There exists A in A which maximizes µ(A) over A and, a fortiori, which maximizes λ(A) over A. Proof. (i) A is convex, bounded, and closed in L2 (0, 1). Hence it is compact in L2 (0, 1) weak.

244

Chapter 5. Metrics via Characteristic Functions (ii) Concavity. For all λ in [0, 1] and A and A in A, L(λA + (1 − λ)A , v) = λL(A, v) + (1 − λ)L(A , v) ≥ λ inf L(A, v) + (1 − λ) inf L(A , v) v∈V

v∈V

⇒ µ(λA + (1 − λ)A ) = inf L(λA + (1 − λ)A , v) ≥ λµ(A) + (1−λ)µ(A ). v∈V

Upper semicontinuity. Let uA ∈ A be a minimizer of L(A, v) and {An } be a sequence converging to A in the L2 (0, 1)-weak topology. By Lemma 3.1, {An } converges to A in the L∞ (0, 1)-weak topology. Then  1 1 µ(An ) − µ(A) ≤ L(An , uA ) − L(A, uA ) = (An − A) |uA |2 dx  1 2 0 1 (An − A) |uA |2 dx = 0. ⇒ lim sup µ(An ) − µ(A) ≤ lim n→∞ 2 0 n→∞ (iii) The existence of solution follows from the concavity and upper semicontinuity of µ(A). Hence it is weakly upper semicontinuous over the weakly compact subset A of L2 (0, 1). The existence of a minimizer of λ(A) follows from identity 5.10.

6

Caccioppoli or Finite Perimeter Sets

The notion of a finite perimeter set has been introduced and developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem, named after the Belgian physicist and professor (1801–1883), who did experimental observations on the geometry of soap films. A modern treatment of this subject can be found in the book of E. Giusti [1]. One of the difficulties in studying the minimal surface problem is the description of such surfaces in the usual language of differential geometry. For instance, the set of possible singularities is not known. Finite perimeter sets provide a geometrically significant solution to Plateau’s problem without having to know ahead of time what all the possible singularities of the solution can be. The characterization of all the singularities of the solution is a difficult problem which can be considered separately. This was a very fundamental contribution to the theory of variational problems, where the optimization variable is the geometry of a domain. This point of view has been expanded, and a variational calculus was developed by F. J. Almgren, Jr. [1]. This is the theory of varifolds. Perhaps the one most important virtue of varifolds is that it is possible to obtain a geometrically significant solution to a number of variational problems, including Plateau’s problem, without having to know ahead of time what all the possible singularities of the solution can be. (cf. F. J. Almgren, Jr. [1, p. viii]). The treatment of this topic is unfortunately much beyond the scope of this book, and the interested reader is referred to the above reference for more details.

6. Caccioppoli or Finite Perimeter Sets

6.1

245

Finite Perimeter Sets

Given an open subset D of RN , consider L1 -functions f on D with distributional gradient ∇f in the space M 1 (D)N of (vectorial) bounded measures; that is,  def f div ϕ

dx : D1 (D; RN ) → R ϕ

→ ∇f, ϕ

D = − D

is continuous with respect to the topology of uniform convergence in D: def

∇f M 1 (D)N = def

def

sup 1

N

∇f, ϕ

D < ∞,  ϕC = sup | ϕ(x)|RN ,

ϕ  ∈D (D;R )  ϕC ≤1 0 

(6.1)

x∈D

where M 1 (D) = D (D) is the topological dual of D0 (D), and    N ∇f ∈ L D0 (D; RN ), R ≡ L D0 (D), R = M 1 (D)N . Such functions are known as functions of bounded variation. The space  def

BV(D) = f ∈ L1 (D) : ∇f ∈ M 1 (D)N

(6.2)

endowed with the norm f BV(D) = f L1 (D) + ∇f M 1 (D)N

(6.3)

is a Banach space. Definition 6.1 (L. C. Evans and R. F. Gariepy [1]). A function f ∈ L1loc (U ) in an open subset U of RN has locally bounded variation if for each bounded open subset V of U such that V ⊂ U , f ∈ BV(V ). The set of all such functions will be denoted by BVloc (U ). Theorem 6.1. Given an open subset U of RN , f belongs to BVloc (U ) if and only if for each x ∈ U there exists ρ > 0 such that B(x, ρ) ⊂ U and f ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. If f ∈ BVloc (U ), then the result is true by specializing to balls in each point of U . Conversely, by assumption, the open set U is covered by a family F(U ) of balls B(x, ρx ) such that B(x, ρx ) ⊂ U and f ∈ BV(B(x, ρx )). Given a bounded open subset V of U such that V ⊂ U , the compact subset V can be covered by a finite number of balls {B(xi , ρi )}, 1 ≤ i ≤ n, in F(U ) for points xi ∈ V , 1 ≤ i ≤ n, V ⊂

n *

Bi ,

def

Bi = B(xi , ρxi ).

i=1

Denote by {ψi ∈ D(Bi )}ni=1 a partition of unity for the family {Bi } such that ∀i, 0 ≤ ψi ≤ 1 in Bi , and

n i=1

ψi = 1 on V .

246

Chapter 5. Metrics via Characteristic Functions

Then for each ϕ

∈ V 1 (V )N ϕ

=

n

ϕ

i on V,

def

ϕ

i = ψi ϕ

,

i=1

∈ V 1 (V )N such that and by construction ϕ

i ∈ V (Bi ∩ V )N . Therefore, for all ϕ  ϕC(V ) ≤ 1  n  n  − f div ϕ

dx = − f div ϕ

i dx = − f div ϕ

i dx 1

V

i=1

V

i=1

 n  f div ϕ

dx ≤ ⇒ V

i=1

Bi ∩V

V ∩Bi

f div ϕ

i dx

 n n ⇒ f div ϕ

dx ≤ ∇f M 1 (Bi ∩V ) ≤ ∇f M 1 (Bi ) < ∞, V

i=1

since V (Bi ∩ V ) ⊂ V (Bi ) and f ∈ BVloc (RN ). 1

N

1

N

i=1

and  ϕi C(Bi ) ≤ 1. Therefore, ∇f M 1 (V ) is finite

For more details and properties see also F. Morgan [1, p. 117], H. Federer [5, sect. 4.5.9], L. C. Evans and R. F. Gariepy [1], W. P. Ziemer [1], and R. Temam [1]. As in the previous section, consider measurable subsets Ω of a fixed bounded open subset D of RN . Their characteristic functions χΩ ∈ X(D) are L1 (D)-functions  χΩ dx = m(Ω) ≤ m(D) < ∞ (6.4) χΩ L1 (D) = D

with distributional gradient 

∀ ϕ ∈ D(D)

N

,

def



∇χΩ , ϕ

D = −

χΩ div ϕ

dx.

(6.5)

D

When Ω is an open domain with boundary Γ of class C 1 , then by the Stokes divergence theorem    div ϕ

dx = − ϕ

· n dΓ = − ϕ

· n dΓ, − Ω∩D

∂(Ω∩D)

Γ∩D

where n is the outward normal field along ∂(Ω ∩ D). Since Γ is of class C 1 the normal field n along Γ belongs to C 0 (Γ), and from the last identity the maximum is   def 2 |n| dΓ = dΓ = HN −1 (Γ ∩ D), PD (Ω) = ∇χΩ M 1 (D) = Γ∩D

Γ∩D

the (N − 1)-dimensional Hausdorff measure of Γ ∩ D. As a result HN −1 (Γ) = PD (Ω) + HN −1 (Γ ∩ ∂D).

(6.6)

Thus the norm of the gradient provides a natural relaxation of the notion of perimeter to the following larger class of domains.

6. Caccioppoli or Finite Perimeter Sets

247

Definition 6.2. Let Ω be a Lebesgue measurable subset of RN . (i) The perimeter of Ω with respect to an open subset D of RN is defined as def

PD (Ω) = ∇χΩ M 1 (D)N .

(6.7)

Ω is said to have finite perimeter with respect to D if PD (Ω) is finite. The family of measurable characteristic functions with finite measure and finite relative perimeter in D will be denoted as def

BX(D) = {χΩ ∈ X(D) : χΩ ∈ BV(D)} . (ii) Ω is said to have locally finite perimeter if for all bounded open subsets D of RN , χΩ ∈ BV(D), that is, χΩ ∈ BVloc (RN ). (iii) Ω is said to have finite perimeter if χΩ ∈ BV(RN ). Theorem 6.2. Let Ω be a Lebesgue measurable subset of RN . Ω has locally finite perimeter, that is, χΩ ∈ BVloc (RN ), if and only if for each x ∈ ∂Ω there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. We use Theorem 6.1. In one direction the result is obvious. Conversely, if x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D 1 (B(x, ρ))    − χΩ div ϕ

dx = − div ϕ

dx = − ϕ

· n dHN −1 = 0 B(x,ρ)

B(x,ρ)

∂B(x,ρ)

since ϕ

= 0 on the boundary of B(x, ρ). Thus χΩ ∈ BV(B(x, ρ)). If x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D1 (B(x, ρ))  − χΩ div ϕ

dx = 0. B(x,ρ)

So again χΩ ∈ BV(B(x, ρ)). Finally, if x ∈ ∂Ω, then by assumption there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)). Therefore, by Theorem 6.1, χΩ ∈ BVloc (RN ). The interest behind this construction is twofold. First, the notion of perimeter of a set is extended to measurable sets; second, this framework provides a first compactness theorem which will be useful in obtaining existence of optimal domains. Theorem 6.3. Assume that D is a bounded open domain in RN with a Lipschitzian boundary ∂D. Let {Ωn } be a sequence of measurable domains in D for which there exists a constant c > 0 such that ∀n,

PD (Ωn ) ≤ c.

(6.8)

Then there exist a measurable set Ω in D and a subsequence {Ωnk } such that χΩnk → χΩ in L1 (D) as k → ∞

and

PD (Ω) ≤ lim inf PD (Ωnk ) ≤ c. k→∞

(6.9)

248

Chapter 5. Metrics via Characteristic Functions

in Moreover ∇χΩnk “converges in measure” to ∇χΩ in M 1 (D)N ; that is, for all ϕ D0 (D, RN ), lim ∇χΩnk , ϕ

M 1 (D)N → ∇χΩ , ϕ

M 1 (D)N .

k→∞

(6.10)

Proof. This follows from the fact that the injection of the space BV(D) endowed with the norm (6.3) into L1 (D) is continuous and compact (cf. E. Giusti [1, Thm. 1.19, p. 17], V. G. Maz’ja [1, Thm. 6.1.4, p. 300, Lem. 1.4.6, p. 62], C. B. Morrey, Jr. [1, Thm. 4.4.4, p. 75], and L. C. Evans and R. F. Gariepy [1]). Example 6.1 (The staircase). In (6.9) the inequality can be strict, as can be seen from the following example (cf. Figure 5.7). For each n ≥ 1, define the set    n  * j j+1 j Ωn = , × 0, 1 − . n n n j=1 Its limit is the set Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x}. It is easy to check that the sets Ωn are contained in the holdall D = ]−1, 2[ × ]−1, 2[ and that √ ∀n ≥ 1, PD (Ωn ) = 4, PD (Ω) = 2 + 2, χΩn → χΩ in Lp (D), 1 ≤ p < ∞. Each Ωn is uniformly Lipschitzian, but the Lipschitz constant and the two neighborhoods of Definition 3.2 in Chapter 2 cannot be chosen independently of n.

1/n

Ωn

Ω Figure 5.7. The staircase.

Corollary 1. Let D be open bounded and Lipschitzian. There exists a constant c > 0 such that for all convex domains Ω in D PD (Ω) ≤ c,

m(Ω) ≤ c

and the set C(D) of convex subsets of D is compact in Lp (D) for all p, 1 ≤ p < ∞.

6. Caccioppoli or Finite Perimeter Sets

249

Proof. If Ω is a convex set with zero volume, m(Ω) = 0, then its perimeter PD (Ω) = 0 (cf. E. Giusti [1, Rem. 1.7 (iii), p. 6]). If m(Ω) > 0, then int Ω = ∅. For convex sets with a nonempty interior the perimeter and the volume enjoy a very nice monotonicity property. If C • denotes the set of nonempty closed convex subsets of RN , then the map Ω → m(Ω) : C • → R is strictly increasing: ∀A, B ∈ C • ,

A  B =⇒ m(A) < m(B)

(cf. M. Berger [1, Vol. 3, Prop. 12.9.4.3, p. 141]). Similarly if P (Ω) denotes the perimeter of Ω in RN , the map Ω → P (Ω) : C • → R is strictly increasing (cf. M. Berger [1, Vol. 3, Prop. 12.10.2, p. 144]). As a result ∀n,

m(Ωn ) ≤ m(D) < ∞ and PD (Ωn ) ≤ P (Ωn ) ≤ P (co D) < ∞,

since the perimeter of a bounded convex set is finite. Then the conditions of the theorem are satisfied and the conclusions of the theorem follow. But we have seen in Theorem 3.5 (iii) that the set C(D) is closed in L1 (D). Therefore, Ω can be chosen convex in the equivalence class. This completes the proof. This theorem and the lower (resp., upper) semicontinuity of the shape function Ω → J(Ω) will provide existence results for domains in the class of finite perimeter sets in D. One example is the transmission problem (4.1) to (4.3) of section 4.1 with the objective function  1 J(Ω) = |y(Ω) − g|2 dx + αPD (Ω), α > 0, (6.11) 2 Ω for some g ∈ L2 (D). We shall come back later to the homogeneous Dirichlet boundary value problem (3.25)–(3.26). It is important to recall that even if a set Ω in D has a finite perimeter PD (Ω), its relative boundary Γ ∩ D can have a nonzero N -dimensional Lebesgue measure. To illustrate this point consider the following adaptation of Example 1.10 in E. Giusti [1, p. 7]. Example 6.2. Let D = B(0, 1) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi } be an ordered sequence of all points in D with rational coordinates. Associate with each i the open ball Bi = {x ∈ D : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}.

Define the new sequence of open subsets of D, Ωn = and notice that for all n ≥ 1, m(∂Ωn ) = 0,

n *

Bi ,

i=1

PD (Ωn ) ≤ 2π,

250

Chapter 5. Metrics via Characteristic Functions

where m = m2 is the Lebesgue measure in R2 and ∂Ωn is the boundary of Ωn . Moreover, since the sequence of sets {Ωn } is increasing, χΩn → χΩ in L1 (D),

Ω=

∞ *

Bi ,

PD (Ω) ≤ lim inf PD (Ωn ) ≤ 2π. n→∞

i=1

Observe that Ω = D and that ∂Ω = Ω ∩ Ω ⊃ D ∩ Ω. Thus 2π m(∂Ω) = m(D ∩ Ω) ≥ m(D) − m(Ω) ≥ 3 since ∞ π π2−2i = . m(D) = π and m(Ω) ≤ 3 1=1 Recall that m(Ωn ) ≤

n i=1

m(Bi ) ≤

n

π2−2i

⇒ m(D) ≤

i=1

∞ i=1

m(Bi ) ≤

∞

π2−2i .

i=1

For p, 1 ≤ p < ∞, the sequence of characteristic functions {χΩn } converges to χΩ in Lp (D)-strong. However, for all n, m(∂Ωn ) = 0, but m(∂Ω) > 0. In fact we can associate with the perimeter PD (Ω) the reduced boundary, ∂ ∗ Ω, which is the set of all x ∈ ∂Ω for which the normal n(x) exists. We quote the following interesting theorems from W. H. Fleming [1, p. 455] (cf. also E. De Giorgi [2], L. C. Evans and R. F. Gariepy [1, Thm. 1, sect. 5.1, p. 167, Notation, pp. 168–169, Lem. 1, p. 208], and H. Federer [2]). Theorem 6.4. Let χΩ ∈ BVloc (RN ). There exist a Radon measure ∂Ω on RN and a ∂Ω-measurable function νΩ : RN → RN such that (i) |νΩ (x)| = 1, ∂Ω almost everywhere and   (ii) RN χΩ div ϕ dx = − RN ϕ · νΩ d∂Ω, for all ϕ ∈ Cc1 (RN ; RN ). Definition 6.3. Let χΩ ∈ BVloc (RN ). The point x ∈ RN belongs to the reduced boundary ∂ ∗ Ω if (i) ∂Ω(Br (x)) > 0, for all r > 0,  (ii) limr0 m(B1r (x)) Br (x) νΩ d∂Ω = νΩ (x), and (iii) |νΩ (x)| = 1. Theorem 6.5. Let Ω have finite perimeter P (Ω). Let ∂ ∗ Ω denote the reduced boundary of Ω. Then (i) ∂ ∗ Ω ⊂ ∂∗ Ω ⊂ Ω• ⊂ ∂Ω and ∂ ∗ Ω = ∂Ω, (ii) P (Ω) = HN −1 (∂ ∗ Ω) (cf. E. Giusti [1, Chap. 4]),

6. Caccioppoli or Finite Perimeter Sets

251

(iii) the Gauss–Green theorem holds with ∂ ∗ Ω, and (iv) m(∂ ∗ Ω) = m(∂∗ Ω) = 0,

HN −1 (∂∗ Ω\∂ ∗ Ω) = 0.

We quote the following density theorem (cf. E. Giusti [1, Thm. 1.24 and Lem. 1.25, p. 23]), which complements Theorem 3.1. Theorem 6.6. Let Ω be a bounded measurable domain in RN with finite perimeter. Then there exists a sequence {Ωj } of C ∞ -domains such that as j goes to ∞  |χΩj − χΩ | dx → 0 and P (Ωj ) → P (Ω). RN

6.2

Decomposition of the Integral along Level Sets

We complete this section by giving some useful theorems on the decomposition of the integral along the level sets of a function. The first one was used by J.´sio [6] (Grad’s model in plasma physics) in 1979 and [9, sect. 4.4, p. 95] in P. Zole 1981, by R. Temam [1] (monotone rearrangements) in 1979, by J. M. Rakotoson and R. Temam [1] in 1987, and more recently in 1995 by M. C. Delfour and ´sio [19, 20, 21, 25] (intrinsic formulation of models of shells). We quote J.-P. Zole the version given in L. C. Evans and R. F. Gariepy [1, Prop. 3, p. 118]. The original theorem can be found in H. Federer [3] and L. C. Young [1]. Theorem 6.7. Let f : RN → R be Lipschitz continuous with |∇f | > 0

a.e.

For any Lebesgue summable function g : RN → R, we have the following decomposition of the integral along the level sets of f : 



∞

0

g dx = {f >t}

t

{f =s}

g dHN −1 |∇f |

1 ds.

The second theorem (W. H. Fleming and R. Rishel [1]) uses a BV function instead of a Lipschitz function. Theorem 6.8. Let D be an open subset of RN . For any f ∈ BV(D) and real t, let def

Et = {x : f (x) < t}. Then ∇f M 1 (D)N =



∞ −∞

P (Et ) dt

(cf., for instance, H. Whitney [1, Chap. 11]). This is known as the co-area formula (see also E. Giusti [1, Thm. 1.23, p. 20] and E. De Giorgi [4]).

252

6.3

Chapter 5. Metrics via Characteristic Functions

Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces

There is a general property enjoyed by functions in BV(D). Theorem 6.9. Let D be a bounded open Lipschitzian domain in RN . (i) BV(D) ⊂ W ε,1 (D),

0 ≤ ε < 1.

(ii) BV(D) ∩ L∞ (D) ⊂ W ε,p (D),

0 ≤ ε < 1p , 1 ≤ p < ∞.

This theorem says that for a Caccioppoli set Ω in D, ∀p ≥ 1, 0 ≤ ε <

1 , p

χΩ ∈ W ε,p (D).

The special case p = 2 was proved by C. Baiocchi, V. Comincioli, E. Magenes, and G. A. Pozzi [1] in the context of the celebrated problem of the dam. They showed that the domain Ω is the hypograph of a continuous monotonically decreasing function on a closed interval is H ε (D) = W ε,2 (D), 0 ≤ ε < 1/2, in R2 . Proof of Theorem 6.9. (i) Given f in BV(D) we want to show that f ∈ W ε,1 (D), 0 < ε < 1. This is equivalent to showing that the double integral   |f (y) − f (x)| dx dy I= |y − x|N +ε D D is finite. Given α > 0 small we break the above integral into two parts:     |f (y) − f (x)| |f (y) − f (x)| dx dy , I = dx dy . I1 = 2 N +ε |y − x| |y − x|N +ε D D∩B(x,α) D
D B(x,α) The second integral I2 is bounded since |y − x| ≥ α. For the first one we change the variable y to t = y − x,   |(χD f )(x + t) − (χD f )(x)| dx dt , I1 = |t|N +ε D B(0,α) and after a change in the order of integration,   1 I1 = dt N +ε dx |(χD f )(x + t) − (χD f )(x)|. |t| B(0,α) D But the integral over D is bounded by |t| ∇f M 1 (D) . The result is true for smooth functions (cf. V. G. Maz’ja [1, Lem. 1.4.6, p. 62]). Pick a sequence of smooth functions such that fn → f in L1 (D),

∇fn → ∇f in M 1 (D)-weak;

then ∇f M 1 (D) ≤ lim sup ∇fn M 1 (D) = lim sup ∇fn L1 (D) n→∞

n→∞

6. Caccioppoli or Finite Perimeter Sets

253

is bounded, and by going to the limit on both sides of the inequality,  dx |(χD fn )(x + t) − (χD fn )(x)| ≤ |t|∇fn L1 (D) , D

we obtain the desired result. Now coming back to the estimate of I1 ,  α1−ε ∇f M 1 (D) , 0 < ε < 1. I1 ≤ |t|1−N −ε dt ∇f M 1 (D) ≤ c(α) 1−ε B(x,α) (ii) The function f belongs to W ε,p (D) if the integral   |f (y) − f (x)|p dx dy < +∞. I= |y − x|N +εp D D For each f ∈ BV(D) ∩ L∞ (D), there exists M > 0 such that |f (x)| ≤ M

a.e. in D.

Then 



I = (2 M )p

dx D

and since

dy D

f (y)−f (x) p 2M |y − x|N +εp

,

f (y) − f (x) ≤ 1, 2M

then for each p, 1 ≤ p < ∞, f (y) − f (x) p f (y) − f (x) ≤ . 2M 2M As a result

 I ≤ (2 M )



p−1

dx D

dy D

|f (y) − f (x)| . |y − x|N +εp

But we have seen in part (i) that for f ∈ BV(D), the double integral is finite if 0 ≤ εp < 1 ⇒ 0 ≤ ε <

1 . p

This complete the proof. For a characteristic function χΩ of a measurable set Ω and ε > 0, we have 0 ≤ ε < 1/p for all p ≥ 1 and   |χΩ (x) − χΩ (y)| dx dy + χΩ pLp (D) . (6.12) χΩ pW ε,p (D) = |x − y|N +pε D D

254

Chapter 5. Metrics via Characteristic Functions 



The W ε,p (D)-norm is equivalent to the W ε ,p (D) for all pairs (p , ε ), p ≥ 1, 0 ≥ ε < 1/p , such that p ε = p ε. At this stage, it is not clear whether that norm is related to some Hausdorff measure of the relative boundary Γ ∩ D of the set. For p = 2 and a characteristic function, χΩ , of a measurable set Ω, we get   2 χΩ H ε (D) = 2 |x − y|−(N +2ε) dx dy + m(Ω ∩ D). (6.13) Ω


D Ω

ε,2

The space X(D)∩W (D) is a closed subspace of the Hilbert space W ε,2 (D) and the square of its norm is differentiable. A direct consequence for optimization problems is that a penalization term of the form χΩ 2W ε,2 (D) is now differentiable and can be used in various minimization problems or to regularize the objective function to obtain existence of approximate solutions. For instance, to obtain existence results when minimizing a function J(Ω) (defined for all measurable sets Ω in D), we can consider a regularized problem in the following form: Jα (Ω) = J(Ω) + αχΩ 2H ε (D) ,

α > 0.

(6.14)

Section 6.1 provided the important compactness Theorem 6.3 for Caccioppoli or finite perimeter sets. For a bounded open holdall D, this also provides a new complete metric topology on X(D) ∩ BV(D) when endowed with the metric def

ρBV (χ1 , χ2 ) = χ2 − χ1 BV(D) = χ2 − χ1 L1 (D) + ∇χ2 − ∇χ1 M 1 (D) . Similarly, the intermediate spaces of Theorem 6.9 between BV(D) and Lp (D) provide little rougher metrics for p ≥ 1 and ε < 1/p on X(D) ∩ W ε,p (D) when endowed with the metric def

ρW ε,p (χ1 , χ2 ) = χ2 − χ1 W ε,p (D) . Theorem 6.10. Given a bounded open holdall D in RN , X(D)∩BV(D) and X(D)∩ W ε,p (D), p ≥ 1, ε < 1/p, are complete metric spaces and X(D) ∩ BV(D) ⊂ X(D) ∩ W ε,p (D) ⊂ X(D).

6.4

Compactness and Uniform Cone Property

In Theorem 6.3 of section 6.1 we have seen a first compactness theorem for a family of sets with a uniformly bounded perimeter. In this section we present a second compactness theorem for measurable domains satisfying a uniform cone property due to D. Chenais [1, 4, 6]. We provide a proof of this result that emphasizes the fact that the perimeter of sets in that family is uniformly bounded. It then becomes a special case of Theorem 6.3. This result will also be obtained as Corollary 2 to Theorem 13.1 in section 13 of Chapter 7. It will use completely different arguments and apply to families of sets that do not even have a finite boundary measure. Theorem 6.11. Let D be a bounded open holdall in RN with uniformly Lipschitzian boundary ∂D. For r > 0, ω > 0, and λ > 0 consider the family   def Ω is Lebesgue measurable and satisfies L(D, r, ω, λ) = Ω ⊂ D : the uniform cone property for (r, ω, λ) . (6.15)

6. Caccioppoli or Finite Perimeter Sets

255

For p, 1 ≤ p < ∞, the set def

X(D, r, ω, λ) = {χΩ : ∀Ω ∈ L(D, r, ω, λ)}

(6.16)

is compact in Lp (D) and there exists a constant p(D, r, ω, λ) > 0 such that ∀Ω ∈ L(D, r, ω, λ),

HN −1 (∂Ω) ≤ p(D, r, ω, λ).

(6.17)

We first show that the perimeter of the sets of the family L(D, r, ω, λ) is uniformly bounded and use Theorem 6.8 to show that any sequence of X(D, r, ω, λ) has a subsequence converging to the characteristic function of some finite perimeter set Ω. We use the full strength of the compactness of the injection of BV(D) into L1 (D) rather than checking directly the conditions under which a subset of Lp (D) is relatively compact. The proof will be completed by showing that the nice representative (Definition 3.3) I of Ω satisfies the same uniform cone property. The proof uses some elements of D. Chenais [4, 6]’s original proof. Proof of Theorem 6.11. Each Ω ∈ L(D, r, ω, λ) satisfies the same uniform cone property (cf. Definition 6.3 (ii)) of Chapter 2 with parameters r, λ, and rλ . By Theorem 6.8 of Chapter 2 it is uniformly Lipschitzian in the sense of Definition 5.2 (iii) of Chapter 2. As a result ρ, the largest radius such that BH (0, ρ) ⊂ {PH (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω}, and the neighborhoods V = BH (0, ρ) and U = B(0, rλ ) ∩ {y : PH y ∈ BH (0, ρ)} can be chosen as specified in Theorem 6.6 (i) of Chapter 2. They are the same for all Ω ∈ L(D, r, ω, λ): for all Ω ∈ L(D, r, ω, λ) and all x ∈ ∂Ω, Vx = V , and there exists Ax ∈ O(N) such that U(x) = x + Ax U . By construction B(x, ρ) ⊂ B(x, rλ ) ∩ {y : PH A−1 x (y − x) ∈ BH (0, ρ)} = x + Ax U = U(x). 

The family B (z, ρ/2) : z ∈ D is an open cover of D. Since D is compact, there exists a finite subcover {Bi }m i=1 , Bi = B (zi , ρ/2), of D. Now for any Ω ∈ L(D, r, ω, λ), m ∂Ω ⊂ D and there is a subcover {Bik }K k=1 of {Bi }i=1 such that ∂Ω ⊂

K *

Bik and ∀k,

1 ≤ k ≤ K,

∂Ω ∩ Bik = ∅.

k=1

Pick a sequence {xk }K k=1 such that xk ∈ ∂Ω ∩ Bik , 1 ≤ k ≤ K, and notice that  ρ Bik = B zik , ⊂ B(xk , ρ) 2 or ∃K ≤ m,

∃{xk }K k=1 ⊂ ∂Ω,

∂Ω ⊂

K * k=1

B(xk , ρ).

256

Chapter 5. Metrics via Characteristic Functions

Thus from the estimate (5.37) in Theorem 5.7 of Chapter 2 with cx = tan ω, HN −1 (∂Ω) ≤

K

K     HN −1 ∂Ω ∩ B(xk , ρ) ≤ HN −1 ∂Ω ∩ U(xk )

k=1

≤

K

k=1

ρN −1 αN −1

'

1 + (tan ω)2 ≤ m ρN −1 αN −1

' 1 + (tan ω)2 ,

k=1

where αN −1 is the volume of the unit (N − 1)-dimensional ball. So the right-hand side of the above inequality is a constant that is equal to p = p(D, r, ω, λ) > 0 and is independent of Ω in L(D, r, ω, λ): ∀Ω ∈ L(D, r, ω, λ),

PD (Ω) = HN −1 (∂Ω) ≤ p.

Now from Theorem 6.3 for any sequence {Ωn } ⊂ L(D, r, ω, λ), there exists Ω such that χΩ ∈ BV(D) and a subsequence, still denoted by {Ωn }, such that χΩn → χΩ in L1 (D) and PD (Ω) ≤ p. (ii) To complete the proof we consider the representative I of Ω (cf. Definition 3.3) and show that I ∈ L(D, r, ω, λ). We need the following lemma. Lemma 6.1. Let χΩn → χΩ in Lp (D), 1 ≤ p < ∞, for Ω ⊂ D, Ωn ⊂ D, and let I be the measure theoretic representative of Ω. Then   ∀x ∈ I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), m B(x, R) ∩ Ωn > 0. Moreover, ∀x ∈ ∂I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R),     m B(x, R) ∩ Ωn > 0 and m B(x, R) ∩ Ωn > 0 and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ I,

∃R > 0,

∀N > 0,

∃n ≥ N,

  m B(x, R) ∩ Ωn = 0.

So there exists a subsequence {Ωnk }, nk → ∞, such that     m B(x, R) ∩ Ω = lim m B(x, R) ∩ Ωnk = 0 k→∞   ⇒ ∃R > 0, m B(x, R) ∩ Ω = 0 =⇒ x ∈ Ω0 = int I = I. But this is in contradiction with the fact that x ∈ I. For x ∈ ∂I = I ∩ I, simultaneously apply the first statement to I and χΩn → χΩ and to I and χ
Ωn → χ
Ω since χ
Ω = χ
I almost everywhere by Theorem 3.3 (i) and choose the larger of the two integers. As for the last property it follows from the fact that the open ball B(x, R) cannot be partitioned into two nonempty disjoint open subsets.

6. Caccioppoli or Finite Perimeter Sets

257

We wish to prove that ∀x ∈ ∂I,

∃Ax ∈ O(N),

∀y ∈ I ∩ B(x, r),

y + Ax C(λ, ω) ⊂ int I.

Since Ωn is Lipschitzian, Ωn is also Lipschitzian and χΩn = χΩn

and χ
Ω = χ
Ωn n

almost everywhere. So we apply the second part of the lemma to x ∈ ∂I, χΩn → χΩ and χ
Ω → χ
Ω . So for each x ∈ ∂I, for all k ≥ 1, there exists nk ≥ k such that n

 r  B x, k ∩ ∂Ωnk = ∅. 2

Denote by xk an element of that intersection:  r  ∀k ≥ 1, xk ∈ B x, k ∩ ∂Ωnk . 2 By construction, xk → x. Next consider y ∈ B(x, r) ∩ I. By Lemma 6.1, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that  r  ∀k ≥ 1, B y, k ∩ Ωnk = ∅. 2 For each k ≥ 1 denote by yk a point of that intersection. By construction, yk ∈ Ωnk → y ∈ I ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this note that y ∈ B(x, r) and that ρ ∃ρ > 0, B(y, ρ) ⊂ B(x, r) and |y − x| + < r. 2 Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r r ρ ρ/ r
−

log(r/ρ) ρ < 0 =⇒ k > 2 + . 2 log 2

So we have constructed a subsequence {Ωnk } such that for k ≥ K, xk ∈ ∂Ωnk → x ∈ ∂I

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ I ∩ B(x, r).

For each k, there exists Ak ∈ O(N) such that yk + Ak C(λ, ω) ⊂ int Ωnk .

258

Chapter 5. Metrics via Characteristic Functions

Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N),

Ak → A,

since O(N) is compact. Now consider z ∈ y + AC(λ, ω). Since z is an interior point ∃ρ > 0,

B(z, ρ) ⊂ y + AC(λ, ω).

So there exists K  ≥ K such that  ρ ∀k ≥ K  , B z, ⊂ yk + Ak C(λ, ω) ⊂ int Ωnk . 2 Therefore, for k ≥ K         ρ ρ m B z, = m B z, ∩ Ωnk , 2 2 and, as k → ∞,        ρ ρ m B z, = m B z, ∩Ω , 2 2 and by Definition 3.3, z ∈ Ω1 and y + AC(λ, ω) ⊂ Ω1 = int I. This proves that I ⊂ D satisfies the uniform cone property and I ∈ L(D, λ, ω, r).

7 7.1

Existence for the Bernoulli Free Boundary Problem An Example: Elementary Modeling of the Water Wave

Consider a fluid in a domain Ω in R3 and assume that the velocity of the flow u satisfies the Navier–Stokes equations ut + Du u − ν ∆u + ∇p = −ρ g in Ω,

div u = 0 in Ω,

(7.1)

where ν > 0 and ρ > 0 are the respective viscosity and density of the fluid, p is the pressure, and g is the gravity constant. The second equation characterizes the incompressibility of the fluid. A standard example considered by physicists is the water wave in a channel. The boundary conditions on ∂Ω are the sliding conditions at the bottom S and on the free boundary Γ, that is, u · n = 0 on S ∪ Γ.

(7.2)

Assume that a stationary regime has been reached so that the velocity of the fluid is no longer a function of the time. Furthermore assume that the motion of the fluid is irrotational. By the classical Hodge decomposition, the velocity can be written in the form u = ∇ϕ + curl ξ.

(7.3)

7. Existence for the Bernoulli Free Boundary Problem

259

As curl u = curl curl ξ = 0 we conclude that ξ = 0, since curl curl is a good isomorphism. Then u = ∇ϕ and the incompressibility condition becomes ∆ϕ = 0 in Ω.

(7.4)

Therefore the Navier–Stokes equation reduces to D(∇ϕ) ∇ϕ + ν ∇(∆ϕ) + ∇p = −∇(ρgz),

(7.5)

but D(∇ϕ) ∇ϕ = so that

 ∇

1 ∇(|∇ϕ|2 ), 2

1 |∇ϕ|2 + p + ρgz 2

(7.6)

 = 0 in Ω.

(7.7)

Then if Ω is connected, there exists a constant c such that 1 |∇ϕ|2 + p + ρgz = c in Ω. 2

(7.8)

In the domain Ω, that Bernoulli condition yields explicitly the pressure p as a function of the velocity |∇ϕ| and the height z in the fluid. The flow is now assumed to be independent of the transverse variable y so that the initial three-dimensional problem in a perfect channel reduces to a two-dimensional one. Since the problem has been reduced to a two-dimensional one, introduce the harmonic conjugate ψ, the so-called stream function, so that the boundary condition ∂ϕ/∂n = 0 takes the form ψ = constant on each connected component of the boundary ∂Ω. On the free boundary at the top of the wave, the pressure is related to the existing atmospheric pressure pa through the surface tension σ > 0 and the mean curvature H of the free boundary. Actually p − pa = −σ H

(7.9)

on the free boundary of the stationary wave, where H is the mean curvature associated with the fluid domain. Also, from the Cauchy conditions, we have |∇ψ| = |∇ϕ| so that the boundary condition (7.8) on the free boundary of the wave becomes 1 |∇ψ|2 + ρgz + σH = pa . 2

(7.10)

In order to simplify the presentation we replace the equation ∆ψ = 0 by ∆ψ = f in Ω to avoid a forcing term on a part of the boundary, and we show that the resulting free boundary problem has the following shape variational formulation. Let D be a fixed, sufficiently large, smooth, bounded, and open domain in R2 . Let a be a real number such that 0 < a < m(D). To find Ω ⊂ D, m(Ω) = a and ψ ∈ H01 (D) such that −∆ψ = f in Ω

(7.11)

260

Chapter 5. Metrics via Characteristic Functions

and ψ = constant and satisfies the boundary condition (7.10) on the free part ∂Ω ∩ D of the boundary. For a fixed Ω, the solution of this problem is a minimizing element of the following variational problem:    1 def 2 |∇ϕ| J(Ω) = inf − f ϕ + ρgz dx + σ PD (Ω), (7.12) 2 ϕ∈H1 (Ω;D) Ω where  def

(7.13) H1 (Ω; D) = u ∈ H01 (D) : u(x) = 0 a.e. x in D \ Ω is the relaxation of the definition of the Sobolev space H01 (Ω)10 for any measurable subset Ω of D. Its properties were studied in Theorem 3.6. With that formulation there exists a measurable Ω∗ ⊂ D, |Ω∗ | = a, such that ∀Ω ⊂ D, |Ω| = a,

J(Ω∗ ) ≤ J(Ω).

(7.14)

By using the methods of Chapter 10 it follows that if ∂Ω∗ is sufficiently smooth, the shape Euler condition dJ(Ω∗ ; V ) = 0 yields the original free boundary problem and the free boundary condition (7.10). The existence of a solution will now follow from Theorem 6.3 in section 6.1. The case without surface tension is physically important. It occurs in phenomena with “evaporation ” (cf. M. Souli and J.´sio [2]). In that case the previous Bernoulli condition becomes P. Zole 2 ∂ξ = g 2 (g ≥ 0) ∂n on the free boundary, so that if we assume (in the channel setting) that there is no cavitation or recirculation in the fluid, then ∂ξ/∂n > 0 on the free boundary and we get the Neumann-like condition ∂ξ/∂n = g together with the Dirichlet condition. In section 7.4 we shall consider the case with surface tension.

7.2

Existence for a Class of Free Boundary Problems

´sio [25, 26, Consider the following free boundary problem, studied in J.-P. Zole 29]: find Ω in a fixed holdall D and a function y on Ω such that −∆y = f in Ω,

y = 0 and

∂y = Q2 on ∂Ω, ∂n

(7.15)

where f and Q are appropriate functions defined in D. To study this type of problem H. W. Alt and L. A. Caffarelli [1] have introduced the following function:   1 def Q2 χϕ>0 dx, (7.16) J(ϕ) = |∇ϕ|2 − f ϕ dx + 2 D D 10 In Chapter 8 the space H 1 (Ω; D) of extensions by zero to D of elements of H 1 (Ω) is defined 0 0 for Ω open. It is then characterized in Lemma 6.1 of Chapter 8 by a capacity condition on the complement D\Ω. This characterization extends to quasi-open sets Ω as defined in section 6 of Chapter 8.

7. Existence for the Bernoulli Free Boundary Problem

261

which is minimized over def

K = {u ∈ H01 (D) : u(x) ≥ 0 a.e. in D},

(7.17)

where χϕ>0 (resp., χϕ=0 ) is the characteristic function of the set {x ∈ D : ϕ(x) > 0}11 (resp., {x ∈ D : ϕ(x) = 0}). The existence of a solution is based on the following lemma. Lemma 7.1. Let {un } and {χn } be two converging sequences such that un → u in L2 (D)-strong, and let the χn ’s be characteristic functions, χn (1 − χn ) = 0, which converge to some function λ in L2 (D)-weak. Then ∀n,

(1 − χn )un = 0

⇒

λ ≥ χu=0 .

(7.18)

Proof. We have (1 − χn )un = 0, and in the limit (1 − λ)u = 0. Thus on the set {x : u(x) = 0} we have λ = 1; elsewhere λ lies between 0 and 1 as the weak limit of a sequence of characteristic functions. Proposition 7.1. Let f and Q be two elements of L2 (D) such that f ≥ 0 almost everywhere. There exists u in K which minimizes the function J over the positive cone K of H01 (D). Proof. Let {un } ⊂ K be a minimizing sequence for the function J over the convex set K. Denote by χn the characteristic function of the set {x ∈ D : un (x) > 0}, which is in fact equal to the subset {x ∈ D : un (x) = 0}. It is easy to verify that the sequence {un } remains bounded in H01 (D). Still denote by {un } a subsequence that weakly converges in H01 (D) to an element u of K. That convergence holds in L2 (D)-strong so that Lemma 7.1 applies and we get λ ≥ χu=0 for any weak limiting element λ of the sequence {χn } (which is bounded in L2 (D)). Denote by j the minimum of J over K. Then J(un ) weakly converges to j. We get   1 1 2 (7.19) |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, n→∞ D 2 D 2    χu=0 Q2 dx ≤ λQ2 dx = lim χn Q2 dx. (7.20) D

D

n→∞

D

Finally, by adding these two estimates we get J(u) ≤ j. Obviously in the upper bound in (7.20), Q2 is a positive function. Moreover (i) the set Ω is given by {x ∈ D : u(x) > 0}, and (ii) the restriction u|Ω of u to Ω is a weak solution of the free boundary problem −∆u = f in Ω,

u = 0 and

∂u = Q2 on ∂Ω. ∂n

(7.21)

11 This set defined up to a set of zero measure is a quasi-open set in the sense of section 6 in Chapter 8.

262

Chapter 5. Metrics via Characteristic Functions

Effectively, the minimization problem (7.16)–(7.17) can be written as a shape optimization problem. First introduce for any measurable Ω ⊂ D the positive cone  def

H01 (Ω)+ = u ∈ H1 (Ω; D) : u(x) ≥ 0 a.e. x ∈ D in the Hilbert space H1 (Ω; D). Then consider the following shape optimization problem: inf {E(Ω) : Ω is measurable subset in D} , where the energy function E is given by    1 def 2 E(Ω) = |∇ϕ| min − f ϕ dx + Q2 dx. ϕ∈H01 (Ω)+ Ω 2 Ω

(7.22)

(7.23)

The necessary condition associated with the minimum could be obtained by the techniques introduced in section 2.1 of Chapter 10. The important difference with the previous formulation (7.16)–(7.17) is that the independent variable in the objective function is no longer a function but a domain. The shape formulation (7.22)– (7.23) now makes it possible to handle constraints on the volume or the perimeter of the domain, which were difficult to incorporate in the first formulation. As we have seen in Theorem 3.6, H1 (Ω; D) endowed with the norm of H01 (D) is a Hilbert space, and H01 (Ω)+ a closed convex cone in H1 (Ω; D), so that for any measurable subset Ω in D, problem (7.23) has a unique solution y in H01 (Ω)+ . Thus we have the following equivalence between problems (7.22)–(7.23) and the minimization (7.16)–(7.17) of J over K. Proposition 7.2. Let u be a minimizing element of J over K. Then def

Ω = {x ∈ D : u(x) > 0} is a solution of problem (7.22) and y = u|Ω is a solution of (7.23). Conversely, if Ω is a measurable subset of D and y is a solution of (7.22)–(7.23) in H01 (Ω)+ , then the element u defined by def

u(x) =

y(x), 0,

if x ∈ Ω, if x ∈ D \ Ω,

belongs to K and minimizes J over K.

7.3 7.3.1

Weak Solutions of Some Generic Free Boundary Problems Problem without Constraint

Problem (7.22)–(7.23) can be relaxed as follows: given any f in L2 (D), G in L1 (D), (P0 )

inf {E(Ω) : Ω ⊂ D measurable},

(7.24)

7. Existence for the Bernoulli Free Boundary Problem

263

where for any measurable subset Ω of D the energy function is now defined by   1 def def min E (ϕ) + G dx, E (ϕ) = (7.25) E(Ω) = |∇ϕ|2 − f ϕ dx, D D 1 ϕ∈H (Ω;D) Ω D 2 where H1 (Ω; D) is defined in (7.13). By Theorem 3.6 H1 (Ω; D) is contained in H•1 (Ω; D), and for any element u in H1 (Ω; D) we have ∇u(x) = 0 for almost all x in D \ Ω so that  1 1 (7.26) ∀ϕ ∈ H (Ω; D), ED (ϕ) = |∇ϕ|2 − f ϕ dx Ω 2    1 G dx . (7.27) ⇒ E(Ω) = min |∇ϕ|2 − f ϕ dx + 1 ϕ∈H (Ω;D) Ω 2 Ω We have the following existence result for problem (P0 ). Theorem 7.1. For any f in L2 (D), G = Q2 in L1 (D), there exists at least one solution to problem (P0 ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the solution to problem (7.25) with Ω = Ωn . If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, the sequence {un } remains uniformly bounded in H01 (D). Taking ϕ = 0 in (7.25) we get  1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that the sequence {χn } weakly converges in L2 (D) to an element λ and that the sequence {un } weakly converges in H01 (D) to an element u. From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Define def

Ω(u) = {x ∈ D : u(x) = 0}.   Then u belongs to H01 Ω(u) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α, since we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1}). In the limit with Ω = Ω(u) we get    1 1 1 2 2 |∇u| − f u dx = |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, (7.28) 2 2 Ω D D 2    G dx ≤ λG dx = lim G dx. (7.29) Ω

D

n→∞

Ωn

By adding (7.28) and (7.29) we get that Ω minimizes E and u minimizes E(Ω).

264 7.3.2

Chapter 5. Metrics via Characteristic Functions Constraint on the Measure of the Domain Ω

Consider an important variation of problem (7.22)–(7.23). Given any f in L2 (D), G in L1 (D), and a real number α, 0 < α < m(D), (P0α )

inf {E(Ω) : Ω ⊂ D measurable and m(Ω) = α} .

(7.30)

We have the following existence result for problem (P0α ). Theorem 7.2. For any f in L2 (D), G = 0, and any real number α, 0 < α < m(D), there exists at least one solution to problem (P0α ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the unique solution to problem (7.25). If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, by picking ϕ = 0 in (7.25), {un } remains bounded in H01 (D),  1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that {χn } weakly converges in L2 (D) to an element λ and that {un } weakly converges in H01 (D) to an element u ∈ H01 (D). In the limit we get   λ(x) dx = lim χn (x) dx = α. n→∞

D

D

From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Define def

Ω(u) = {x ∈ D : u(x) = 0}. Then u belongs to H1 (Ω(u); D) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α (as we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1})). In the limit we get for Ω = Ω(u)    1 1 1 |∇u|2 − f u dx = |∇u|2 − f u dx ≤ lim inf |∇un |2 − f un dx, (7.31) n→∞ 2 2 2 Ω D D   λG dx = lim G dx, (7.32) n→∞

D

so that

 Ω

1 |∇u|2 − f u dx + 2

Ωn

 λG dx ≤ D

inf

Ω ⊂D m(Ω )=α

If G ≥ 0 almost everywhere in D, we get E(Ω) ≤

inf

Ω ⊂D m(Ω )=α

{E(Ω )} ,

E(Ω ).

7. Existence for the Bernoulli Free Boundary Problem

265

but Ω does not necessarily satisfy the constraint m(Ω) = α. Note that for any measurable set Ω such that Ω ⊂ Ω ⊂ D, we have   1 1 2 |∇u| − f u dx = |∇u|2 − f u dx 2  Ω Ω 2 so that in expression (7.31) Ω can be enlarged to any such Ω . The inclusion of Ω in Ω implies the inclusion of H1 (Ω; D) in H1 (Ω ; D). So if G ≤ 0 almost everywhere in D, then it is readily seen that from (7.25) E(Ω ) ≤ E(Ω). In view of the previous assumption on G, now G = 0 almost everywhere in D. To conclude the proof we just have to select Ω with m(Ω ) = α. That measurable set Ω is admissible and minimizes the objective function in (7.30), and we have E(Ω ) = E(Ω) =

inf

{E(Ω )}.

Ω ⊂D m(Ω )=α

Corollary 1. Assume that G = 0 and f in L2 (D) and f = Q2 in L1 (D). Then the following problem has an optimal solution: (P0α− )

inf

Ω⊂D m(Ω)≤α

E(Ω).

(7.33)

Proof. The proof is similar to the proof of the theorem where the minimizing sequence is chosen such that m(Ωn ) ≤ α, so that in the weak limit we get  λ(x) dx ≤ α. m(Ω) ≤ D

7.4

Weak Existence with Surface Tension

Problems (P0 ), (P0α ), and (P0α− ) have optimal solutions, but as they are associated with the homogeneous Dirichlet boundary condition, u in H1 (Ω; D), the optimal domain Ω is in general not allowed to have holes, that is to say, roughly speaking, that the topology of Ω is a priori specified. In many examples it turns out that the solution u physically corresponds to a potential and the classical homogeneous Dirichlet boundary condition is not the appropriate one. The physical condition is that the potential u should be constant on each connected component of the boundary ∂Ω in D. When Ω is a simply connected domain in R2 , then ∂Ω has a single connected component so the constant can be taken as zero. In general this constant can be fixed only in one connected component; in the others the constant is an unknown of the problem. The minimization problems (P0 ), (P0α ), and (P0α− ) associated with the Hilbert space H•1 (Ω; D) fail (in the sense that the previous techniques for existence of an optimal Ω fail). The main reason is that Lemma 7.1 is no longer true when un is replaced by ∇un weakly converging in L2 (D)N . The key idea is to recover the equivalent of Lemma 7.1 by imposing the strong L2 (D)-convergence of the sequence {un }. In practice {un } corresponds to the sequence of characteristic functions

266

Chapter 5. Metrics via Characteristic Functions

χΩn of a minimizing sequence {Ωn }. To obtain the strong L2 (D)-convergence of a subsequence we add a constraint on the perimeters. Consider the family of finite perimeter sets in D of Definition 6.2 (ii) in section 6.1, def

BPS(D) = {Ω ⊂ D : χΩ ∈ BV(D)}, where BV(D) is defined in (6.2). Introduce the following problem indexed by σ > 0: (Pσα )

inf

Ω⊂D m(Ω)=α def

E(Ω) =

Eσ (Ω), 

min

ϕ∈H•1 (Ω;D)

Ω

def

Eσ (Ω) = E(Ω) + σPD (Ω), 1 |∇ϕ|2 − f ϕ dx + 2

(7.34)





G dx .

(7.35)

Ω

Theorem 7.3. Let f in L2 (D), G in L1 (D), σ > 0, and 0 ≤ α < m(D). Then problem (Pσα ) has at least one optimal solution Ω in BPS(D). Proof. Let {Ωn } be a minimizing sequence for (Pσα ) and let {χn } be the corresponding sequence of characteristic functions associated with {Ωn }. By picking ϕ = 0 in (7.35),    −1 2 |G| dx + σPD (D) . cf L2 (D) + PD (Ωn ) ≤ σ D

By Theorem 6.3 there exist a subsequence of {Ωn }, still indexed by n, and Ω ⊂ D with finite perimeter such that χn → χ = χΩ in L1 (D)-strong and α = m(Ωn ) = m(Ω). For each n, let un in H•1 (Ωn ; D) be the unique minimizer of (7.35). That sequence remains bounded in H01 (D). Pick a subsequence, still indexed by n, such that {un } weakly converges to an element u in H01 (D). From the identity (1 − χn )∇un = 0 almost everywhere in D, we get in the limit (1 − χΩ )∇u = 0 almost everywhere in D, so that the limiting element u belongs to H•1 (Ω; D). Hence we have    1 1 f u dx |∇u|2 − f u dx = |∇u|2 dx − Ω 2 D 2 Ω   (7.36) 1 2 ≤ lim inf |∇un | dx − f un dx, n→∞ D 2 Ωn     f u dx = lim f un dx, G dx = lim G dx, (7.37) Ω

so that

 Ω

n→∞

Ωn

1 |∇u|2 − f u dx + 2

Ω

n→∞

Ωn

 G dx + σPD (Ω) ≤ Ω

inf

Ω⊂D m(Ω)=α

Eσ (Ω).

(7.38)

Chapter 6

Metrics via Distance Functions 1

Introduction

In Chapter 5 the characteristic function was used to embed the equivalence classes of measurable subsets of D into Lp (D) or Lploc (D), 1 ≤ p < ∞, and induce a complete metric on the equivalence classes of measurable sets. This construction is generic and extends to other set-dependent functions embedded in an appropriate function space. The Hausdorff metric is the result of such a construction, where the distance function plays the role of the characteristic function. The distance function embeds equivalence classes of subsets A of a closed holdall D with the same closure A into the space C(D) of continuous functions. When D is bounded the C 0 -norm of the difference of the distance functions of two subsets of D is the Hausdorff metric. The Hausdorff topology has many much desired properties. In particular, for D bounded, the set of equivalence classes of nonempty subsets A of D is compact. Yet, the volume and perimeter are not continuous with respect to the Hausdorff topology. This can be fixed by changing the space C(D) to the space W 1,p (D) since distance functions also belong to that space. With that metric the volume is now continuous. The price to pay is the loss of compactness when D is bounded. But other sequentially compact families can easily be constructed. By analogy with Caccioppoli sets we introduce the sets for which the elements of the Hessian matrix of second-order derivatives of the distance function are bounded measures. They are called sets of bounded curvature. Their closure is a Caccioppoli set, and they enjoy other interesting properties. Convex sets belong to that family. General compactness theorems are obtained for such families under global or local conditions. This chapter also deals with the family of open sets characterized by the distance function to their complement. They are discussed in parallel along with the distance functions to a set. The properties of distance functions and Hausdorff and Hausdorff complementary metric topologies are studied in section 2. The projections, skeletons, cracks, and differentiability properties of distance functions are discussed in section 3. W 1,p topologies are introduced in section 4 and related to characteristic functions. The 267

268

Chapter 6. Metrics via Distance Functions

compact families of sets of bounded and locally bounded curvature are characterized in section 5. The notion of reach and Federer’s sets of positive reach are studied in section 6. The smoothness of smooth closed submanifolds is related to the smoothness of the square of the distance function in a neighborhood. Approximation of distance functions by dilated sets/tubular neighborhoods and their critical points are presented in section 7. Convex sets are characterized in section 8 along with the special family of convex sets. Both sets of positive reach and convex sets will be further investigated in Chapter 7. Finally section 9 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.

2

Uniform Metric Topologies

2.1

Family of Distance Functions Cd (D)

In this section we review some properties of distance functions and present the general approach that will be followed in this chapter. Definition 2.1. Given A ⊂ RN the distance function from a point x to A is defined as def

dA (x) =

inf |y − x|,

A = ∅,

+∞,

A = ∅,

y∈A

(2.1)

and the family of all distance functions of nonempty subsets of D as  def

Cd (D) = dA : ∀A, ∅ = A ⊂ D .

(2.2)

When D = RN the family Cd (RN ) is denoted by Cd . By definition dA is finite in RN if and only if A = ∅. Recall the following properties of distance functions. Theorem 2.1. Assume that A and B are nonempty subsets of RN . (i) The map x → dA (x) is uniformly Lipschitz continuous in RN : ∀x, y ∈ RN ,

|dA (y) − dA (x)| ≤ |y − x|

(2.3)

0,1 (RN ).1 and dA ∈ Cloc

(ii) There exists p ∈ A such that dA (x) = |p − x| and dA = dA in RN . (iii) A = {x ∈ RN : dA (x) = 0}. 1 A function f belongs to C 0,1 (RN ) if for all bounded open subsets D of RN its restriction to loc D belongs to C 0,1 (D).

2. Uniform Metric Topologies

269

(iv) dA = 0 in RN ⇐⇒ A = RN . (v) A ⊂ B ⇐⇒ dA ≥ dB . (vi) dA∪B = min{dA , dB }. (vii) dA is (Fr´echet) differentiable almost everywhere and |∇dA (x)| ≤ 1

a.e. in RN .

(2.4)

Proof. (i) For all z ∈ A and x, y ∈ RN |z − y| ≤ |z − x| + |y − x|, dA (y) = inf |z − y| ≤ inf |z − x| + |y − x| = dA (x) + |y − x| y∈A

y∈A

and hence the Lipschitz continuity. (ii) As the function y → |y − x| is continuous (hence upper semicontinuous), dA (x) = inf |y − x| = inf |y − x| = dA (x). y∈A

y∈A

Since the function |y − x| ≥ 0 is bounded below, its infimum over y ∈ A is finite. Let {yn } ⊂ A be a minimizing sequence dA (x) = inf |y − x| = lim |yn − x|. y∈A

n→∞

Then {yn − x} and hence {yn } are bounded sequences. Hence there exists a subsequence converging to some y ∈ A and by continuity dA (x) = |y − x|. (iii) By the continuity of dA , d−1 A {0} is closed and A ⊂ d−1 A {0}

⇒ A ⊂ d−1 A {0}.

Conversely, if dA (x) = 0, there exists y in A such that 0 = dA (x) = |y − x| and necessarily x = y ∈ A. (iv) follows from (iii). (v) From (ii) for x ∈ A, dA (x) = 0 and dB (x) ≤ dA (x) = 0 necessarily implies dB (x) = 0 and x ∈ B. Conversely, if A ⊂ B, then dB (x) = dB (x) = inf |y − x| ≤ inf |y − x| = dA (x) = dA (x). y∈B

y∈A

(vi) is obvious. (vii) follows from Rademacher’s theorem.

2.2

Pomp´ eiu–Hausdorff Metric on Cd (D)

Let D be a nonempty subset of RN and associate with each nonempty subset A of D the equivalence class  def

[A]d = B : ∀B, B ⊂ D and B = A ,

270

Chapter 6. Metrics via Distance Functions

since from Theorem 2.1 (v) dA = dB if and only if A = B. So A is the invariant closed representative of the class [A]d . Consider the set  def

Fd (D) = [A]d : ∀A, ∅ = A ⊂ D that can be identified with the set 

A : ∀A, ∅ = A closed ⊂ D . In general, there is no open representative in the class [A]d since dA = dA ≤ dint A , where int A = A denotes the interior of A. By the definition of [A]d the map [A]d → dA : Fd (D) → Cd (D) ⊂ C(D) is injective. So Fd (D) can be identified with the subset of distance functions Cd (D) in C(D). The distance function plays the same role as the set X(D) in Lp (D) of equivalence classes of characteristic functions of measurable sets. When D is bounded, C(D) is a Banach space when endowed with the norm f C(D) = sup |f (x)|.

(2.5)

x∈D

As for the characteristic functions of Chapter 5, this induces a complete metric def

ρ([A]d , [B]d ) = dA − dB C(D) = sup |dA (x) − dB (x)|

(2.6)

x∈D

on Fd (D), which turns out to be equal to the classical Pomp´eiu–Hausdorff metric2 



def

ρH ([A]d , [B]d ) = max sup dA (x), sup dB (y) x∈B

y∈A

(cf. J. Dugundji [1, Chap. IX, Prob. 4.8, p. 205] for the definition of ρH ). Indeed for x ∈ D, xA ∈ A, and xB ∈ B |x − xA | ≤ |x − xB | + |xB − xA | ⇒ dA (x) ≤ |x − xB | + dA (xB ) ≤ |x − xB | + sup dA (y) y∈B

⇒

dA (x) ≤ dB (x) + sup dA (y) y∈B

⇒

dA (x) − dB (x) ≤ sup dA (y). y∈B

2 The “´ ´iu [1] in his thesis presented ecart mutuel” between two sets was introduced by D. Pompe in Paris in March 1905. This is the first example of a metric between two sets in the literature. It was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914.

2. Uniform Metric Topologies

271

By interchanging the roles of A and B  ∀x ∈ D,



|dA (x) − dB (x)| ≤ max sup dA (z), sup dB (y) . z∈B

y∈A

The converse inequality follows from the fact that A ⊂ D and B ⊂ D. 0 When D is open but not necessarily bounded, we use Cloc (D) (cf. (2.19) in section 2.6.1 of Chapter 2), the space C(D) of continuous functions on D endowed with the Fr´echet topology of uniform convergence on compact subsets K of D, which is defined by the family of seminorms ∀K compact ⊂ D,

def

qK (f ) = max |f (x)|. x∈K

(2.7)

The Fr´echet topology on C(D) is metrizable since the topology induced by the family of seminorms {qK } is equivalent to the one generated by the subfamily {qKk }k≥1 , where the compact sets {Kk }k≥1 are chosen as follows:   1 def Kk = x ∈ D : d
D (x) ≥ and |x| ≤ k , k ≥ 1. (2.8) k It is equivalent to the topology defined by the metric def

δ(f, g) =

∞ 1 qKk (f − g) . 2k 1 + qKk (f − g)

(2.9)

k=1

It will be shown below that Cd (D) is a closed subset of Cloc (D) and that this will induce the following complete metric on Fd (D): def

ρδ ([A]d , [B]d ) = δ(dA , dB ) =

∞ 1 qKk (dA − dB ) , 2k 1 + qKk (dA − dB )

(2.10)

k=1

which is a natural extension of the Hausdorff metric to an unbounded domain D. Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cd (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) defines a complete metric topology on Fd (D). (ii) When D is a bounded open subset of RN , the set Cd (D) is compact in C(D) and the metrics ρ and ρH are equal. Proof. (i) We give the proof of C. Dellacherie [1, Thm. 2, p. 42 and Rem. 1, p. 43]. The basic constructions will again be used in Chapter 7. Consider a sequence {An } of nonempty subsets of D such that dAn converges to some element f of Cloc (D). We wish to prove that f = dA for def

A = {x ∈ D : f (x) = 0}

272

Chapter 6. Metrics via Distance Functions

and that the closed subset A of D is nonempty. Fix x in D; then ∀n, ∃yn ∈ An ,

|yn − x| = inf |z − x| = dAn (x) z∈An

⇒ lim |yn − x| = lim dAn (x) = f (x). n→∞

n→∞

Hence {yn − x} and {yn } ⊂ D are bounded and there exists a subsequence, still indexed by n, which converges to some y ∈ D yn → y, and |y − x| = f (x). In particular, f (y) = 0, since in the inequality f (y) ≤ f (y) − dAn (y) + dAn (y) − dAn (yn ) + dAn (yn ), the last term is zero and dAn is Lipschitz continuous with constant equal to 1: |f (y)| ≤ |f (y) − dAn (y)| + |y − yn |, and both terms go to zero. By the definition of A, y ∈ A and A is not empty. Therefore for each x ∈ D, there exists y ∈ A such that f (x) = |y − x| ≥ inf |z − x| = dA (x). z∈A

Next we prove the inequality in the other direction. By construction for any An ⊂ D ∀x, y ∈ D,

|dAn (x) − dAn (y)| ≤ |x − y|

and |f (x) − f (y)| ≤ |f (x) − dAn (x)| + |dAn (x) − dAn (y)| + |dAn (y) − f (y)|. By uniform convergence the first and last terms converge to zero, and by Lipschitz continuity of dAn , ∀x, y ∈ D, |f (x) − f (y)| ≤ |x − y|. Hence for all x ∈ D and y ∈ A, f (y) = 0 and f (x) ≤ f (y) + |x − y| = |x − y| ⇒ ∀x ∈ D,

f (x) ≤ inf |x − y| = dA (x). y∈A

This proves the reverse equality. (ii) Observe that on a compact set D and for any A ⊂ D def

dA (x) = inf |y − x| ≤ sup |y − x| ≤ c = sup |y − x| < ∞, y∈A

∀x, y ∈ D,

y∈A

y,x∈D

|dA (y) − dA (x)| ≤ |y − x|.

2. Uniform Metric Topologies

273

The compactness of Cd (D) now follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2 and part (i). By construction, ρ and ρδ are metrics on Fd (D). By definition, for A and B in the compact D, ρ(A, B) = max |dA (x) − dB (x)| x∈D   ≥ max max |dA (x) − dB (x)|, max |dA (x) − dB (x)| x∈B x∈A   ≥ max max dA (x), max dB (x) = ρH (A, B). x∈B

x∈A

Conversely, for any x ∈ D and y in A, dA (x) − dB (x) ≤ |y − x| − dB (x) and there exists xB ∈ B such that dB (x) = |x − xB |. Therefore, ∀y ∈ A, dA (x) − dB (x) ≤ |y − x| − |x − xB | ≤ |y − xB | ⇒ dA (x) − dB (x) ≤ inf |y − xB | = dA (xB ) ≤ max dA (x). y∈A

x∈B

Similarly dB (x) − dA (x) ≤ max dB (x), x∈A

and for all x in D

  |dB (x) − dA (x)| ≤ max max dA (x), max dB (x) ⇒ ρ(A, B) ≤ ρH (A, B). x∈B

x∈A

When D is bounded the family Fd (D) enjoys more interesting properties. Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of RN . Define for a subset S of RN the sets  def

H(S) = dA ∈ Cd (D) : S ⊂ A ,  def

I(S) = dA ∈ Cd (D) : A ⊂ S ,  def

J(S) = dA ∈ Cd (D) : A ∩ S = ∅ . (i) Let S be a subset of RN . Then H(S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I(S) is closed in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then J(S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J(S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then I(S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded holdall D, the map  def

[A]d → #(A) : F# (D) = [A]d : ∀A, ∅ = A ⊂ D and #(A) < ∞ → R

274

Chapter 6. Metrics via Distance Functions is lower semicontinuous, where #(A) is the number of connected components as defined in Definition 2.2 of Chapter 2. For a fixed number c > 0 the subset {dA ∈ Cd (D) : #(A) ≤ c} of Cd (D) is compact in C(D). In particular, the subset {dA ∈ Cd (D) : A is connected} of Cd (D) is compact in C(D).

Proof. (i) If S ⊂ D, then H(S) = ∅ and there is nothing to prove. Assume that S ⊂ D. From Theorem 2.1 (v), S ⊂ A ⇒ dS ≥ dA . So for any sequence {dAn } in H(S) converging to dA in Cloc (D), ∀n,

dS ≥ dAn

⇒ dS ≥ dA

⇒ S⊂S⊂A

⇒ dA ∈ H(S).

(ii) We shall use the same technique for I(S) as for H(S), but here we need S = S to conclude. For D ∩ S = ∅, J(S) = ∅ and there is nothing to prove. Assume that D ∩ S = ∅ and consider a sequence {dAn } in J(S) converging to dA in Cloc (D). Assume that A ∩ S = ∅. Then A ⊂ D implies A ∩ [S ∩ D] = A ∩ S = ∅. By assumption, S ∩ D is compact and ∃δ > 0, ∀x ∈ S ∩ D, dA (x) ≥ δ, ∃N ≥ 1, ∀n ≥ N, dAn − dA C(D) ≤ δ/2. So for all x ∈ D ∩ S dAn (x) ≥ dA (x) − |dAn (x) − dA (x)| ≥ δ − δ/2 > 0 ⇒ An ∩ S = An ∩ [D ∩ S] = ∅, and this contradicts the fact that An ∈ J(S). (iii) By definition, for any S, J(S) = I(S),



 J(S) = dA ∈ Cd (D) : A ∩ S = ∅ = dA ∈ Cd (D) : A ⊂ S = I(S). Since S is closed, J(S) is closed from part (ii) and J(S) is open. Similarly, replacing S by S in the previous identity, I(S) = J(S). So from part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. From part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. (iv) Let {An } and A be nonempty subsets of D such that dAn converges to dA in C(D). Assume that #(A) = k is finite. Then there exists a family of disjoint open sets G1 ,. . . , Gk such that A ⊂ G = ∪ki=1 Gi

and ∀i, A ∩ Gi = ∅.

In view of the definitions of I(S) and J(S) in part (i) A ∈ U = ∩ki=1 J(Gi ) ∩ I(G).

2. Uniform Metric Topologies

275

But U is not empty and open as the finite intersection of k + 1 open sets. As a result there exist ε > 0 and an open neighborhood of [A]d : Nε ([A]d ) = {[B]d : dB − dA  < ε} ⊂ U. Hence since dAn converges to dA , there exists n > 0 such that, for all n ≥ n, [An ]d ∈ U, and necessarily An ⊂ G,

∀n ≥ n,

An ∩ Gi = ∅, ∀i

⇒ #(An ) ≥ #(A).

Therefore, [A]d → #(A) is lower semicontinuous. This theorem has many interesting corollaries. For instance part (iii) says something about the function that gives the number of connected components of A (cf. T. J. Richardson [1] for an application to image segmentation).

2.3

Uniform Complementary Metric Topology and Cdc (D)

In the previous section we dealt with a theory of closed sets since the equivalence class of the subsets of D was completely determined by their unique closure. In partial differential equations the underlying domain is usually open. To accommodate this point of view, consider the set of open subsets Ω of a fixed nonempty open holdall D in RN , endowed with the Hausdorff topology generated by the distance functions d
Ω to the complement of Ω. This approach has been used in ´sio [6, sect. 1.3, p. 405] in the conseveral contexts, for instance, in J.-P. Zole ˇ ´ k [2] uses the family Cdc (D) for text of free boundary problems. Also, V. Sver a open sets Ω such that #([Ω]) ≤ c for some fixed c > 0. His main result is that in dimension 2 the convergence of a sequence {Ωn } to Ω of such sets implies the convergence of the corresponding projection operators {PΩn : H01 (D) → H01 (Ωn )} to PΩ : H01 (D) → H01 (Ω), where the projection operators are directly related to homogeneous Dirichlet linear boundary value problems on the corresponding domains {Ωn } and Ω. In dimension 1 the constraint on the number of components can be dropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 8. By analogy with the constructions of the previous section, consider for a nonempty subset D of RN the family

 d
A : ∅ = A and A ⊂ D . By definition, A ⊃ D

⇒ d
A = 0 in D,

and, by Lipschitz continuity, d
A = 0 in D and d
A = 0 on ∂D = D ∩ D. If int D = ∅, then d
A = 0 in RN and for all sets A in the family int A = ∅. If int D = ∅, associate with each A the open set def

Ω = int A = A

⇒ Ω = A and d
Ω = d
A .

276

Chapter 6. Metrics via Distance Functions

By definition and the previous considerations, A ⊃ D

⇒ Ω = A ⊂ D = int D.

So finally, for int D = ∅, 



d
A : ∅ = A and A ⊂ D = d
Ω : Ω = RN and Ω open ⊂ int D . The invariant set int A is the open representative in the equivalence class   def [A]cd = B : B = A = ∅ , but, in general, there is no closed representative in that class since d
int A = d
A = d
A ≤ d
A , as in the case of dA , where dA = dA ≤ dint A . From this analysis it will be sufficient to consider the family of open subsets of an open holdall D. Definition 2.2. Let D be a nonempty open subset of RN . Define the family of functions  def

Cdc (D) = d
Ω : ∀Ω open subset of D and Ω = RN corresponding to the family of open sets  def

G(D) = Ω ⊂ D : ∀Ω open and Ω = RN .

(2.11)

(2.12)

For D bounded define the Hausdorff complementary metric ρcH on G(D): def

ρcH (Ω2 , Ω1 ) = d
Ω2 − d
Ω1 C(D) .

(2.13)

Note that for Ω1 ⊂ D and Ω2 ⊂ D, d
Ω1 = d
Ω2 = 0 in D. Therefore, d
Ω1 = d
Ω2 in D ⇐⇒ Ω1 = Ω2 ⇐⇒ Ω1 = Ω2 . Theorem 2.4. Let D be a nonempty open subset of RN . (i) The set Cdc (D) is closed in Cloc (D). (ii) If, in addition, D is bounded, then Cdc (D) is compact in C0 (D) and (G(D), ρcH ) is a compact metric space. (iii) (Compactivorous property) Let {Ωn } and Ω be sets in G(D) such as d
Ωn → d
Ω

in Cloc (D).

Then for any compact subset K ⊂ Ω, there exists an integer N (K) > 0 such that ∀n ≥ N (K), K ⊂ Ωn .

2. Uniform Metric Topologies

277

Proof. (i) Let {Ωn } be a sequence of open subsets in D such that {d
Ωn } is a Cauchy sequence in Cloc (D). For all n, d
Ωn = 0 in D and {d
Ωn } is also a Cauchy sequence in Cloc (RN ). By Theorem 2.2 (i) the sequence {d
Ωn } converges in Cloc (RN ) to some distance function dA . By construction Ωn ⊂ D =⇒ Ωn ⊃ D =⇒ d
Ωn ≤ d
D =⇒ dA ≤ d
D =⇒ A ⊃ D =⇒ A ⊂ D. By choosing the open set Ω = A in D we get d
Ωn → d
Ω

in Cloc (RN ) and C0,loc (D).

(ii) To prove the compactness we use the compactness of Cd (D) from Theorem 2.2 (ii) and the fact that Cdc (D) is closed in C0 (D). Observe that since Ωn ⊃ D, d
Ωn = 0 in D, d
Ωn ∈ C0 (D), and Ωn = [Ωn ∩ D] ∪ [Ωn ∩ D] ⊂ [Ωn ∩ D] ∪ D ⊂ Ωn ⇒ d
Ωn = d[
Ωn ∩D]∪
D = min{d
Ωn ∩D , d
D }. There exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that d
Ωn ∩D → dA

in C0 (D)

⇒

d
Ωn = min{d
Ωn ∩D , d
D } → min{dA , d
D } = dA∪
D

⇒

d
Ωn → dA∪
D

in C(D)

in C(RN )

since d
Ωn = 0 = dA∪
D in D. The result now follows by choosing the open set def

Ω = [A ∪ D] = A ∩ D ⊂ D. (iii) Define

def

m = inf inf |x − y| = inf d
Ω (x). x∈K y∈
Ω

x∈K

Since K ⊂ Ω ⊂ D, there exist x ˆ ∈ K, yˆ ∈ ∂Ω such that x − yˆ|. m = inf inf |x − y| = |ˆ x∈K y∈∂Ω

Necessarily, m > 0 since Ω is open, x ˆ ∈ K ⊂ Ω, and yˆ ∈ ∂Ω. Now ∃N > 0, ∀n ≥ N, d
Ωn − d
Ω C(K) < m/2 ⇒ ∀x ∈ K, d
Ωn (x) ≥ d
Ω (x) − |d
Ωn (x) − d
Ω (x)| ≥ m − m/2 > 0 ⇒ x = Ωn

⇒ x ∈ Ωn = int Ωn = Ωn

Notation 2.1. It will be convenient to write

⇒ K ⊂ Ωn .

Hc

Ωn → Ω for the Hausdorff complementary convergence of open sets of G(D) d
Ωn → d
Ω

in Cloc (D).

278

Chapter 6. Metrics via Distance Functions

We have the analogue of Theorem 2.3 for the distance function to the complement of an open subset of D. Theorem 2.5. Let D be a nonempty open (resp., bounded open) subset of RN . Define for a subset S of RN and open subsets Ω of D the following families:  def

H c (S) = d
Ω ∈ Cdc (D) : S ⊂ Ω ,  def

I c (S) = d
Ω ∈ Cdc (D) : Ω ⊂ S ,  def

J c (S) = d
Ω ∈ Cdc (D) : Ω ∩ S = ∅ . (i) Let S be a subset of RN . Then H c (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I c (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then J c (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J c (S) is open in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then I c (S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded open holdall D, the map    def

Ω → #(Ω) : G# (D) = Ω : Ω open ⊂ D and # Ω < ∞ → R is lower semicontinuous, where #(Ω) is the number of connected components of Ω as defined in Definition 2.2 of Chapter 2. For a fixed number c ≥ 0, the subset   {d
Ω ∈ Cdc (D) : # Ω ≤ c} is compact in C(D). In particular, the subset {d
Ω ∈ Cdc (D) : Ω is hole-free} is compact in C(D), where hole-free is in the sense of Definition 2.2 of Chapter 2.

2.4

c Families Cdc (E; D) and Cd,loc (E; D)

Now and then, it is desirable to avoid having the empty set as a solution of a shape optimization problem. This can occur for the C(D)-topology on Cdc (D). One way to get around this is to assume that each set of the family contains a fixed nonempty set or, more generally, that each set of the family contains a translated and rotated image of a fixed nonempty set. Theorem 2.6. Let D, ∅ = D ⊂ RN , be bounded open, let E, ∅ = E ⊂ D, and let def

Cdc (E; D) = {d
Ω : E ⊂ Ω open ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open ⊂ D N

d
Ω :

(2.14)

& .

(2.15)

3. Projection, Skeleton, Crack, and Differentiability

279

c (E; D) are compact in C(D). (i) If E is open, Cdc (E; D) and Cd,loc c (ii) If E is closed, Cdc (E; D) and Cd,loc (E; D) are open in C(D).

Proof. (a) We first deal with Cdc (E; D) using the fact that

 Cdc (E; D) = d
Ω : Ω ⊂ E and Ω open ⊂ D = I c (E) and Theorem 2.5. If E is open, E is closed and I c (E) is closed and hence compact since D is bounded. If E is closed, E is open, E ∩ D is compact, and I c (E) is open. c (D) (b) Assume that E is open. Let {d
Ωn } be a Cauchy sequence in CD converging to some d
Ω . Denote by xn ∈ RN and An ∈ O(N) the translation and rotation of E such that xn + An E ⊂ Ωn and d
xn +An E ≤ d
Ω . Since, for all n, xn + An E ⊂ D and D is bounded, there exist x ∈ RN and A ∈ O(N) and subsequences, still indexed by n, such that xn → x and An → A. But d
xn +An E (z) = d
E (An (z − xn )) since A−1 = A in O(N) and d
E (An (z − xn )) → d
E (A(z − x)). Finally, d
x+AE = d
E (A(z − x)) ≤ d
Ω and, since x + AE and Ω are both closed, x + AE ⊂ Ω, c c which implies x + AE ⊃ Ω and d
Ω ∈ Cd,loc (E; D). Therefore Cd,loc (E; D) as c a closed subset of the compact set Cd (D) is compact. We leave the proof that c (E; D) is open when E is closed to the reader. Cd,loc

3

Projection, Skeleton, Crack, and Differentiability

In this section we study the connection between the gradient of dA , the set of projections onto A, and the characteristic function of A. We partition the set of singularities of the gradient of dA into the skeleton 3 and set of cracks of A. Definition 3.1. Let A ⊂ RN , ∅ = A, and denote by  def

Sing (∇dA ) = x ∈ RN : ∇dA (x) 

(3.1)

the set of singularities of the gradient of dA . (i) Given x ∈ RN , a point p ∈ A such that |p − x| = dA (x) is called a projection onto A. The set of all projections onto A will be denoted by  def

ΠA (x) = p ∈ A : |p − x| = dA (x) . (3.2) When ΠA (x) is a singleton, its element will be denoted by pA (x). 3 Our

definition of a skeleton does not exactly coincide with the one used in morphological mathematics, where it is defined as the closure Sk (A) of our skeleton Sk (A) (cf., for instance, `re [1], and J. Serra [1] for the pioneering applications in mining G. Matheron [1] or A. Rivie engineering in 1968 and later in image processing).

280

Chapter 6. Metrics via Distance Functions

(ii) The skeleton of A is the set of all points of RN whose projection onto A is not unique. It will be denoted by  def

Sk (A) = x ∈ RN : ΠA (x) is not a singleton . (3.3) (iii) The set of cracks is defined as def

Ck (A) = Sing ∇dA \Sk (A).

(3.4)

Note that Sk (A) = Sk (A), Ck (A) = Ck (A), and Sing (A) = Sing (A) since dA = dA by Theorem 2.1 (ii). Even for a set A with smooth boundary ∂A, the gradient ∇dA (x) may not exist far from the boundary as shown in Figure 6.1, where ∇dA (x) exists everywhere outside A except on Sk (A), a semi-infinite line. Yet, ∇dA (x) exists close to the boundary. An example of the set of cracks is given by the eyes, nose, mouth, and rays of the smiling sun in Figure 5.1 of Chapter 5 that belong to the boundary of the set as opposed to the skeleton Sk (A) that lies outside of A. Sk (A)

Sk (A) A

Figure 6.1. Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). The level sets of dA generate dilations/tubular neighborhoods of A. Definition 3.2. Let ∅ = A ⊂ RN and let h > 0. (i) The open h-tubular neighborhood , the closed h-tubular neighborhood , and the h-boundary of A are respectively defined as   def

def

Uh (A) = x ∈ RN : dA (x)| < h , Ah = x ∈ RN : dA (x)| ≤ h , and d−1 A {h}. We shall also use the terminology open or closed dilated sets. (ii) For 0 < s < h < ∞   def

def

Us,h (A) = x ∈ RN : s < dA (x)| < h and As,h = x ∈ RN : s ≤ dA (x)| ≤ h . This terminology is justified by the fact that a dilated set is a relatively nice set.

3. Projection, Skeleton, Crack, and Differentiability

281

Theorem 3.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A).

(3.5)

In particular, for 0 < s < r, Us,r (A) = As,r . Proof. By definition A ⊂ Ur (A) ⊂ Ar and Ur (A) ⊂ Ar . For any x ∈ Ar , dA (x) ≤ r. / A and there If dA (x) < r, then x ∈ Ur (A) ⊂ Ur (A). If dA (x) = r > 0, then x ∈ exists p ∈ ∂A such that dA (x) = |x − p| = r > 0. Consider the points   x−p x−p 1 def r →p+r = x. xn = p + 1 − n |x − p| |x − p| Then p ∈ ΠA (xn ), and dA (xn ) = r (1 − 1/n) < r, dA (xn ) → dA (x) = r, and x ∈ Ur (A). Therefore, since Ur (A) is open, Ar = Ur (A) = Ur (A) ∪ ∂Ur (A) implies that ∂Ur (A) = d−1 A {r}. Moreover, since Ur (A) is closed, Ur (A) = int Ur (A) ∪ ∂Ur (A) = Ar ∪ ∂Ur (A) implies that Ar = Ur (A). The proof is the same for Us,r (A) = As,r . We now give several important technical results that will be used later for sets of positive reach and in Chapter 7. We first give a few additional definitions. Definition 3.3. (i) A function f : U → R defined in a convex subset U of RN is convex if for all x and y in U and all λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y). It is concave if the function −f is convex. A function f : U → R defined in a convex subset U of RN is semiconvex (resp., semiconcave) if ∃c ≥ 0,

fc (x) = c |x|2 + f (x)

(resp., fc (x) = c |x|2 − f (x))

is convex in U . (ii) A function f : U → R defined in a subset U of RN is locally convex (resp., locally concave) if it is convex (resp., concave) in every convex subset of U . A function f : U → R defined in a subset U of RN is locally semiconvex (resp., locally semiconcave) if it is semiconvex (resp., semiconcave) in every convex subset of U . When U is convex the local definitions coincide with the global ones. The next theorem is a slight extension of a result in J. H. G. Fu [1, Prop. 1.2]. Theorem 3.2. Let A ⊂ RN be such that A = ∅. (i) Given h > 0, the function def

kA,h (x) =

|x|2 − 2 h dA (x), |x|2 − d2A (x) − h2 ,

if dA (x) ≥ h, if dA (x) < h,

282

Chapter 6. Metrics via Distance Functions is convex (and continuous) in RN and the function x →

kA,h (x) |x|2 = − dA (x) : RN \Uh (A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (A). (ii) The function def

x → fA (x) =

 1  2 |x| − d2A (x) : RN → R 2

is convex and continuous, d2A (x) is the difference of two convex functions   d2A (x) = |x|2 − |x|2 − d2A (x) , (3.6)  N ∇fA and ∇d2A belong to BVloc (RN )N , and ∇dA ∈ BVloc RN \∂A . Proof. (i) For all p ∈ A define the convex function p (x) =

|x − p| − h, 0,

|x − p| ≥ h, |x − p| < h.

Since p is nonnegative, 2p is convex and |x|2 − 2 h |x − p| + |p|2 + h2 − 2 x · p, 0,

2p (x) =

|x − p| ≥ h, |x − p| < h.

By subtracting the constant term |p|2 + h2 and the linear term −2 x · p from 2p , we get the new convex function mp (x) =

|x|2 − 2 h |x − p|, |x|2 − |p − x|2 − h2 ,

|x − p| ≥ h, |x − p| < h.

Then the function def

k(x) = sup mp (x) = p∈A

|x|2 − 2 h dA (x),

& dA (x) ≥ h,

|x|2 − d2A (x) − h2 ,

dA (x) < h

is convex in RN . Moreover, it is finite for each x ∈ RN and hence continuous in x. Indeed, if x is such that dA (x) ≥ h, by Theorem 2.1 (ii) there exists p¯ ∈ A such that |x − p¯| = dA (x) ≥ h and |x − p¯| = dA (x) =

inf p∈A |x−p|≥h

|x − p|

3. Projection, Skeleton, Crack, and Differentiability

283

and k(x) = |x|2 − 2 h dA (x). If dA (x) < h, then there exists p¯ ∈ A such that |x − p¯| = dA (x) < h and |x − p¯| = dA (x) =

inf

|x − p|

p∈A |x−p|
and k(x) = |x|2 − d2A (x) − h2 . We recover the result of J. H. G. Fu [1, Prop. 1.2] by observing that the restriction of k to RN \Uh (A) is locally convex. In addition the function |x|2 − d2A (x) is locally convex in Uh (A). (ii) Since for all h > 0, |x|2 − d2A (x) is locally convex in Uh (A) it is locally convex and continuous in RN = ∪h>0 Uh (A) and hence convex in RN . Therefore, from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240], ∇fA ∈ BVloc (RN )N and ∇d2A ∈ BVloc (RN )N . Finally, RN \∂A is the union of the two disjoint open sets RN \A and int A. On int A, ∇dA = 0. For any point x in RN \A, there exists ρ, 0 < 3 ρ < dA (x), such that the open ball B(x, 2 ρ) is contained in RN \Uρ (A), where kA,ρ is locally convex by part (i). Hence kA,ρ is convex in B(x, 2 ρ) and ∇kA,ρ , and ∇dA belong to BVloc (B(x, 2ρ))N and, by Definition 6.1 in Chapter 5, ∇dA belong to BV(B(x, ρ))N . Finally, by Theorem 6.1 in Chapter 5, ∇dA belongs to BVloc (RN \A)N . We now give some basic results. Theorem 3.3. Let A, ∅ = A ⊂ RN , and let x ∈ RN . (i) The set ΠA (x) is nonempty, compact, and ∀x ∈ /A

ΠA (x) ⊂ ∂A

and

∀x ∈ A

ΠA (x) = {x}.

(ii) For all x and v in RN , the Hadamard semiderivative4 of d2A always exists, d2A (x + tv) − d2A (x) = min 2(x − p) · v t0 t p∈ΠA (x)

dH d2A (x; v) = lim

= 2(x · v − σΠA (x) (v)), fA (x + tv) − fA (x) = σΠA (x) (v) = σco ΠA (x) (v), dH fA (x; v) = lim t0 t where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular, 

Sk (A) = x ∈ RN : ∇d2A (x)  ⊂ RN \A. 4A

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Definition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.7)

284

Chapter 6. Metrics via Distance Functions Given v ∈ RN , for all x ∈ RN \∂A the Hadamard semiderivative of dA exists, dH dA (x; v) =

min 2 p∈ΠA (x)

x−p · v, |x − p|

(3.8)

and for x ∈ ∂A, dH dA (x; v) exists if and only if lim

t0

dA (x + tv) t

exists.

(3.9)

(iii) The following statements are equivalent: (a) d2A (x) is (Fr´echet) differentiable at x. (b) d2A (x) is Gateaux differentiable at x. (c) ΠA (x) is a singleton. (iv) ∇fA (x) exists if and only if ΠA (x) = {pA (x)} is a singleton. In that case 1 pA (x) = ∇fA (x) = x − ∇d2A (x). 2 For all x and y in RN , ∀p(x) ∈ ΠA (x),

 1 2  1 2 |y| − d2A (y) ≥ |x| − d2A (x) + p(x) · (y − x), (3.10) 2 2

or equivalently, ∀p(x) ∈ ΠA (x),

d2A (y) − d2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.11)

For all x and y in RN ∀p(x) ∈ ΠA (x), ∀p(y) ∈ ΠA (y),

(p(y) − p(x)) · (y − x) ≥ 0.

(3.12)

(v) The functions pA : RN \Sk (A) → RN

and

∇d2A : RN \Sk (A) → RN

are continuous. For all x ∈ RN \Sk (A) 1 pA (x) = x − ∇d2A (x) = ∇fA (x). 2 In particular, for all x ∈ A, ΠA (x) = {x} and ∇d2A (x) = 0, 

Sk (A) = x ∈ RN : ∇d2A (x)  and ∇dA (x)  ⊂ RN \A,

 Ck (A) = x ∈ RN : ∇d2A (x) ∃ and ∇dA (x)  ⊂ ∂A, and Sing (∇dA ) = Sk (A) ∪ Ck (A).

(3.13)

(3.14)

3. Projection, Skeleton, Crack, and Differentiability

285

(vi) The function   ∇dA : RN \ Sk (A) ∪ ∂A → RN   is continuous. For all x ∈ int A, ∇dA (x) = 0 and for all x ∈ RN \ Sk (A) ∪ A ∇dA (x) =

∇d2A (x) x − pA (x) = 2 dA (x) |x − pA (x)|

|∇dA (x)| = 1.

and

(3.15)

If ∇dA (x) exists for x ∈ A, then ∇dA (x) = 0. In particular, ∇dA = 0 almost everywhere in A. (vii) The functions d2A and dA are differentiable almost everywhere and m(Sk (A)) = m(Ck (A)) = m(Sing (∇dA )) = 0. For ∂A = ∅ Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.16) (3.17)

If ∂A = ∅, then either A = RN or A = ∅. (viii) Given A ⊂ RN , ∅ = A (resp., ∅ = A), χA (x) = 1 − |∇dA (x)|,

χint
A (x) = |∇dA (x)| in RN \Sing (∇dA )

(resp., χ
A (x) = 1 − |∇d
A (x)|, χint A (x) = |∇d
A (x)| in RN \Sing (∇d
A )) and the above identities hold for almost all x in RN . def

(ix) Given x ∈ RN , α ∈ [0, 1], p ∈ ΠA (x), and xα = p + α (x − p), then dA (xα ) = |xα − p| = α |x − p| = α dA (x)

and

∀α ∈ [0, 1], ΠA (xα ) ⊂ ΠA (x).

In particular, if ΠA (x) is a singleton, then ΠA (xα ) is a singleton and ∇d2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = A, then for all 0 < α ≤ 1 ∇dA (xα ) exists and ∇dA (xα ) = ∇dA (x). Remark 3.1. In general, for each v ∈ RN ∃ˆ p = p(v) ∈ ΠA (x),

dH d2A (x; v) = 2(x − pˆ) =

inf p∈ΠA (x)

2(x − p) · v,

since ΠA (x) is compact. However, when ΠA (x) is not a singleton, pˆ is not necessarily unique since 2(x − p) depends on the direction v. For points x outside of A and Sk (A) the norm of ∇dA (x) is equal to 1. When A is sufficiently smooth, ∇dA (x) coincides with the outward unit normal to A at the point pA (x). When A is not smooth, this normal is not always unique as shown in Figure 6.2.

286

Chapter 6. Metrics via Distance Functions x pA (x) = pA (y) ∇dA (x) A

∇dA (y)

y Figure 6.2. Nonuniqueness of the exterior normal.

It is useful to consider examples to illustrate the subtleties of the theorem. The set of cracks Ck (A) of a set A has zero measure, but its boundary ∂A can have a nonempty interior and a nonzero measure. Example 3.1. Let B1 (0) be the open ball of radius 1 at the origin in RN and let A = {x ∈ B1 (0) : x has rational coordinates}. Then A = B1 (0), A = RN , ∂A = B1 (0), ∂A = ∂B1 (0), ∇dA (x) = ∇d2A (x) = 0 in B1 (0)

x = pA (x) in B1 (0), |x| Ck (A) = ∂B1 (0).

and ∇dA (x) =

Sk (A) = ∅ and

In general ∂A ⊂ ∂A and this inclusion can be strict as illustrated by this example. Also note that the boundary ∂A has a nonempty interior int ∂A = B1 (0) and a nonzero Lebesgue measure (nonzero volume in dimension N = 3). Thus the boundary of two members of the equivalence class [A] can be different. The boundary ∂A of a closed or an open set A has no interior; that is, ∂A is nowhere dense. Yet, its boundary ∂A still does not necessarily have a zero measure. If A is closed, then A = int A ∪ ∂A. If int A = ∅, there exist x ∈ ∂A and r > 0 such that Br (x) ⊂ ∂A ⊂ A and x ∈ int A, a contradiction. If A is open, A is closed, A = int A ∪ ∂A, and ∂A has no interior. Example 3.2 (Modified version of Example 1.10 in E. Giusti [1, p. 7]). Let B1 (0) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi : i ∈ N} be the sequence of all points in B1 (0) with rational coordinates for some choice of ordering. Associate with each i the open ball Bi = {x ∈ R2 : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}

⇒ Bi ⊂ B1 (0).

3. Projection, Skeleton, Crack, and Differentiability

287

Consider the increasing sequence of open subsets of B1 (0) def

Ωn =

n *

def

Bi  Ω =

i=1

∞ *

Bi

i=1

def

and the closed set A = Ω. We get A = A = Ω,

A = Ω,

A = Ω = B1 (0),

∂A = B1 (0) ∩ Ω.

As a result, we have the following decreasing sequence of closed subsets of B1 (0): def

Dn = B1 (0) ∩ Ωn  ∂A = B1 (0) ∩ Ω 2 n n  1 m(Bi ) = π − π ⇒ m(Dn ) ≥ m(B1 (0)) − 2i i=1 i=1 2 ∞  2 1 ⇒ m(∂A) ≥ π − π = π. i 2 3 i=1 Proof of Theorem 3.3. (i) Existence. The function z → |z − x|2 : RN → R

(3.18)

is continuous and, a fortiori, upper semicontinuous, and hence inf |a − x|2 = inf |a − x|2 .

a∈A

a∈A

If A is bounded, there exists p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . If A is unbounded, the function (3.18) is coercive with respect to A, |a − x|2 = +∞, |a| a∈A, |a|→∞ lim

and we also have the existence of p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . By definition, ΠA (x) ⊂ A. For x ∈ RN \A, dA (x) > 0. Assume that there exists p ∈ ΠA (x) such that p ∈ int A. Therefore, there exists r, 0 < r < dA (x)/2, such that Br (p) ⊂ A. Choose the point r x−p 2 |x − p|

⇒ |y − p| = r/2 ⇒ y ∈ Br (p) ⊂ A   r r x−p = 1− (p − x) ⇒ y−x=p−x+ 2 |x − p| 2dA (x)   r ⇒ ∃y ∈ A such that |y − x| = 1 − |p − x| < dA (x); 2dA (x) y =p+

this contradicts the minimality of dA (x). Hence x ∈ ∂A.

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(ii) Semidifferentiability of fA . The semiderivative in a direction v can be readily computed by using a theorem on the differentiability of the min with respect to a parameter (cf. Chap. 10, sect. 2.3, Thm. 2.1). It is sufficient to prove the semidifferentiability of d2A in any direction v, that is, for each x and v the existence of the limit d2 (x + tv) − d2A (x) def . dd2A (x; v) = lim A t0 t Recall that for t ≥ 0,

 ΠA (x + t v) = pt ∈ A¯ : |x + tv − pt |2 = d2A (x + tv) and consider for t > 0 the quotient qt =

d2A (x + tv) − d2A (x) . t

For all p ∈ ΠA (x) and pt ∈ ΠA (x + t v) |x + tv − pt |2 − |x − p|2 t |x + tv − p|2 − |x − p|2 = t |v|2 + 2 v · (x − p), ≤ t

qt =

and for all p ∈ ΠA (x) lim sup qt ≤ 2 v · (x − p) t0

⇒ q¯ = lim sup qt ≤ 2 t0

inf p∈ΠA (x)

v · (x − p).

In the other direction choose a sequence tk > 0 such that tk → 0 and qtk → q = lim inf t0 . The corresponding sequence ptk in A is uniformly bounded since |ptk | ≤ |x + tk v − ptk | + |x + tk v| ≤ dA (x + tk v) − dA (x) + dA (x) + |x + tk v| ≤ tk |v| + dA (x) + |x + tk v|, and we can find p0 ∈ A¯ and a subsequence of {ptk }, still denoted by {ptk }, such that ptk → p0 . But by continuity dA (x + tk v) → dA (x) and |x + tk v − ptk | → |x − p0 | ⇒ ∃p0 ∈ A¯ such that dA (x) = |x − p0 | ⇒ p0 ∈ ΠA (x). Therefore, |x + tk v − ptk |2 − |x − p0 |2 tk |x + tk v − ptk |2 − |x − ptk |2 ≥ = tk |v|2 + 2 v · (x − ptk ) tk ⇒ q ≥ 2 v · (x − p0 ) ≥ 2 inf v · (x − p).

qtk =

p∈ΠA (x)

3. Projection, Skeleton, Crack, and Differentiability

289

Hence dd2A (x; v) = 2

inf p∈ΠA (x)

v · (x − p).

Finally, by Theorem 2.4 (ii), dH d2A (x; v) = dd2A (x; v) since d2A is locally Lipschitzian. The same remark applies to ddA (x; v) in RN \∂A. (iii) From part (ii), d2A (x) is Gateaux differentiable at x if and only if ΠA (x) is a singleton. The (Fr´echet) differentiability is a standard part of Rademacher’s theorem, but we reproduce it here for completeness as a lemma. Lemma 3.1. Let f : RN → R be a locally Lipschitzian function. Then f is (Fr´echet) differentiable at x, that is, lim

y→x

f (y) − f (x) − ∇f (x) · (y − x) =0 |y − x|

if and only if it is Gateaux differentiable at x, that is, ∀v ∈ RN ,

f (x + tv) − f (x) exists t0 t

df (x; v) = lim

and the map v → df (x; v) is linear and continuous. Proof. It is sufficient to prove that the Gateaux differentiability implies that f (y) − f (x) − ∇f (x) · (y − x) → 0. y→x |y − x| lim

The function f is locally Lipschitzian and there exist δ1 > 0 and c1 > 0 such that f is uniformly Lipschitzian in the open ball B(x, δ1 ) with Lipschitz constant c1 . Therefore, |∇f (x)| ≤ c1 . The unit sphere S(0, 1) = {v ∈ RN : |v| = 1} is compact and can be covered by the family of open balls {B(v, ε/(4c1 )) : v ∈ S(0, 1)} for an arbitrary ε > 0. Hence there exists a finite subcover {B(vn , ε/(4c1 )) : 1 ≤ n ≤ N } of S(0, 1). As a result, given any v ∈ S(0, 1), there exists n, 1 ≤ n ≤ N , such that |v − vn | ≤ ε/(4c1 ). Define the function g(x, v, t) =

f (x + tv) − f (x) − ∇f (x) · v t

for x ∈ RN , v ∈ S(0, 1), and t > 0. Since f (x) is Gateaux differentiable ∃δ, 0 < δ < δ1 , ∀n, 1 ≤ n ≤ N, ∀t < δ,

|g(x, vn , t)| < ε/2.

Hence for |y − x| < δ   g x, y − x , |y − x| |y − x|   y−x ≤ |g(x, vn , |y − x|)| + g x, , |y − x| − g(x, vn , |y − x|) |y − x| y−x ε ε − vn ≤ + 2 c1 ≤ |g(x, vn , |y − x|)| + 2 c1 = ε, |y − x| 2 (4c1 ) and we conclude that f (x) is differentiable at x.

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(iv) From part (ii), ∇fA (x) exists if and only if ΠA (x) is a singleton. Inequality (3.10) follows directly from the inequality ∀x and y ∈ RN , ∀ p(x) ∈ ΠA (x),

d2A (y) ≤ |p(x) − y|2

since d2A (y) − |y|2 ≤ |p(x) − x + x − y|2 − |y|2 ≤ |p(x) − x|2 + |x − y|2 + 2 (p(x) − x) · (x − y) − |y|2 ≤ |p(x) − x|2 − |x|2 + 2 (p(x) − x) · (x − y) + |x − y|2 + |x|2 − |y|2 and − 2fA (y) ≤ −2fA (x) + 2 p(x) · (x − y) − 2 x · (x − y) + |x − y|2 + |x|2 − |y|2 ≤ −2fA (x) − 2 p(x) · (y − x) ⇒ fA (y) ≥ fA (x) + 2 p(x) · (y − x). Inequality (3.11) is (3.10) rewritten. Inequality (3.12) follows by adding inequality (3.10) to the same inequality with x and y permuted. (v) From the equivalences in (ii), when d2A is differentiable at x, then ΠA (x) = {pA (x)} is a singleton, and from part (i) ∇d2A (x) = 2(x − pA (x)), which yields the expression for pA (x). When x ∈ A, ΠA (x) = {x}, and by substitution ∇d2A (x) = 0. Finally, it is sufficient to prove the continuity of pA . Given x ∈ RN \Sk (A), consider a sequence {xn } ⊂ RN \Sk (A) such that xn → x and the associated sequence {pA (xn )} ⊂ A. For all n |pA (xn ) − x| ≤ |pA (xn ) − xn | + |xn − x| ≤ |pA (x) − xn | + |xn − x| ≤ |pA (x) − x| + 2|xn − x| < dA (x) + 2r,

def

r = sup |xn − x| < ∞. n

So the sequence {pA (xn )} ⊂ A is bounded and there exists p ∈ A and a subsequence, still denoted by {pA (xn )}, such that pA (xn ) → p. Therefore dA (x) ← dA (xn ) = |xn − pA (xn )| → |x − p| ⇒ ∃p ∈ A,

dA (x) = |x − p|

⇒ pA (x) = p

and pA is continuous at x since p = pA (x) is independent of the choice of the converging subsequence. To complete the proof of identities (3.14), recall that the projection of each point of A onto A being a singleton, Sk (A) ⊂ RN \A. Therefore, for any x ∈ Sk (A), we have dA (x) > 0 and if ∇d2A (x) does not exist, then ∇dA (x) cannot exist. This sharpens the characterization (3.7) of Sk (A). From this we readily get the characterization of Ck (A) = Sing (∇dA )\Sk (A) and Ck (A) ⊂ A. This last

3. Projection, Skeleton, Crack, and Differentiability

291

property can be improved to Ck (A) ⊂ ∂A by noticing that for x ∈ int A, dA (x) = 0 and ∇d2A (x) = ∇dA (x) = 0. (vi) For all x ∈ int A, dA (x) = 0 and hence ∇dA (x) = 0 is constant and, a fortiori, continuous in int A. For x ∈ RN \(Sk (A) ∪ A), dA (x) > 0 and for all v ∈ RN and t > 0 sufficiently small dA (x + tv)2 − dA (x)2 1 dA (x + tv) − dA (x) = t t dA (x + tv) + dA (x) dd2 (x; v) 1 1 ⇒ ddA (x; v) = A = ∇d2 (x) · v ⇒ ∇dA (x) = ∇d2 (x). 2dA (x) 2dA (x) A 2dA (x) A By continuity of ∇d2A (x) from part (iv) and continuity of dA , ∇dA (x) is continuous since dA (x) > 0. Moreover, from part (iv), ∇dA (x) = 2(x − pA (x))/2dA (x) and |∇dA (x)| = 1. If ∇dA (x) exists in some x ∈ A, then dA (x) = 0 and for all v ∈ RN , the differential quotient converges: dA (x + tv) dA (x + tv) − dA (x) = lim = ∇dA (x) · v t0 t t ⇒ ∀v, 0 ≤ ±∇dA (x) · v ⇒ ∇dA (x) = 0.

0 ≤ lim

t0

Since ∇dA exists almost everywhere, ∇dA = 0 almost everywhere in A. (vii) The function dA is Lipschitzian and the function d2A is locally Lipschitzian. By Rademacher’s theorem they are both differentiable almost everywhere. The other properties follow from parts (iii), (iv), and (v). By definition Sk (∂A) = {x ∈ RN : ∇d2∂A (x)} ⊂ RN \∂A = RN \A∪RN \A. If x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = dA (x), ∇d2A (x), and x ∈ Sk (A); if x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = d
A (x), ∇d2
A (x), and x ∈ Sk (A). Conversely, if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = dA (x), ∇d2∂A (x), and x ∈ Sk (∂A); if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = d
A (x), ∇d2∂A (x), and x ∈ Sk (∂A). Finally, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ ∂A ∪ ∂A = ∂(∂A). The property that Ck (∂A) = Ck (A) ∪ Ck (A) is a consequence of the following property for x ∈ ∂A: ∇d∂A (x) exists if and only if ∇dA (x) and ∇d
A (x) exist. Indeed, for x ∈ ∂A, d∂A (x) = dA (x) = d
A (x) = 0. From part (v), if ∇d∂A (x) exists, it is 0. But ∂A ⊂ A implies d∂A ≥ dA and for all v ∈ RN and t > 0 0≤

dA (x + tv) − dA (x) d∂A (x + tv) − d∂A (x) ≤ →0 t t

⇒ ∇dA (x) = 0

and similarly ∇d
A (x) = 0. Conversely, from part (v), if ∇dA (x) and ∇d
A (x) exist, they are 0. Moreover, d∂A (x) = max{dA (x), d
A (x)}, where   dA (x + tv) − dA (x) d
A (x + tv) − d
A (x) d∂A (x + tv) − d∂A (x) 0≤ = max , . t t t Since both ∇dA (x) = 0 and ∇d
A (x) = 0, then ∇d∂A (x) = 0. Finally, it is easy to check that ∂A ∪ ∂A = ∂(∂A). The other properties for ∂A = ∅ are obvious.

292

Chapter 6. Metrics via Distance Functions

(viii) From part (i), if ∇dA (x) exists for x ∈ A, it is equal to 0. Hence χA (x) = 1 = 1 − |∇dA (x)| on A\Ck (A). From the definition of Sk (A), ∇dA (x) exists and |∇dA (x)| = 1 for all points in int A\Sk (A). Hence from part (i), χA (x) = 0 = 1 − |∇dA (x)| = 0 on A\Sk (A). From this, we get the first line of identities in RN \Sing (∇dA ). Since m(Sing (∇dA )) = 0, the above identities are satisfied almost everywhere in RN . We get the same results with A in place of A. (ix) If x ∈ A, ΠA (x) = {x} and there is nothing to prove. If x ∈ / A, then dA (x) = |x − p| > 0 and p = x for all p ∈ ΠA (x). First dA (xα ) ≤ |xα − p| = α |x − p|. If the inequality is strict, then there exists pα ∈ A such that |xα − pα | = dA (xα ) and |x − pα | ≤ |x − xα | + |xα − pα | = (1 − α)|x − p| + dA (xα ) < |x − p| = dA (x). This contradicts the minimality of dA (x) with respect to A. Therefore, dA (xα ) = α |x − p| = |xα − p|

⇒ p ∈ ΠA (xα ).

Now for any pα ∈ ΠA (xα ), pα ∈ A and |pα − x| ≤ |pα − xα | + |xα − x| = α |x − p| + (1 − α) |p − x| = |p − x| = dA (x) and pα ∈ ΠA (x). Hence ΠA (xα ) ⊂ ΠA (x).

4 4.1

W 1,p -Metric Topology and Characteristic Functions Motivations and Main Properties

0,1 Since distance functions are locally Lipschitzian, they belong to Cloc (RN ) and hence 1,p N to Wloc (R ) for all p ≥ 1. Thus the constructions of section 2.1 can be repeated 1,p with Wloc (D) in place of Cloc (D) to generate new W 1,p -metric topologies on the family Cd (D). One big advantage is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of the closure of the sets and hence the convergence of volumes (cf. Theorem 3.3 (viii)). In the uniform Hausdorff topology, the convergence of volumes and perimeters is usually lost. In general, the measure of a closed set is only upper semicontinuous with respect to that topology. This includes Lebesgue and Hausdorff measures. The next example shows that the Hausdorff metric convergence is not sufficient to get the Lp -convergence of the characteristic functions of the closure of the corresponding sets in the sequence. The volume function is only upper semicontinuous with respect to the Hausdorff topology. The perimeters do not converge either as illustrated by the example of the staircase of Example 6.1 and Figure 5.7 in Chapter 5, where the volumes converge but not the perimeters.

4. W 1,p -Metric Topology and Characteristic Functions

293

Example 4.1. Denote by D = ] − 1, 2[ × ] − 1, 2[ the open unit square in R2 , p ≥ 1 a real number, and for each n ≥ 1 define the sequence of closed sets   2k 2k + 1 def ≤ x1 ≤ , 0 ≤ k < n, 0 ≤ x2 ≤ 1 . An = (x1 , x2 ) ∈ D : 2n 2n

set A4

set A8

set S

Figure 6.3. Vertical stripes of Example 4.1. This defines n vertical stripes of equal width 1/2n each distant of 1/2n (cf. Figure 6.3). Let S = [0, 1] × [0, 1] be the closed unit square. Clearly, for all n ≥ 1, m(An ) = ∀x ∈ S,

1 , 2

dAn (x) ≤

PD (An ) = 2n + 1, 1 , 4n

∇dAn Lp (D) ≥ 2−1/p ,

where m(An ) is the surface and PD (An ) the perimeter of An . Hence dAn → dS

in C(D),

m(S) = 1,

PD (S) = 4.

But m(An ) = m(An ) =

1  1 = m(S) 2

PD (An )  PD (S),

χAn 

⇒ χAn  χS in Lp (D), 1 χS in Lp (D)-weak. 2

Since the characteristic functions do not converge, the sequence {∇dAn } does not converge in Lp (D)N and {dAn } does not converge in W 1,p (D). Theorem 4.1. Let µ be a measure5 and let D ⊂ RN be bounded open such that µ(D) < ∞. (i) The map dA → µ(A) : Cd (D) → R

(4.1)

is upper semicontinuous with respect to the topology of uniform convergence (Hausdorff topology on Cd (D)). 5 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

294

Chapter 6. Metrics via Distance Functions

(ii) The map d
A → µ(int A) : Cdc (D) → R

(4.2)

is lower semicontinuous with respect to the topology of uniform convergence (Hausdorff complementary topology on Cdc (D)). Note that µ can be the Lebesgue measure but also any of the Hausdorff measures. Proof. We prove (ii). The proof of (i) is similar and simpler. It is sufficient to work with open sets Ω = int A = A. Let {Ωn } be a sequence of open subsets of D such that d
Ωn → d
Ω in C(D) for some open Ω ⊂ D. Given ε > 0, there exists N such that for all n > N d
Ωn − d
Ω }C(D) ≤ ε

⇒ Ωn ⊂ (Ω)ε .

By definition of a measurable set µ(Ωn ) = µ(Ωn ∩ D) = µ(D) − µ(D ∩ Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ) ⇒ lim inf µ(Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ). n→∞

But D ∩ (Ω)ε is monotonically decreasing to ∩ε>0 D ∩ (Ω)ε = D ∩ Ω and hence lim inf µ(Ωn ) ≥ µ(D) − lim µ(D ∩ (Ω)ε ) = µ(D) − µ(D ∩ Ω) = µ(Ω) n→∞

ε→0

since D ∩ Ω = Ω. Theorem 4.2. Let D be an open (resp., bounded open) subset of RN . 1,p (D) (resp., W 1,p (D)) on Cd (D) and Cdc (D) (i) The topologies induced by Wloc are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cd (D) is closed in Wloc def

ρD ([A2 ], [A1 ]) =

∞ dA2 − dA1 W 1.p (B(0,n)) 1 n 2 1 + dA2 − dA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ], [A1 ]) = dA2 − dA1 W 1,p (D) ) defines a complete metric structure on F(D). For p, 1 ≤ p < ∞, the map 1,p dA → χA = 1 − |∇dA | : Cd (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and nonempty subsets A1 and A2 of D χA2 − χA1 Lp (K) ≤ ∇dA2 − ∇dA1 Lp (K) ≤ dA2 − dA1 W 1,p (K) .

4. W 1,p -Metric Topology and Characteristic Functions

295

1,p (D) (resp., W01,p (D)) for p, 1 ≤ p < ∞, and (iii) Cdc (D) is closed in Wloc def

ρD (Ω2 , Ω1 ) =

∞ d
Ω2 − d
Ω1 W 1.p (B(0,n)) 1 n 2 1 + d
Ω2 − d
Ω1 W 1.p (B(0,n)) n=1 def

(resp., ρD (Ω2 , Ω1 ) = d
Ω2 − d
Ω1 W 1,p (D) ) defines a complete metric structure on the family G(D) of open subsets of D. For p, 1 ≤ p < ∞, the map 1,p d
Ω → χΩ = |∇d
Ω | : Cdc (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and open subsets Ω1 = RN and Ω2 = RN of D χΩ2 − χΩ1 Lp (K) ≤ ∇d
Ω2 − ∇d
Ω1 Lp (K) ≤ d
Ω2 − d
Ω1 W 1,p (K) . Proof. (i) The proof is similar to the proof of Theorem 2.3 in Chapter 5. It is sufficient to prove it for D bounded open. Since D is bounded it is contained in a sufficiently large ball of radius c. Therefore, dA (x) = inf |x − y| ≤ |x| + |y| ≤ 2c y∈A

since y ∈ A ⊂ D, and by Theorem 2.1 (vii) |∇dA (x)| ≤ 1 a.e. in D. For all p > 1 the injection of W 1,p (D) into W 1,1 (D) is continuous since dA W 1,1 (D) ≤ dA W 1,p (D) m(D)1/q with 1/p + 1/q = 1. Conversely, we have the continuity in the other direction. For p > 1 and any dA and dB in Cd (D),  |dA − dB |p + |∇dA − ∇dB |p dx D  |dA − dB | |dA − dB |p−1 + |∇dA − ∇dB | |∇dA − ∇dB |p−1 dx = D  p−1 p−1 |dA − dB | + |∇dA − ∇dB | dx ≤ max{(2c) , 2 } D

⇒ dA − dB pW 1,p (D) ≤ (2 max{c, 1})p−1 dA − dB W 1,1 (D) . Therefore, for any ε > 0, pick δ = εp /(2 max{c, 1})p−1 and dA − dB W 1,1 (D) ≤ δ

⇒ dA − dB W 1,p (D) ≤ ε.

(ii) It is sufficient to prove it for D bounded open. Let {dAn } be a Cauchy sequence in Cd (D) which converges to some f in W 1,p (D)-strong. By Theorem 2.2 (ii),

296

Chapter 6. Metrics via Distance Functions

Cd (D) is compact in C(D) and there exist a subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(D) and, a fortiori, in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D)-strong, f = dA and dAn converges to dA in W 1,p (D)-strong. Therefore Cd (D) is closed in W 1,p (D). For the Lipschitz continuity, recall that the distance function dA is differentiable almost everywhere in RN for A = ∅. In view of Theorem 3.3 (viii) χA = 1 − |∇dA (x)| almost everywhere in RN . Given two nonempty subsets A1 and A2 of D

⇒ χ A1 ≤ χ A2

|∇dA2 | ≤ |∇dA1 | + |∇dA2 − ∇dA1 |  + |∇dA2 − ∇dA1 | ⇒ |χA1 − χA2 |p dx ≤ dA2 − dA1 pW 1,p (D) D

for 1 ≤ p < ∞ and with the ess-sup norm for p = ∞. (iii) Again it is sufficient to prove the result for D bounded. In that case Ω = ∅ for all open subsets Ω of D. Let {Ωn } be a sequence of open subsets of D such that {d
Ωn } is Cauchy in W 1,p (D). By assumption Ωn ⊂ D, Ωn ⊃ D, ∀n ≥ 1,

d
Ωn = 0 in D

⇒ d
Ωn ∈ W01,p (D),

and the Cauchy sequence converges to some f ∈ W01,p (D). By Theorem 2.4 (ii), Cdc (D) is compact in C(D) and there exist a subsequence, still denoted by {d
Ωn }, and an open set Ω ⊂ D such that d
Ωn → d
Ω in C(D) and hence in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D), f = d
Ω and the Cauchy sequence d
Ωn converges to d
Ω in W01,p (D). The other part of the proof is similar to that of part (ii).

4.2

Weak W 1,p -Topology

We have the following general result. Theorem 4.3. Let D be a bounded open domain in RN . (i) If {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {dAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely if {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cd (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞.6 (iv) Parts (i) to (iii) also apply to Cdc (D). 6 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E
, any closed ball is a compact metrizable space for the weak topology. Since Cd (D) is a bounded subset of the normed reflexive separable Banach space W 1,p (D), 1 ≤ p < ∞, the weak compactness of ´ [1, Vol. II, Chap. XII, Cd (D) coincides with the weak sequential compactness (cf. J. Dieudonne sect. 12.15.9, p. 75]).

4. W 1,p -Metric Topology and Characteristic Functions

297

Proof. (i) For D bounded there exists a constant c > 0 such that for all dA ∈ Cd (D) dA (x) ≤ c and |∇dA (x)| ≤ 1 a.e. in D. If {dAn } weakly converges in W 1,p (D), then {dAn } weakly converges in Lp (D),

{∇dAn } weakly converges in Lp (D)N .

By Lemma 3.1 (iii) in Chapter 5 both sequences weakly converge for all p ≥ 1, and hence {dAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {dAn } converges in C(D), then by Theorem 2.2 (i) there exists dA ∈ Cb (D) such that dAn → dA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N ,     ∇dAn · ϕ dx = − dAn div ϕ dx → − dA div ϕ dx = ∇dA · ϕ dx. D

D

D

D

By density of D(D) in L2 (D), ∇dAn → ∇dA in L2 (D)N -weak and hence dAn → dA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D) there exist a subsequence, still indexed by n, and dA such that dAn → dA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit dA = f . Therefore, all convergent subsequences in C(D) converge to the same limit, so the whole sequence converges in C(D). (iii) Consider an arbitrary sequence {dAn } in Cd (D). From Theorem 2.2 (ii) Cd (D) is compact and there exist a subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cd (D) is compact in W 1,p (D)-weak. Theorem 4.4. Let D be a closed subset of RN and {An }, ∅ = An ⊂ D, be a sequence such that dAn → dA in Cloc (D) for some A, ∅ = A ⊂ D. Then ∀x ∈ RN \A, lim χAn (x) = χA (x) = 0 and ∀x ∈ RN , lim sup χAn (x) ≤ χA (x), n→∞

n→∞

and for any compact subset K of D   χAn dx ≤ χA dx. lim sup n→∞

K

K

Corollary 1. Let D = ∅ be a bounded open subset of RN and {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets of D converging to an open subset Ω, ∅ = Ω ⊂ D, of ¯ Then D in the Hausdorff complementary topology: d
Ωn → d
Ω in C(D). ∀x ∈ Ω, lim χΩn (x) = χΩ (x) = 1, n→∞

and

and

n→∞



 χΩn dx ≥

lim inf n→∞

∀x ∈ RN , lim inf χΩn (x) ≥ χΩ (x),

D

χΩ dx. D

298

Chapter 6. Metrics via Distance Functions

Proof of Theorem 4.4. For all x ∈ / A, dA (x) > 0 and ∃Nx > 0, ∀n ≥ Nx , dA − dAn  ≤ dA (x)/2 ⇒ ∃Nx > 0, ∀n ≥ Nx , dAn (x) ≥ dA (x)/2 > 0 ⇒ ∃Nx > 0, ∀n ≥ Nx ,

x∈ / An and χAn (x) = 0.

So for all x ∈ /A lim χAn (x) = χA (x) = 0.

n→∞

Finally, for all x ∈ A and all n ≥ 1 χAn (x) ≤ 1 = χA (x) and the result follows trivially by taking the limsup of each term. We conclude that lim sup χAn ≤ χA , n→∞

and by using the analogue of Fatou’s lemma for the limsup we get for all compact subsets K of D    lim sup χAn dx ≤ χA dx. lim sup χAn dx ≤ n→∞

K n→∞

K

K

In order to get the Lp -convergence of the characteristic functions of the closure of the sets in the sequence, we need the Lp -convergence of the gradients of the distance functions which are related to the characteristic functions of the closure of the sets (cf. Theorem 3.3 (viii)). We have seen in Example 4.1 that the weak convergence of the characteristic functions is not sufficient to obtain the strong convergence of the sequence {dAn } to dA in W 1,2 (D). However, if we assume that {χAn } is strongly convergent, it converges to the characteristic function χB of some measurable subset B of D. Is this sufficient to conclude that χB = χA ? The answer is negative. The counterexample is provided by Example 6.2 in Chapter 5, where dAn  dD in W 1,2 (D)-weak

and

χAn → χB in L2 (D)-strong

for some B ⊂ D such that m(D) = π >

π ≥ m(B) 3

⇒ χD = χB .

Remark 4.1. In view of part (ii) of Theorem 4.2 an optimization problem with respect to the characteristic functions χΩ of open sets Ω in D for which we have the continuity with respect to χΩ can be transformed into an optimization problem with respect to d
Ω in W 1,1 (D) since χΩ = |∇d
Ω |. This would apply to the transmission problem (4.3)–(4.6) in section 4.1 of Chapter 5.

5. Sets of Bounded and Locally Bounded Curvature

5

299

Sets of Bounded and Locally Bounded Curvature

From Theorem 6.1 in Chapter 5 and Theorem 3.2, it is readily seen that  N ∇dA ∈ BVloc RN

⇐⇒

∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N .

The global properties of ∇dA depend only on its local properties around ∂A. This local property is fairly general and is verified for sets with corners, that is, sets with discontinuities in the orientation of the normal along the boundary. This suggests introducing a new family of sets and motivates the study of their general properties, We shall later see that convex sets have this property and later in Chapter 7 that the larger family of sets with positive reach (cf. section 6) also does. One remarkable property is that such sets are Caccioppoli sets (cf. section 6 in Chapter 5). We give the definitions for an arbitrary set A, but they can be specialized to A and ∂A (associated with the distance functions d
A and d∂A ) that would require a finer terminology such as exterior, interior, or boundary curvature to distinguish them. Definition 5.1. (i) Given a bounded open holdall D ⊂ RN and A, ∅ = A ⊂ D, the set A is said to be of bounded curvature with respect to D if ∇dA ∈ BV(D)N .

(5.1)

The family of sets with bounded curvature will be denoted by  def

BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N . (ii) A set A, A = ∅, in RN is said to be of locally bounded curvature if ∇dA ∈ BVloc (RN )N . The family of sets with locally bounded curvature will be denoted by  def

BCd = dA ∈ Cd (RN ) : ∇dA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, it is sufficient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. A, ∅ = A ⊂ RN , is of locally bounded curvature if and only if ∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Theorem 6.1 and Definition 6.1 in Chapter 5, a function belongs to BVloc (RN ) if and only if for each x ∈ RN it belongs to BV(B(x, ρ)) for some ρ > 0. The equivalence now follows from Theorem 3.2.

300

Chapter 6. Metrics via Distance Functions

Recall that the relaxation of the perimeter of a set was obtained from the norm of the gradient of its characteristic function for Caccioppoli sets. For distance functions χA = 1 − |∇dA | almost everywhere and the gradient of dA also has a jump discontinuity along the boundary ∂A of magnitude at most 1. So it is not too surprising that a set A with bounded curvature is a Caccioppoli set. Theorem 5.2. (i) Let D ⊂ RN be a bounded open holdall and let A, ∅ = A ⊂ D. If ∇dA ∈ BV(D)N , then ∇χA M 1 (D) ≤ 2 D2 dA M 1 (D) ,

χA ∈ BV(D),

and A has finite perimeter with respect to D. (ii) For any subset A of RN , A = ∅, ∇dA ∈ BVloc (RN )N

⇒

χA ∈ BVloc (RN )

and A has locally finite perimeter. Proof. Given ∇dA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇dA in L1 (D)N

and Duk M 1 (D) → D 2 dA M 1 (D)

as k goes to infinity, and since |∇dA (x)| ≤ 1, this sequence can be chosen in such a way that |uk (x)| ≤ 1 for all k ≥ 1. This follows from the use of mollifiers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N    − χA div V dx = (|∇dA |2 − 1)div V dx = |∇dA |2 div V dx. D

D

D

For each uk   |uk |2 div V dx = −2 D

∗

D

 [Duk ]uk · V dx = −2

uk · [Duk ]V dx, D

where ∗ Duk is the transpose of the Jacobian matrix Duk and   |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore,   χ div V dx = |∇dA |2 div V dx ≤ 2D2 dA M 1 V  A C(D) D

D

as k goes to infinity, where D2 dA is the Hessian matrix of second-order partial derivatives of dA . Finally, ∇χA ∈ M 1 (D)N .

5. Sets of Bounded and Locally Bounded Curvature

5.1

301

Examples

It is useful to consider the following three simple illustrative examples (cf. Figure 6.4). By convexity, they are all of locally bounded curvature. Example 5.1 (Half-plane in R2 ). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that (0, 0), x1 < 0, dA (x1 , x2 ) = max{x1 , 0}, ∇dA (x1 , x2 ) = (1, 0), x1 > 0,  ϕ dH1 , ∂12 dA = ∂21 dA = ∂22 dA = 0, ∂11 dA , ϕ = ∂A  ∆dA , ϕ = ϕ dH1 ⇒ ∆dA = H1 . ∂A

Thus ∆dA is the one-dimensional Hausdorff measure of ∂A.

Figure 6.4. ∇dA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly dA (x) = max{0, |x| − R}, 

∇dA (x) = 

2π

∂11 dA , ϕ =

2

R cos (θ)ϕ dθ + 0





2π


A

x/|x|, (0, 0),

|x| > R, |x| < R,

x22 ϕ dx, (x21 + x22 )3/2

x21 ϕ dx, 2 2 3/2 0
A (x1 + x2 )  2π  x2 x1 R cos(θ) sin(θ)ϕ dθ + ϕ dx, ∂12 dA , ϕ = ∂21 dA , ϕ = 2 + x2 )3/2 (x 0
A 1 2   1 ϕ dH1 + ϕ dx, ∆dA , ϕ = 2 2 1/2 ∂A
A (x1 + x2 ) ∂22 dA , ϕ =

R sin2 (θ)ϕ dθ +

302

Chapter 6. Metrics via Distance Functions

where H1 is the one-dimensional Hausdorff measure. ∆dA contains the onedimensional Hausdorff measure of the boundary ∂A plus a term that corresponds to the volume integral of the mean curvature over the level sets of dA in A. Example 5.3 (Unit square in R2 ). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1}. Since A is symmetrical with respect to both axes, it is sufficient to specify dA in the first quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence

  min{x2 − 1, 0}, x ∈ D2 ∩ Q1 , dA (x) = |x − c1 |, x ∈ D3 ∩ Q1 ,   min{x1 − 1, 0}, x ∈ D1 ∩ Q1 ,

∇dA (x) =

 (0, 1),    x−c1 

x ∈ D2 ∩ Q1 and x2 > 1, |x−c1 | , x ∈ D3 ∩ Q1 ,

 (1, 0),    (0, 0),

x ∈ D1 ∩ Q1 and x1 > 1, x ∈ Q1 , x1 < 1 and x2 < 1,

 (x2 − ci,2 )2 ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D1 i=1 D3 ∩Qi  4  (x1 − ci,1 )2 ∂22 dA , ϕ ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D2 i=1 D3 ∩Qi 4  (x2 − ci,2 )(x1 − ci,1 ) ∂12 dA , ϕ = ∂21 dA , ϕ = ϕ dx, |x − ci |2 i=1 D3 ∩Qi  4  1 ϕ dx + ∆dA , ϕ = ϕ dH1 . |x − ci | ∂A i=1 D3 ∩Qi ∂11 dA , ϕ =

4 

Notice that the structure of the Laplacian is similar to that observed in the previous examples. The norm D2 dA M 1 (Uh (A)) is decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A.

6. Reach and Federer’s Sets of Positive Reach

303

Example 5.4. Let N = 2. For the finite square and the ball of finite radius, lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorff measure.

6

Reach and Federer’s Sets of Positive Reach

The material in this section is mostly taken from the pioneering 1959 work of H. Federer [3, sect. 4], where he introduced the notion of reach and the sets of positive reach.

6.1

Definitions and Main Properties

By definition, A∩Sk (A) = ∅. For a ∈ int A, there exists r > 0 such that Br (a) ⊂ A. Thus Sk (A) ∩ Br (a) = ∅ and sup {r > 0 : Sk (A) ∩ Br (a) = ∅} > 0.

(6.1)

/ Sk (A) or a ∈ Sk (A). If a ∈ / Sk (A), there exists r > 0 If a ∈ ∂A, then either a ∈ such that Br (a) ∩ Sk (A) = ∅. Thus Br (a) ∩ Sk (A) = ∅ and (6.1) is verified. If a ∈ Sk (A), then for all r > 0, Br (a) ∩ Sk (A) = ∅. This leads to the following definition. Definition 6.1. Given A, ∅ = A ⊂ RN , and a point a ∈ A def

reach (A, a) =

0, if a ∈ ∂A ∩ Sk (A), sup {r > 0 : Sk (A) ∩ Br (a) = ∅} , otherwise, def

reach (A) = inf {reach (A, a) : a ∈ A} . The set A is said to have positive reach if reach (A) > 0. Remark 6.1. (1) Since Sk (A) = Sk (A), reach (A) = reach (A). (2) For any set A, Sk (A) ⊂ Sk (∂A) and ∀a ∈ ∂A,

0 ≤ reach (∂A, a) ≤ reach (A, a) ≤ +∞.

(3) For all a ∈ / ∂A ∩ Sk (A), reach (A, a) > 0. Lemma 6.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅.

304

Chapter 6. Metrics via Distance Functions

(iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. Proof. (i) ⇒ (ii) If there exists a ∈ A such that reach (A, a) = +∞, then RN = ∪r>0 Br (a) ⊂ RN \Sk (A) and Sk (A) = ∅. (ii) ⇒ (iii) If Sk (A) = ∅, for all a ∈ A reach (A, a) = +∞. (iii) ⇒ (iv) If for all a ∈ A reach (A, a) = +∞, then reach (A) = inf a∈A reach (A, a) = +∞. (iv) ⇒ (i) is obvious. Remark 6.2. We shall see later that all nonempty convex sets A have positive reach and that reach (A) = +∞. Bounded domains and submanifolds of class C 2 also have positive reach. Theorem 6.1. Let A, ∅ = A ⊂ RN , be such that Sk (A) = ∅. (i) The function a → reach (A, a) : A → R is continuous. (ii) If reach (A) > 0, then for all h, 0 < h < reach (A), Uh (A) ∩ Sk (A) = ∅. Conversely, if there exists h > 0 such that Uh (A) ∩Sk (A) = ∅, reach (A) ≥ h, and A has positive reach. (iii) If reach (A) > 0, then A ∩ Sk (A) = ∅. If A ∩ Sk (A) = ∅ and A is bounded, then reach (A) > 0. Remark 6.3. When A is unbounded and A ∩ Sk (A) = ∅, we can possibly have reach (A) = 0 even if for all a ∈ A, reach (A, a) > 0 as readily seen from the following example: def

A = {(x1 , x2 ) : |x2 | ≥ ex1 } , where Sk (A) = {(x1 , 0) : x1 ∈ R} . Proof. (i) We prove that the function is both lower and upper semicontinuous. For h, reach (A, a) > h, there exists r, h < r < reach (A, a), such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (r − h)/2 > 0. Then r − δ = (r + h)/2 > h and for all a ∈ Bδ (a)∩A, Br−δ (a ) ⊂ Br (a). Thus Br−δ (a )∩Sk (A) ⊂ Br (a)∩Sk (A) = ∅, reach (A, a ) ≥ r − δ > h, and reach (A, a) is lower semicontinuous. Similarly, for k, reach (A, a) < k, there exists r, reach (A, a) > r > k, such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (k − r)/2 > 0. Then r + δ = (r + k)/2 < k and for all a ∈ Bδ (a) ∩ A, Br (a) ⊂ Br+δ (a ). Thus ∅ = Br (a) ∩ Sk (A) ⊂ Br+δ (a ) ∩ Sk (A), reach (A, a ) ≤ r + δ < k, and reach (A, a) is upper semicontinuous. (ii) By definition of reach (A) as an infimum. Conversely, for all 0 < r < h and all x ∈ A, Br (x) ∩ Sk (x) = ∅, and reach (A, x) ≥ r. Since this is true for all 0 < r < h, reach (A, x) ≥ h and reach (A) = inf x∈A reach (A, x) ≥ h > 0. (iii) By separation of a closed set and a compact set. Theorem 6.2. Let ∅ = A ⊂ RN . (i) Associate with a ∈ A the sets  def

P (a) = v ∈ RN : ΠA (a + v) = {a} ,

 def

Q(a) = v ∈ RN : dA (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta A)∗ .

6. Reach and Federer’s Sets of Positive Reach

305

(ii) Given a ∈ A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : ΠA (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), ΠA (a + tv) = {a} and dA (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (A). (iii) If there exists h such that 0 < h ≤ reach (A), then for all x ∈ Uh (A)\A and 0 < t < h, ΠA (x(t)) = {pA (x)} and dA (x(t)) = t, where def

x(t) = pA (x) + t ∇dA (x) = pA (x) + t

x − pA (x) . dA (x)

(6.2)

(iv) Given b ∈ A, x ∈ RN \Sk (A), a = pA (x), such that reach (A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (A, a)

(6.3)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (A) such that dA (x) ≤ r and reach (A, pA (x)) ≥ q, y ∈ RN \Sk (A) such that dA (y) ≤ r and reach (A, pA (y)) ≥ q, then |pA (y) − pA (x)| ≤

q |y − x|. q−r

(6.4)

(vi) If 0 < reach (A) < +∞, then for all h, 0 < h < reach (A), reach (A) |y − x|, ∀x, y ∈ Ah = {x ∈ RN : dA (x) ≤ h}, reach (A) − h   2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + reach (A) |y − x|, ∀x, y ∈ Ah ; (6.5) reach (A) − h

 for all 0 < s < h < reach (A) and all x, y ∈ As,h = x ∈ RN : s ≤ dA (x) ≤ h ,   reach (A) 1 2+ |y − x| . |∇dA (y) − ∇dA (x)| ≤ s reach (A) − h

|pA (y) − pA (x)| ≤

If reach (A) = +∞, ∀x, y ∈ RN , |pA (y) − pA (x)| ≤ |y − x|, 2 ∇d (y) − ∇d2 (x) ≤ 4 |y − x|; A A for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ dA (x)}, |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(6.6)

306

Chapter 6. Metrics via Distance Functions

(vii) If reach (A) > 0, then, for all 0 < r < reach (A), Ur (A) is a set of class C 1,1 , Ur (A) = Ar , ∂Ur (A) = {x ∈ RN : dA (x) = r}, and Ar = Ur (A) = {x ∈ RN : dA (x) ≥ r}. (viii) For all a ∈ A,

 dA (a + th) =0 . h ∈ R : lim inf t0 t

 Ta A =

N

In particular, if a ∈ int A, Ta A = RN . Remark 6.4. For part (viii), more properties of Ta A and the normal cone (−Ta A)∗ (cf. Definitions 2.3 and 2.4 in Chapter 2) can be found in H. Federer [3, sect. 4]. Proof. (i) By definition for all a ∈ A, a ∈ P (A) a ∈ Q(A)

⇐⇒ ⇐⇒

∀b ∈ A\{a}, ∀b ∈ A\{a},

|a + v − b| > |v|, |a + v − b| ≥ |v|

and P (a) ⊂ Q(a). To prove the convexity, we need the identity |a + v − b|2 − |v|2 = (a − b) · (2v + a − b). The convexity of P (a) and Q(a) now follow from the following identities: for all λ ∈ [0, 1] and v, w ∈ RN , |a + (λv + (1 − λ)w) − b|2 − |(λv + (1 − λ)w|2 = (a − b) · (2((λv + (1 − λ)w) + a − b) = λ (a − b) · (2v + a − b) + (1 − λ) (a − b) · (2w + a − b)     = λ |a + v − b|2 − |v|2 + (1 − λ) |a + w − b|2 − |v|2 . Finally, for h ∈ Ta A, there exist sequences {bn } ⊂ A and {εn > 0}, εn  0, such that (bn − a)/εn → h and bn → a. For all v ∈ Q(a), 2

bn − a |bn − a|2 ·v ≤ |bn − a| ≤ 0 εn εn

⇒ h·v ≤0

⇒ h ∈ [−Ta A]∗ .

def

(ii) Let S = {t > 0 : ΠA (a + tv) = {a}}. Since, by assumption, r(a, v) = sup S > 0, then S = ∅ and v = 0. For each t ∈ S, ΠA (a + tv) = {a} and dA (a + tv) = t |v|. In particular for all 0 < s < t, dA (a + sv) ≤ s |v|. For any p ∈ ΠA (a + sv) ⊂ A, dA (a + sv) = |a + sv − p| ≤ s |v| and dA (a + tv) ≤ |a + tv − p| = |a + sv − p + (t − s)v| ≤ |a + sv − p| + (t − s)|v| ≤ s |v| + (t − s)v| = t |v| = dA (a + tv). As a result p ∈ ΠA (a + tv) = {a}, p = a, and necessarily ΠA (a + sv) = {a} and dA (a + sv) = s |v|. This proves that for all t ∈ S, {s : 0 < s ≤ t} ⊂ S and hence for all 0 ≤ t < r(a, v).

6. Reach and Federer’s Sets of Positive Reach

307

Assume that when r = r(a, v) < ∞, a + rv ∈ / Sk (A). Then there exists δ > 0 such that Bδ (a+rv)∩(Sk (A)∪A) = ∅. By continuity of the projection, ΠA (a+rv) = = r |v|> 0 so that a+rv ∈ / A. By Theorem 3.3 (vi), the function {a} and dA (a+rv)   ∇dA : RN \ Sk (A) ∪ ∂A → R is continuous and for all x ∈ RN \ Sk (A) ∪ A , ∇dA (x) = (x − pA (x))/|x − pA (x)|. For 0 < t < r, pA (a + tv) = a, dA (a + tv) = t|v|, and ∇dA (a + tv) = v/|v|. By continuity, ∇dA (a + rv) = v/|v|. Since ∇dA is continuous in Bδ (a+rv), the Caratheodory conditions are verified and there exists ε, 0 < ε < δ, such that the differential equation x (t) = ∇dA (x(t)),

x(0) = a + rv = a + r |v|

v |v|

has a solution in (−ε, ε). This yields d dA (x(t)) = ∇dA (x(t)) · x (t) = ∇dA (x(t)) · ∇dA (x(t)) = 1 dt ⇒ dA (x(t)) = r|v| + t in (−ε, ε). For −ε < p < q < ε, the length of the curve x(t) is given by  q  q d dA (x(t)) dt = dA (x(q)) − dA (x(p)) ≤ |x(q) − x(p)| |x (t)| dt = p p dt so that the length of the curve x(t) between the points x(p) and x(q) is less than or equal to the distance |x(q) − x(p)| between x(p) and x(q). Therefore x(t) is a line between x(p) and x(q): x(q) − x(p) , q−p

x(q) − x(p) q−p v x(q) − x(p) ⇒ = x (0) = ∇dA (x(0)) = ∇dA (a + rv) = q−p |v| v v = a + rv + t ⇒ ∀t, 0 < t < ε, x(t) = x(0) + (t − 0) |v| |v| ⇒ |x(t) − a| = r|v| + t = dA (x(t)) ⇒ pA (x(t)) = a and ΠA (a + r|v| + t) = {a} x(t) = x(p) + (t − p)

x (t) =

and we get the contradiction sup S ≥ r + t/|v| > r = sup S. Therefore, a + rv ∈ / Sk (A). (iii) If there exists 0 < h ≤ reach (A), then for all a ∈ A and all 0 < s < h, Bs (a) ∩ Sk (A) = ∅. For x ∈ Uh (A)\A, 0 < dA (x) < h and for all p ∈ ΠA (x) (dA (x) + h)/2 < h ≤ reach (A)

⇒ B(dA (x)+h)/2 (p) ∩ Sk (A) = ∅. def

Since x ∈ B(dA (x)+h)/2 (p), ΠA (x) is a singleton and p = pA (x). From part (ii), S = def

{t > 0 : ΠA (pA (x) + t ∇dA (x)) = {pA (x)}} = ∅ and rA = r(pA (x), ∇dA (x)) ≥ dA (x) > 0. If rA = +∞, then for all t ≥ 0 we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. If rA < +∞, then x(rA ) ∈ Sk (A) and for all ρ > 0, BrA +ρ (pA (x)) ∩ Sk (A) = ∅. This implies that, for all ρ > 0, reach (pA (x), A) ≤ rA + ρ and

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reach (A) ≤ reach (pA (x), A) ≤ rA . Thus, for all 0 < t < h, t < h ≤ reach (A) ≤ rA and we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. (iv) If b = a, the inequality is verified. For b = a, let def

v =

x−a , |x − a|

def

S = {t > 0 : ΠA (a + tv) = {a}} .

By assumption pA (x) = a and x = a + |x − a| v

⇒ |x − a| ∈ S,

sup S ≥ |x − a| > 0,

and, from part (ii), a + sup S v ∈ Sk (A). If reach (A, a) > sup S, then for ρ, reach (A, a) > ρ > sup S, Bρ (a) ∩ Sk (A) = ∅

⇒ Bsup S (a) ∩ Sk (A) = ∅

and this contradicts the fact that a + sup S v ∈ Sk (A). Therefore, reach (A, a) ≤ sup S. Given t, 0 < t < reach (A, a), define xt = a + tv. Then t ∈ S and t < sup S. Thus ΠA (xt ) = {a}, pA (xt ) = a, and dA (xt ) = |xt − a| = t. For b ∈ A, |xt − b| ≥ |xt − a| = t > 0 and 2 a + t x − a − b ≥ t2 |x − a|

⇒ (a − b) · (x − a) ≥ −|a − b|2

|x − a| . 2t

To conclude, choose a sequence {tn }, 0 < tn < reach (A, a), such that tn → reach (A, a) in the above inequality. (v) Apply the inequality of part (iii) twice to the triplets (a, x, b) and (b, y, a): |x − a| r ≥ −|a − b|2 , 2 reach (A, a) 2q |y − b| r ≥ −|a − b|2 . (b − a) · (y − b) ≥ −|a − b|2 2 reach (A, b) 2q

(a − b) · (x − a) ≥ −|a − b|2

By adding the two inequalities r (x − a + b − y) · (a − b) ≥ − |a − b|2 q

 ⇒ (x − y) · (a − b) ≥

⇒ |pA (x) − pA (y)| = |a − b| ≤

r 1− q

 |a − b|2

q |x − y|. q−r

(vi) Given 0 < h < r < reach (A), for all a ∈ A, 0 < h < r < reach (A) ≤ reach (A, a)

and Br (a) ∩ Sk (A) = ∅.

Therefore Ah ⊂ Ur (A) = ∪a∈A Br (a) ⊂ RN \Sk (A) and from (iv) ∀x, y ∈ Ah , ∀r, h < r < reach (A),

|pA (x) − pA (y)| ≤

r |x − y|. r−h

6. Reach and Federer’s Sets of Positive Reach

309

The function r → r/(r − h) : (h, ∞) → R is strictly decreasing to 1. If reach (A) = +∞, then for all a ∈ A, reach (A, a) = +∞ and by letting r go to infinity ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤ |x − y|.

If reach (A) is finite, then for all a ∈ A, reach (A, a) ≥ reach (A) > r and by letting r go to reach (A) ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤

reach (A) |x − y|. reach (A) − h

The inequalities for ∇d2A follow from identity (3.13) in Theorem 3.3 (v). Finally, the Lipschitz continuity of ∇dA follows from identity (3.15) in Theorem 3.3 (vi). For all pairs 0 < s < h < reach (A) and all x, y ∈ {x ∈ RN : s ≤ dA (x) ≤ h} y − pA (y) x − pA (x) |∇dA (y) − ∇dA (x)| = − dA (y) dA (x) |(y − pA (y)) dA (x) − (x − pA (x)) dA (y)| = dA (x) dA (y) |(y − pA (y)) (dA (x) − dA (y))| |[(y − pA (y)) − (x − pA (x))] dA (y)| + ≤ dA (x) dA (y) dA (x) dA (y) 1 ≤ (|dA (x) − dA (y)| + |y − x| + |pA (y) − pA (x)|) dA (x)   1 reach (A) ≤ 2+ |y − x| . s reach (A) − h When reach (A) = +∞, for all s > 0 and all x, y ∈ {x ∈ RN : s ≤ dA (x)} |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(vii) For 0 < r < reach (A), there exists h such that 0 < r < h < reach (A). N The open set Ur (A) is defined via the r-level set d−1 A {r} = {x ∈ R : dA (x) = r} of −1 the function dA . From part (vi), for each x ∈ dA {r}, there exists a neighborhood V (x) = Bρ (x), ρ = min{r, h − r}/2, of x in Uh (A)\A such that dA ∈ C 1,1 (V (x)). Therefore, ∇dA exists and is not zero on the level set ∂Ar = d−1 A {r} since |∇dA | = 1. From Theorem 4.2 in Chapter 2, the set Ur (A) is of class C 1,1 , Ur (A) = Ar , and ∂Ur (A) = d−1 A {r}, and Ar = Ur (A). (viii) Given h ∈ Ta A (cf. Definition 2.3 in Chapter 2), there exist sequences {an } ⊂ A and {εn }, εn  0, such that (an − a)/εn → h. For each n, dA (a + th) dA (a + εn h) |a + εn h − an | an − a → 0 ⇒ lim inf = 0. 0≤ ≤ = h − t0 εn εn εn t

310

Chapter 6. Metrics via Distance Functions

Conversely, there exists {tn > 0}, tn  0, such that dA (a + tn h)/tn → 0. This means that there exists a sequence of projections {an } ⊂ A such that h − an − a = |a + tn h − an | = dA (a + tn h) → 0 tn tn tn and h ∈ Ta A. The next theorem summarizes several characterizations of sets of positive reach. Note that condition (ii) is a global condition on the smoothness of d2A in the tubular neighborhood Uh (A). Theorem 6.3. Given ∅ = A ⊂ RN , the following conditions are equivalent: (i) A has positive reach, that is, reach (A) > 0. (ii) There exists h > 0 such that d2A ∈ C 1,1 (Uh (A)). (iii) There exists h > 0 such that for all s, 0 < s < h, dA ∈ C 1,1 (Us,h (A)). (iv) There exists h > 0 such that dA ∈ C 1 (Uh (A)\A). (v) There exists h > 0 such that for all x ∈ Uh (A), ΠA (x) is a singleton. Proof. (i) ⇒ (ii) For all h, 0 < h < reach (A), and all a ∈ A, Bh (a) ∩ Sk (A) = ∅. Hence for all x ∈ Uh (A) ΠA (x) is a singleton and ∇d2A (x) exists. The functions d2A and ∇d2A are bounded and uniformly continuous in Uh (A): ∀x ∈ Uh (A),

d2A (x) ≤ h2 ,

∇d2A (x) = |2(x − pA (x)| = 2 dA (x) ≤ 2h,

|d2A (y) − d2A (x)| = (|dA (y) − dA (x)) |dA (y) − dA (x)| ≤ 2h |y − x|,   2 reach (A) 2 ∀x, y ∈ Uh (A), ∇dA (y) − ∇dA (x) ≤ 2 1 + |y − x| reach (A) − h

∀x, y ∈ Uh (A),

from inequality (6.5) of Theorem 6.2 (vi). By definition, d2A ∈ C 1,1 (Uh (A)). (ii) ⇒ (iii) From the proof of Theorem 6.2 (vi). (iii) ⇒ (iv) By assumption, for all 0 < s < h, dA ∈ C 1 (Us,h (A)). Since Uh (A)\A = ∪0<s 0 and ∇d2A (x) = 2 dA (x)∇dA (x) exists; from Theorem 3.3 (v), for all x ∈ A, ∇d2A (x) = 0. This means that ΠA (x) is a singleton in Uh (A) = Uh (A)\A ∪ A. (v) ⇒ (i) By assumption, for all a ∈ A and all x ∈ Bh (a), Bh (a) ∩ Sk (A) = ∅. This implies that for all a ∈ A, reach (A, a) ≥ h, which finally implies that reach (A) ≥ h > 0.

6.2

C k -Submanifolds

For arbitrary closed submanifolds M of RN of codimension greater than or equal to one, ∇dM has a discontinuity along M and generally does not exist. In that case the smoothness of M is related to the existence and the smoothness of ∇d2M in a neighborhood of M that is, a fortiori, locally of positive reach. The following analysis of the smoothness of M was given by J.-B. Poly and G. Raby [1] in 1984.

6. Reach and Federer’s Sets of Positive Reach

311

Theorem 6.4. Let A be a closed nonempty subset of RN and k ≥ 2 be an integer (k = ∞ and ω included7 ). Then sing k A = A ∩ sing k d2A , where sing k d2A = RN \reg k d2A , sing k A = RN \reg k A, and  def

reg k d2A = x ∈ RN : d2A is C k in a neighborhood of x ,  def

reg k A = x ∈ A : A is a C k -submanifold of RN in a neighborhood of x . It is more interesting to rephrase their theorem in terms of regularities than singularities. Theorem 6.5. Let ∅ = A ⊂ RN and let k ≥ 2 be an integer (k = ∞ and ω included), and x ∈ A.8 (i) d2A is C k in a neighborhood of a point x ∈ A if and only if A is a C k submanifold in a neighborhood of x. (ii) Under conditions (i), the dimension of A in x is equal to the rank of DpA (x) (dimension of the subspace DpA (RN )) and DpA (x) is the orthogonal projector onto the tangent space to A in x. (iii) If ∂A = ∅, then A = RN . If ∂A = ∅ and d2A is C 2 in a neighborhood of x ∈ A, then there exists an open neighborhood V (x) of x such that (a) if x ∈ int A, then dA = dRN = 0 in V (x); (b) if x ∈ ∂A, then dA = d∂A in V (x), int A ∩ V (x) = ∅, d
A = 0 in V (x), A is a C 2 -submanifold of dimension less than or equal to N − 1 in V (x), and m(∂A ∩ V (x)) = 0. Proof. (i) This follows from Theorem 6.4. (ii) For y ∈ V (x) 1 ∇fA (y) = y − ∇d2A (y) = pA (y),  def 1  2 2 2 |y| − dA (y) fA (y) = ⇒ 1 2 2 D fA (y) = I − D2 d2A (y) = DpA (y). 2 Since pA is a projection, pA (pA (y)) = pA (y) implies that DpA ◦ pA DpA = DpA and (DpA )2 = DpA on A. Therefore, if R(x) denotes the image of DpA (x), DpA (x) is an orthogonal projector onto R(x) and I − DpA (x) is an orthogonal projector onto R(x)⊥ and ∀v ∈ RN , 7ω

v = DpA (x)v + [I − DpA (x)]v ∈ R(x) ⊕ R(x)⊥ .

indicates the analytical case. that A can have several connected components with the same smoothness k but different dimension. When A = RN , dA is identically zero. Another set of results for H¨ olderian sets was obtained by G. M. Lieberman [1] (see also V. I. Burenkov [1]) by introducing a regularized distance function. 8 Note

312

Chapter 6. Metrics via Distance Functions

Consider the C 1 -mapping Tx (y) = (DpA (x)(pA (y) − x), [I − DpA (x)](y − pA (y))) : W (x) → R(x) ⊕ R(x)⊥ . def

Tx is a C 1 -diffeomorphism from a neighborhood U (x) ⊂ V (x) of x onto a neighborhood W (0) of 0 since DTx is continuous in V (x) and def

DTx (y) = (DpA (x)DpA (y), [I − DpA (x)](I − DpA (y))), def

DTx (x) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x))) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x)) = (DpA (x), I − DpA (x)) = I. By construction Tx (U (x) ∩ A) = V (0) ∩ R(x), ⊥

V (0) ∩ R(x) = {(0, [I − DpA (x)](y − pA (y))) : ∀y ∈ U (x)} , and U (x) ∩ A is a C 1 -submanifold in RN of codimension r, where N − r is the rank of DpA (x). (iii) If ∂A = ∅, A ⊂ A and A is both open and closed. Since A = ∅, then A = RN . Let ∂A = ∅. If x ∈ ∂A, then A = ∅. We claim that int A = ∅. If this is not true, there exists a neighborhood V (x) of x where d2A is C 2 and int A ∩ V (x) = ∅ where dA = 0 and D2 d2A = 0. This implies that DpA = I in int A ∩ V (x). In A ∩ V (x), D2 d2A is continuous and dA > 0. Therefore, ∇dA and D2 dA exist, |∇dA | = 1, and 1 2 2 D dA = ∇dA ∗∇dA + dA D 2 dA 2 ⇒ DpA = I − ∇dA ∗∇dA − dA D 2 dA and DpA ∇dA = 0 since |∇dA |2 = 1 implies that D2 dA ∇dA = 0. As a result the rank of DpA is less than or equal to N − 1 in A ∩ V (x) and equal to N in int A ∩ V (x). This contradicts the continuity of D2 d2A and DpA in x. Therefore, A∩V (x) = ∂A∩V (x), and d∂A = dA in V (x) and, by the previous argument in A ∩ V (x), A is a C 2 submanifold of dimension less than or equal to N − 1 in a neighborhood of x. At this juncture, several interesting remarks can be made. (a) The equivalence in (i) fails in the direction (⇒) for d2A ∈ C 1,1 as seen from Example 6.1 of J.-B. Poly and G. Raby [1]. (b) The equivalence in (i) also fails in the direction (⇐) for a C 1,
submanifold, 0 ≤  < 1, as seen from Example 6.2 of S. G. Krantz and H. R. Parks [1]. (c) The square d2A is really pertinent for C k -submanifolds of codimension greater than or equal to one and not for sets of class C k . Indeed, when ∂A = ∅ and d2A is C k , k ≥ 2, in a neighborhood of each point of A, then A is necessarily a C k -submanifold of codimension greater than or equal to one. In particular, A = RN , A = ∂A, and m(A) = 0.

6. Reach and Federer’s Sets of Positive Reach

313

Example 6.1. Consider the half-space in RN  def

A = x ∈ RN : x · eN ≤ 0 , dA (x) =

0, x · eN ,

x · eN ≤ 0, x · eN > 0,

∇d2A (x) =

x · eN ≤ 0, x · eN > 0,

0, 2 (x · eN ) eN ,

and d2A is locally C 1,1 . For any point x ∈ ∂A = {x ∈ RN : x · eN = 0}, A ∩ V (x) is not even a manifold in any neighborhood V (x) of x. The equivalence in (i) also fails in the direction (⇐) for C 1,
, 0 ≤  < 1, as seen from Example 6.2, that shows that for a set of class C 1,
, 0 ≤  < 1, we do not get bA ∈ C 1,λ in a neighborhood of the boundary ∂A. Example 6.2. Consider the two-dimensional domain Ω defined as the epigraph of the function f : def

Ω = {(x, z) : f (x) > z, ∀x ∈ R} ,

def

1

f (x) = |x|2− n

for some arbitrary integer n ≥ 1. Ω is a set of class C 1,1−1/n and ∂Ω is a C 1,1−1/n submanifold of dimension 1. In view of the presence of the absolute value in (0, 0), it is the point where the smoothness of ∂Ω is minimum. We claim that Sk (∂Ω) = Sk (Ω) = {(0, y) : y > 0}

and ∂Ω ∩ Sk (∂Ω) = (0, 0) = ∅.

Because the skeleton is a line touching ∂Ω in (0, 0), d2∂Ω and d2Ω cannot be C 1 in any neighborhood of (0, 0). Indeed, for each (0, y)

 d2∂Ω (0, y) = inf x2 + |f (x) − y|2 . x∈R

The point (0, y), y > 0, belongs to Sk (∂Ω) since there exist two different points x ˆ that minimize the function def

F (x, y) = x2 + +|f (x) − y|2 .

(6.7)

Since f is symmetric with respect to the y-axis it is sufficient to show that there exists a strictly positive minimizer x ˆ > 0. A locally minimizing point x ˆ ≥ 0 must satisfy the conditions    1 1 1  2− n x ˆ1− n = 0, x, y) = 2ˆ x + 2 |ˆ x| −y 2− (6.8) Fx (ˆ n       2 1 2 1 1  1 1 F (ˆ 3− x ˆ2− n − 2 − 1− yx ˆ− n ≥ 0. (6.9) x, y) = 1 + 2 − 2 x n n n n Equation (6.8) can be rewritten as . 1   / 1 1 2ˆ x1− n x ˆn + x ˆ2− n − y (2 − 1/n) = 0,

314

Chapter 6. Metrics via Distance Functions

and x ˆ = 0 is a solution. The second factor can be written as an equation in the def 1/n new variable X = x ˆ and def

g(X) =

n X + X 2n−1 − y = 0. 2n − 1

ˆ since It has exactly one solution X ∀X,

n dg (X) = + (2n − 1)X 2n−2 > 0, dX 2n − 1

ˆ > 0 for g(0) = −y, and g(X) goes to infinity as X goes to infinity. In particular, X n ˆ ˆ y > 0 and X = 0 for y = 0. For y > 0 and x ˆ=X 1  x, y) = 0 F (ˆ 2 x and

1

⇒ y=x ˆ2− n +

1 n x ˆn 2n − 1

(6.10)

   2 1  2 1 F (ˆ 3− x ˆ2− n x, y) =1 + 2 − 2 x n n     2 1 1 n 2− n 1− x ˆ − 2− + n n 2n − 1   2   2 2 2n − 1 2n − 1 1− 1 n−1 1 + x ˆ n > 0. =1 − x ˆ2− n = + n n n n

Therefore, x ˆ is a local minimum. To complete the proof for y > 0 we must compare ˆ = 0 with F (0, y) = y 2 for the solution corresponding to x 1

F (ˆ x, y) = x ˆ2 + |ˆ x2− n − y|2 for the solution x ˆ > 0. Again using identity (6.10) 2 2 n F (ˆ x, y) = x ˆ + x ˆn, 2n − 1 2  2  n 2 2 2− 1 1 1 n n x ˆ n = x x ˆ2 ˆ2− n + x ˆn + 2 F (0, y) = x ˆ n+ 2n − 1 2n − 1 2n − 1   2 2   1 1 2n 1 −1 x ˆ2 + x x ˆ2 + x ⇒ F (0, y) − F (ˆ x, y) = ˆ2− n = ˆ2− n > 0. 2n − 1 2n − 1 

2

This proves that for y > 0, x ˆ > 0 is the minimizing positive solution. Therefore, for all n ≥ 1, f (x) = |x|2−1/n yields a domain Ω for which the strictly positive part of the y-axis is the skeleton. It is interesting to note that, for each point x, the points (x, f (x)) ∈ ∂Ω are the projections of the point   1 1 n (0, y) = 0, |x|2− n + |x| n 2n − 1

6. Reach and Federer’s Sets of Positive Reach

315

and that the square of the distance function is equal to  2 n 2 xn , b2Ω (0, y) = x2 + 2n − 1 1 ∇bΩ (x, f (x)) = ' (f  (x), −1) 1 + f  (x)2 1 1 −n = 2n−1 2 2− 2 ((2 − 1/n) x |x| , −1). 1+ |x| n n Finally for all n ≥ 1, f ∈ / C 1,1 . For n = 1, f ∈ C 0,1 , and for n > 1, f ∈ C 1,1−1/n . This gives an example in dimension 2 of a domain Ω of class C 1,λ , 0 ≤ λ < 1, with skeleton converging to the boundary.

6.3

A Compact Family of Sets with Uniform Positive Reach

The following family of sets of positive reach and the associated compactness theorem were first given in H. Federer [3, Thm. 4.13, Rem. 4.14]. Theorem 6.6. Given an open (resp., bounded open) holdall D in RN and h > 0, the family  def

Cd,h (D) = dA : ∅ = A ⊂ D, reach (A) ≥ h

(6.11)

is closed in Cloc (D) (resp., compact in C(D)). Proof. It is sufficient to prove the theorem for a bounded open D. Given any C(D)Cauchy sequence {dAn } ⊂ Cd,h (D), there exists dA ∈ Cd (D) such that dAn → dA in C(D). Choose W open such that W ⊂ Uh (A). Then supx∈W dA (x) < h. Choose r, supx∈W dA (x) < r < h. Then W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A). Choose N such that ∀n > N,

sup dAn (x) < r < h.

x∈W

Then for all n > N , W ⊂ Ur (An ). From Theorem 6.2 (v), for all n > N and ∀x, y ∈ (An )r ,

 2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + n n

h h−r

 |y − x|.

(6.12)

But W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A), d2An → d2A in C(D), and ∀n > N,

d2An ∈ C 1,1 (W ) ⊂ C 1 (W ).

From Theorem 2.5 in Chapter 2, the embedding C 1,1 (W ) ⊂ C 1 (W ) is compact. Therefore, for all open subsets such that W ⊂ Uh (A), d2An → d2A ∈ C 1 (W ), d2A ∈ C 1 (Ur (A)), and reach (A) ≥ r. Since this is true for all r < h, we finally get reach (A) ≥ h and dA ∈ Cd,h (D).

316

Chapter 6. Metrics via Distance Functions The compactness of Theorem 6.7 can also be expressed as follows.

Theorem 6.7. Let D be a fixed bounded open subset of RN . Let {An }, An = ∅, be a sequence of subsets of D. Assume that there exists h > 0 such that ∀n,

d2An ∈ C 1,1 (Uh (An )).

(6.13)

Then there exist a subsequence {Ank } and A ⊂ D, A = ∅, such that reach (A) ≥ h and for all r, 0 < r < h, d2A ∈ C 1,1 (Ur (A)), d2An → d2A in C 1 (Ur (A)) k

and

dAnk → dA in C(D).

(6.14)

This condition in an h-tubular neighborhood is generic of other such conditions that will be expressed in different function spaces.

7

Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

The W 1,p -topology turns out to be an appropriate setting for the approximation of the closure of a set by its dilated sets as defined in Definition 3.2. Theorem 7.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A), dUr (A) = dAr (x) = ∀x ∈ RN ,

dA (x) − r, 0,

if dA (x) ≥ r, if dA (x) < r,

0 ≤ dA (x) − dAr (x) ≤ r,

dUr (A) = dAr (x) → dA (x) uniformly in R

N

dUr (A) = dAr → dA in

1,p Wloc (RN )

and

(7.1) (7.2) (7.3)

as r → 0,

χAr → χA in Lploc (RN )

(7.4)

for all p, 1 ≤ p < ∞. Proof. The first line of properties follows from Theorem 3.1. For r > 0, A ⊂ Ur (A) ⊂ Ar and dA (x) ≥ dAr (x) for all x ∈ RN . For x ∈ RN such that dA (x) < r, dAr (x) = 0 and 0 ≤ dA (x) − dAr (x) < r. For x ∈ RN such that dA (x) ≥ r and all a ∈ A and ar ∈ Ar , |x − a| ≤ |x − ar | + |x − ar | ⇒ dA (x) ≤ |x − ar | + dA (ar ) ≤ |x − ar | + r ⇒ dA (x) ≤ dAr (x) + r ⇒ 0 ≤ dA (x) − dAr (x) ≤ r. We show that dA (x) − dAr (x) = r. Assume that dA (x) − dAr (x) < r. There exists a unique p ∈ A such that dA (x) = |x − p| ≥ r > 0. Consider the point def

xr = p + r

x−p |x − p|

⇒ |x − xr | = dA (x) − r

and |xr − p| = r.

7. Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

317

By definition, dA (xr ) ≤ |xr − p| = r and xr ∈ Ar . As a result, dA (x) < dAr (x) + r ≤ |x − xr | + r = dA (x) − r + r = dA (x) yields a contradiction. This establishes identity (7.2) and for all x ∈ RN , |dAr (x) − dA (x)| ≤ r → 0 as r goes to zero. When ∇dAr (x) exists at a point ∈ Ar , ∇dAr (x) = 0 (cf. Theorem 3.3 (vi)). This means that ∇dAr = 0 almost everywhere in Ar and that the gradient of dAr is almost everywhere equal to ∇dAr (x) =

∇dA (x), 0,

dA (x) > r, dA (x) ≤ r.

For any compact K ⊂ RN       χAr − χA dx. |∇dAr − ∇dA | dx = |∇dA | χAr dx = (1 − χA ) χAr dx = K

K

K

K

Since {K ∩ Ar } is a decreasing family of closed sets as r → 0 and ∩r>0 Ar = A, m(K ∩ Ar ) → m(K ∩ A) (cf. W. Rudin [1, Thm. 1.1.9 (e), p. 16]). Finally, 1,1 ∇dAr − ∇dA L1 → 0 and dAr → dA in Wloc (RN ). S. Ferry [1] proved in 1975 that if N = 2 or 3, then d−1 A {r} is an (N − 1)manifold for almost all r, and that if A is a finite polyhedron in RN , then d−1 A {r} is an (N − 1)-manifold for all sufficiently small r. On the other hand, there is a Cantor set K in R4 such that d−1 K {r} is not a 3-manifold for any r between 0 and 1. The set K generalizes to higher dimensions. Generalizing those results, J. H. G. Fu [1] investigated the regularity properties of ∂Ar and proved the following theorem involving the notion of critical point of dA and of Lipschitz manifold. In order to make things simple we use a specific characterization of the critical points of dA as a definition (cf. J. H. G. Fu [1, 2] and Figure 6.5). Definition 7.1. Given ∅ = A ⊂ RN , x ∈ RN \A is said to be a critical point of dA if x ∈ co ΠA (x). The set of all critical points of dA will be denoted by crit (dA ). Definition 7.2. An m-dimensional Lipschitz manifold is a paracompact metric space M such that there is a system of open sets {Uα } covering M and, for each α, a bi-Lipschitzian homeomorphism ϕα of Uα onto an open subset of Rm . Theorem 7.2. Let A ⊂ RN be compact. The subset C(A) = dA (crit (dA )) ⊂ R is compact:    1/2 N C(A) ⊂ 0, diam (A) and H(N −1)/2 (C(A)) = 0 2N + 2 (the assertion about the measure of C(A) is nontrivial only for N ≤ 3). For all r∈ / C(A), Ar has positive reach and ∂Ar is a Lipschitz manifold.

318

Chapter 6. Metrics via Distance Functions

critical points of A

Figure 6.5. Set of critical points of A.

8

Characterization of Convex Sets

8.1

Convex Sets and Properties of dA

In the convex case the squared distance function is differentiable everywhere and this property can be used to characterize the convexity of a set. Theorem 8.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅. (iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. (v) For all x ∈ RN , ΠA (x) is a singleton.9 (vi) A is convex. (vii) dA is convex. In particular, ∀x, ∀y ∈ RN ,

|pA (y) − pA (x)| ≤ |y − x|

(8.1)

1,1 2,∞ (RN ) (and, a fortiori, to Wloc (RN )). Moreover, if A is and d2A belongs to Cloc convex, then A = A and d
A = d
A .

Remark 8.1. The convexity of dA implies the convexity of A but not of A. The set A of all points of the unit open ball with rational coordinates has a convex closure but is not convex. 9 This part of the theorem is related to deeper results on the convexity of Chebyshev sets in metric spaces. A subset A of a metric space X is called a Chebyshev set provided that every point x of X has a unique projection pA (x) in A. The reader is referred to V. L. Klee [1] for details and background material.

8. Characterization of Convex Sets

319

Proof. The equivalence of properties (i) to (v) follows from Lemma 6.1. (v) ⇒ (vi) Given x, y ∈ A and λ ∈ [0, 1], consider the point xλ = λx + (1 − λ)y and the continuous function λ → dA (xλ ) : [0, 1] → R is continuous. There exists µ ∈ [0, 1] such that dA (xµ ) = supλ∈[0,1] dA (xλ ). If supλ∈[0,1] dA (xλ ) = 0, then xλ ∈ A. If supλ∈[0,1] dA (xλ ) > 0, then x = y, dA (xµ ) > 0, µ ∈ (0, 1), and xµ ∈ / A. Since Sk (A) = ∅, ∇dA (xµ ) exists and the maximum is characterized by d = 0 ⇒ ∇dA (xµ ) · (x − y) = 0. dA (xλ ) dλ λ=µ By assumption there exists a unique projection pµ = pA (xµ ) ∈ A such that dA (xµ ) = |xµ − pµ | and by Theorem 3.3 (vi) ∇dA (xµ ) =

xµ − pµ . |xµ − pµ |

From inequality (6.6) in Theorem 6.2 (vi) |pµ − x| = |pA (xµ ) − pA (x)| ≤ |xµ − x| = (1 − µ) |x − y|, |pµ − y| = |pA (xµ ) − pA (y)| ≤ |xµ − y| = µ |x − y| ⇒ |x − y| ≤ |pµ − x| + |pµ − y| ≤ |x − y|. Therefore the point pµ lies on the segment between x and y: there exists µ ∈ [0, 1], µ = µ, such that pµ = xµ and xµ − pµ = xµ − xµ = (µ − µ ) (x − y). Finally, µ − µ xµ − xµ xµ − pµ |x − y| = · (x − y) = · (x − y) = ∇dA (xµ ) · (x − y) = 0  |µ − µ | |xµ − xµ | |xµ − pµ | and we get the contradiction x = y. (vi) ⇒ (v) By definition for each x ∈ RN d2A (x) = inf |z − x|2 = inf |z − x|2 , z∈A

z∈A

and since A is closed and convex and z → |z − x|2 is strictly convex and coercive, there exists a unique minimizing point pA (x) in A and ΠA (x) is a singleton. (vi) ⇒ (vii) Given x and y in RN , there exist x and y in A such that dA (x) = |x−x| and dA (y) = |y−y|. By convexity of A, for all λ, 0 ≤ λ ≤ 1, λ x+(1−λ) y ∈ A and dA (λ x + (1 − λ) y) ≤|λ x + (1 − λ) y − (λ x + (1 − λ) y)| ≤λ |x − x| + (1 − λ) |y − y| = λ dA (x) + (1 − λ) dA (y) and dA is convex in RN . (vii) ⇒ (vi) If dA is convex, then ∀λ ∈ [0, 1], ∀x, y ∈ A,

dA (λx + (1 − λ)y) ≤ λdA (x) + (1 − λ)dA (y).

320

Chapter 6. Metrics via Distance Functions

But x and y in A imply that dA (x) = dA (y) = 0 and hence ∀λ ∈ [0, 1], dA (λx + (1 − λ)y) = 0. Thus λx + (1 − λ)y ∈ A and A is convex. Properties (8.1) follow from Theorem 6.2 and the identity on the complements follows from Theorem 3.5 (ii) in Chapter 5. The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 8.2. For all subsets A of RN such that ∂A = ∅ and A is convex, (i) ∇dA belongs to BVloc (RN )N , (ii) the Hessian matrix D2 dA of second-order derivatives is a matrix of signed Radon measures that are nonnegative on the diagonal, and (iii) dA has a second-order derivative almost everywhere and for almost all x and y in RN , dA (y) − dA (x) − ∇dA (x) · (y − x) − 1 (y − x) · D2 dA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. From the equivalence of (vi) and (vii) in Theorem 8.1 dA is convex and continuous and the result follows from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240, Thm. 2, p. 239, and Aleksandrov’s Thm., p. 242].

8.2

Semiconvexity and BV Character of dA

Since the BV properties in Theorem 8.2 arise solely from the convexity of the distance function dA , the BV property will extend to sets A such that dA is semiconvex. Theorem 8.3. Let A be a nonempty subset of RN such that ∀x ∈ ∂A, ∃ρ > 0 such that dA is semiconvex in B(x, ρ). Then ∇dA ∈ BVloc (RN )N . Proof. For all x ∈ int A, dA = 0 and there exists ρ > 0 such that B(x, ρ) ⊂ int A and the result is trivial. For each x ∈ RN \A, dA (x) > 0. Pick h = dA (x)/2. Then for all y ∈ B(x, h) and all a ∈ A, |y − a| ≥ |x − a| − |y − x| ≥ dA (x) − |y − x| > 2h − h = h and B(x, h) ⊂ RN \Ah . By Theorem 3.2 (i) the function |x|2 /(2h) − dA (x) is locally convex in RN \Ah and, a fortiori, convex in B(x, h). Finally, by assumption for all x ∈ ∂A, there exists ρ > 0 such that dA is semiconvex. Hence for each x ∈ RN , ∇dA belongs to BV(B(x, ρ/2)) for some ρ > 0. Therefore, ∇dA belongs to BVloc (RN )N .

8. Characterization of Convex Sets

321

However, it is never semiconcave in RN for nontrivial sets. Theorem 8.4. (i) Given a subset A of RN , A = ∅, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN

if and only if ∃h > 0, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x)

is locally convex in Uh (A). (ii) Given a subset A of RN , ∅ = A = RN , c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN .

Proof. (i) If fc is convex in RN , it is locally convex in Uh (A). Conversely, assume that there exist h > 0 and c ≥ 0 such that fc is locally convex in Uh (A). From Theorem 3.2 (i), 1 2 |x| − dA (x) h is locally convex in RN \Ah/2 . Therefore, for c = max{c, 1/h}, the function fc is locally convex in RN and hence convex on RN . (ii) Assume the existence of a c ≥ 0 for which fc is convex in RN . For each y ∈ A, the function x → Fc (x, y) = c |x − y|2 − dA (x) is also convex since it differs from fc (x) by the linear term x → c (|y|2 − 2 x · y). Since ∅ = A = RN , there exist x ∈ A and p ∈ A such that 1 . 2c For any t > 0, define xt = p − t (x − p) and λ = t/(1 + t) ∈ ]0, 1[ and observe that 0 < dA (x) = |x − p| ≤

def

xλ = λ x + (1 − λ) xt = p and Fc (xλ , p) = 0. But λ Fc (x, p) + (1 − λ) Fc (xt , p) 1 t = [c |x − p|2 − dA (x)] + [c |xt − p|2 − dA (xt )] 1+t 1+t t ≤ [c dA (x)2 − dA (x) + c tdA (x)2 ] 1+t t ≤ dA (x) [(1 + t) c dA (x) − 1] 1+t   dA (x) t t − 1 t 1 dA (x) (1 + t) − 1 = , ≤ 1+t 2 1+t 2

322

Chapter 6. Metrics via Distance Functions

since by construction dA (x) > 0 and c dA (x) < 1/2. Therefore for t, 0 < t < 1, the above quantity is strictly negative and we have constructed two points x and xt and a λ, 0 < λ < 1, such that Fc (λ x + (1 − λ) xt , p) = 0 > λ Fc (x, p) + (1 − λ) Fc (xt , p). This contradicts the convexity of the function x → Fc (x, p) and, a fortiori, of fc .

8.3

Closed Convex Hull of A and Fenchel Transform of dA

Another interesting property is that the closed convex hull of A is completely characterized by the convex envelope of dA that can be obtained from the double Fenchel transform of dA . Theorem 8.5. Let A be a nonempty subset of RN : (i) The Fenchel transform d∗A (x∗ ) = sup x∗ · x − dA (x) x∈RN

of dA is given by

d∗A (x∗ ) = σA (x∗ ) + IB(0,1) (x∗ ),

(8.2)

where σA (x∗ ) is the support function of A and IB(0,1) (x∗ ) =

x∈ / B(0, 1), x ∈ B(0, 1),

+∞, 0,

is the indicator function of the closed unit ball B(0, 1) at the origin. (ii) The Fenchel transform of d∗A is given by d∗∗ A = dco A = dco A .

(8.3)

In particular, dco A is the convex envelope of dA . The function dA and the indicator function IA are both zero on A and the Fenchel transform of IA , ∗ = σA = σA , IA

coincides with d∗A on B(0, 1). Proof. (i) From the definition, d∗A (x∗ ) = sup x∗ · x − dA (x) = sup x∈RN

x∈RN

.

x∗ · x − inf |x − p|

/

p∈A

∗

= sup sup x · x − |x − p| = sup sup x∗ · x − |x − p| x∈RN p∈A

p∈A x∈RN

∗

∗

= sup sup x · p + x · (x − p) − |x − p| p∈A x∈RN ∗

= sup x · p + sup [ x∗ · (x − p) − |x − p| ] x∈RN

p∈A ∗

= sup x · p + sup [ x∗ · x − |x| ] = σA (x∗ ) + IB(0,1) (x∗ ). p∈A

x∈RN

8. Characterization of Convex Sets

323

(ii) We now use the property that σA = σco A : ∗∗ ∗∗ · x∗ − d∗A (x∗ ) = sup x∗∗ · x∗ − σA (x∗ ) − IB(0,1) (x∗ ) d∗∗ A (x ) = sup x x∗ ∈RN x∗ ∈RN . / ∗∗ ∗ ∗ = sup x · x − σA (x ) = sup x∗∗ · x∗ − sup x∗ · x |x∗ |≤1

= sup

|x∗ |≤1

inf (x

∗∗

|x∗ |≤1 x∈co A

x∈co A

∗

− x) · x

≤ sup (x∗∗ − p(x∗∗ )) · x∗ ≤ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ). |x∗ |≤1

But in fact we have a saddle point. If x∗∗ ∈ co A, pick x∗ = 0 and ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗ ≥ 0 = dco A (x∗∗ ).

|x∗ |≤1 x∈co A

For x∗∗ ∈ / co A rewrite the function as (x∗∗ − x) · x∗ = (x∗∗ − p(x∗∗ )) · x∗ + (p(x∗∗ ) − x) · x∗ . Since p(x∗∗ ) is the minimizing point of the function |x−x∗∗ |2 over the closed convex set co A it is completely characterized by the variational inequality ∀x ∈ co A,

(p(x∗∗ ) − x∗∗ ) · (x − p(x∗∗ )) ≥ 0.

(8.4)

By choosing the following special x∗ , x∗ =

x∗∗ − p(x∗∗ ) , |x∗∗ − p(x∗∗ )|

we get ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗

|x∗ |≤1 x∈co A

≥ |x∗∗ − p(x∗∗ )| + (p(x∗∗ ) − x) ·

x∗∗ − p(x∗∗ ) |x∗∗ − p(x∗∗ )|

≥ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ) in view of (8.4). Therefore sup

inf (x∗∗ − x) · x∗ = dco A (x∗∗ ) = inf

|x∗ |≤1 x∈co A

sup (x∗∗ − x) · x∗ .

x∈co A |x∗ |≤1

Finally from the previous theorem dco A is a convex function that coincides with the convex envelope since it is the bidual of dA .

8.4

Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) Cd,loc

Theorem 8.6. Let D be a nonempty open subset of RN . The subfamily def

Cd (D) = {dA ∈ Cd (D) : A = ∅ convex}

(8.5)

324

Chapter 6. Metrics via Distance Functions

of Cd (D) is closed in Cloc (D). It is compact in C(D) when D is bounded. The subfamily def

Cdc (D) = {d
Ω ∈ Cdc (D) : Ω open and convex}

(8.6)

and for E open, ∅ = E ⊂ D, the subfamilies def

Cdc (E; D) = {d
Ω : E ⊂ Ω open and convex ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open and convex ⊂ D N

d
Ω :

&

(8.7) (8.8)

of Cdc (D) are closed in Cloc (D). If D is bounded, they are compact in C(D). Proof. The set of all convex functions in Cloc (D) is a closed convex cone with vertex at 0. Hence its intersection with Cd (D) is closed. Any Cauchy sequence in Cd (D) converges to some convex dA . From part (i) A is convex. But dA = dA and the nonempty convex set A can be chosen as the limit set. For the complementary distance function, it is sufficient to prove the compactness for D bounded. Given any sequence {d
Ωn }, there exist an open subset Ω of D and a subsequence, still indexed by n, such that d
Ωn → d
Ω in C(D). If Ω is empty, there is nothing to prove. If Ω is not empty, consider two points x and y in Ω and λ ∈ [0, 1]. There exists r > 0 such that B(x, r) ⊂ Ω and B(y, r) ⊂ Ω

⇒ d
Ω (x) ≥ r and d
Ω (y) ≥ r.

There exists N > 0 such that for all n > N , d
Ωn − d
Ω C(D) < r/2 and d
Ωn (x) ≥ d
Ω (x) − r/2 = r/2 ⇒ B(x, r/2) ⊂ Ωn , d
Ωn (y) ≥ d
Ω (y) − r/2 = r/2 ⇒ B(y, r/2) ⊂ Ωn . By convexity of Ωn , xλ = λx + (1 − λ)y ∈ Ωn , B(xλ , r/2) ⊂ λ B(x, r/2) + (1 − λ)B(y, r/2) ⊂ Ωn ⇒ ∀n > N, d
Ωn (xλ ) ≥ r/2 ⇒ d
Ω (xλ ) ≥ r/2 ⇒ B(xλ , r/2) ⊂ Ω. Therefore xλ ∈ Ω, Ω is convex, and Cdc (D) is compact. Finally, Cdc (E; D) = c c (E; D) = Cd,loc (E; D) ∩ Cdc (D) are closed since Cdc (D) is Cdc (E; D) ∩ Cdc (D) and Cd,loc c c closed and Cd (E; D) and Cd,loc (E; D) are closed by Theorem 2.6.

9

Compactness Theorems for Sets of Bounded Curvature

We have seen two compactness conditions in Theorems 6.6 and 6.7 of section 6.3 for sets of positive reach from a condition in a tubular neighborhood. In this section we give compactness theorems for families of sets of global and local bounded curvature.

9. Compactness Theorems for Sets of Bounded Curvature

325

For the family of sets with bounded curvature, the key result is the compactness of the embeddings

 (9.1) BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N → W 1,p (D), 

BCcd (D) = d
Ω ∈ Cdc (D) : ∇d
Ω ∈ BV(D)N → W 1,p (D) (9.2) for bounded open Lipschitzian subsets D of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(9.3)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(9.4)

for bounded open Lipschitzian subsets D of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 5, we give a first version involving global conditions on a fixed bounded open Lipschitzian holdall D. In the second version, the sets are contained in a bounded open holdall D with local conditions in their tubular neighborhood or the tubular neighborhood of their boundary.

9.1

Global Conditions in D

Theorem 9.1. Let D be a nonempty bounded open Lipschitzian holdall in RN . The embedding (9.1) is compact. Thus for any sequence {An }, ∅ = An , of subsets of D such that (9.5) ∃c > 0, ∀n ≥ 1, D2 dAn M 1 (D) ≤ c, there exist a subsequence {Ank } and A = ∅, such that ∇dA ∈ BV(D)N and dAnk → dA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

n→∞

and

D2 dA M 1 (D) ≤ c.

(9.6)

Proof. Given c > 0 consider the set  def

Sc = dA ∈ Cd (D) : D 2 dA M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {dAn }, there exist a subsequence, still denoted by {dAn }, and f ∈ BV(D)N such that ∇dAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cd (D) is compact in C(D) for bounded D and there exist another subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D) and, a fortiori, in L1 (D). Therefore, dAnk converges in W 1,1 (D) and also in

326

Chapter 6. Metrics via Distance Functions

L1 (D). By uniqueness of the limit, f = ∇dA and dAnk converges in W 1,1 (D) to dA . For Φ ∈ D1 (D)N ×N as k goes to infinity   −→ −→ ∇dAnk · div Φ dx → ∇dA · div Φ dx D D   −→ −→ ∇dA · div Φ dx = lim ∇dAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D dA M 1 (D) ≤ c, and ∇dA ∈ BV(D) . This proves the compactness of the embedding for p = 1 and properties (9.6). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cd (D) in Theorem 4.2 (i). 2

N

When D is bounded open, Cdc (D) is compact in C(D) and closed in W01,p (D), 1 ≤ p < ∞, and we have the analogue of the previous compactness theorem. Theorem 9.2. Let D, ∅ = D ⊂ RN , be bounded open Lipschitzian. The embedding (9.2) is compact. Thus for any sequence {Ωn } of open subsets of D such that ∃h > 0, ∃c > 0, ∀n,

D2 d
Ωn M 1 (D) ≤ c,

(9.7)

there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇d
Ω ∈ BV(D)N and d
Ωn → d
Ω in W01,p (D)

(9.8)

k

for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D)N ×N , D2 d
Ωn , ϕ → D2 d
Ω , ϕ,

D 2 d
Ω M 1 (D) ≤ c,

and

χΩ ∈ BV(D). (9.9)

Proof. Given c > 0 consider the set  def

Scc = d
Ω ∈ Cdc (D) : D 2 d
Ω M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {d
Ωn } there exist a subsequence, still denoted by {d
Ωn }, and f ∈ BV(D)N such that ∇d
Ωn → f in L1 (D)N . But by Theorem 2.4 (ii), Cdc (D) is compact in C0 (D) for bounded D and there exist another subsequence {d
Ωn } and d
Ω ∈ Cdc (D) such that d
Ωn → d
Ω k

k

in C0 (D) and, a fortiori, in L1 (D). Therefore, d
Ωn converges in W01,1 (D) and also k

in L1 (D). By uniqueness of the limit, f = ∇d
Ω and d
Ωn converges in W01,1 (D) k to d
Ω . For all Φ ∈ D1 (D)N ×N as k goes to infinity   −→ −→ ∇d
Ωn · divΦ dx → ∇d
Ω · divΦ dx k D D   −→ −→ ∇d
Ω · divΦ dx = lim ∇d
Ωn · divΦ dx ≤ cΦC(D) , ⇒ D

k→∞

D

k

D 2 d
Ω M 1 (D) ≤ c, ∇d
Ω ∈ BV(D)N , and χΩ ∈ BV(D). This proves the compactness of the embedding for p = 1 and properties (9.9). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cdc (D) in Theorem 4.2 (i).

9. Compactness Theorems for Sets of Bounded Curvature

9.2

327

Local Conditions in Tubular Neighborhoods

The global conditions (9.5) and (9.7) can be weakened to local ones in a neighborhood of each set of the sequence, and the Lipschitzian condition on D can be removed since only the uniform boundedness of the sets of the sequence is required. Theorem 9.3. Let D, ∅ = D ⊂ RN , be a bounded open holdall and {An }, ∅ = An , be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 dAn M 1 (Uh (∂An )) ≤ c.

(9.10)

Then there exist a subsequence {Ank } and a subset A, ∅ = A, of D such that ∇dA ∈ BVloc (RN )N , and for all p, 1 ≤ p < ∞, dAnk → dA in W 1,p (Uh (D))-strong.

(9.11)

Moreover, for all ϕ ∈ D0 (Uh (A)) lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

k→∞

D2 dA M 1 (Uh (A)) ≤ c,

(9.12)

and χA belongs to BVloc (RN ). Proof. First notice that, since An is bounded and nonempty, ∂An = ∅. Further Uh (∂An ) can be replaced by Uh (An ) in condition (9.10) since dAn = 0 in An . So it is sufficient to prove the theorem for that case. Lemma 9.1. For any A, ∂A = ∅ and h > 0, D2 dA M 1 (Uh (A)) = D2 dA M 1 (Uh (∂A)) . The proof of the lemma will be given after the proof of the theorem. (i) The assumption An ⊂ D implies Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that dAn → dA in C(Uh (D)) and another subsequence, still denoted by {dAn }, such that dAn → dA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and x in Uh (D) dAn (x) ≤ dA (x) + ε,

dA (x) ≤ dAn (x) + ε.

Therefore, An ⊂ Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ),

(9.13)

Uh−ε (A) ⊂ Uh−2 ε (An ) ⊂ An .

(9.14)

From (9.10) and (9.13) ∀n ≥ N,

D2 dAn M 1 (Uh−ε (A)) ≤ c.

328

Chapter 6. Metrics via Distance Functions

In order to use the compactness of the embedding (9.4) as in the proof of Theorem 9.1, we would need Uh−ε (A) to be Lipschitzian. To get around this, we construct a bounded Lipschitzian set between Uh−2ε (A) and Uh−ε (A). Indeed, by definition, Uh−ε (A) = ∪x∈A B(x, h − ε)

and Uh−2ε (A) ⊂ Uh−ε (A),

and by compactness, there exists a finite sequence of points {xi }ni=1 in A such that |xi − xj | < (h − ε)/2 for all i = j so that no two balls are tangent, and def

Uh−2ε (A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a finite number of nontangent balls, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {dAn }, and f ∈ BV(UB )N such that ∇dAn → f in L1 (UB )N . Since Uh (D) is bounded, Cd (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, dAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇dA on UB and dAn converges to dA in W 1,1 (UB ). By Definition 3.3 and Theorem 5.1, ∇dA and ∇dAn all belong to BVloc (RN )N since they are BV in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χA ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral  |∇dAn − ∇dA |2 dx Uh (D)   2 = |∇dAn − ∇dA | dx + |∇dAn − ∇dA |2 dx. Uh−2ε (A)

Uh (D)\Uh−2ε (A)

From part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of Uh−2ε (A). From (9.14) for all n ≥ N, |∇dAn (x)| = 1 a.e. in An ⊃ Uh−3ε (An ) ⊃ Uh−2ε (A), |∇dA (x)| = 1 a.e. in A ⊃ Uh−2ε (A). The second integral reduces to   |∇dAn − ∇dA |2 dx = Uh (D)\Uh−2ε (A)

2 (1 − ∇dAn · ∇dA ) dx,

Uh (D)\Uh−2ε (A)

which converges to zero since ∇dAn  ∇dA in L2 (Uh (D))N -weak in part (i) and the fact that |∇dA | = 1 almost everywhere in Uh (D)\Uh−2ε (A). Therefore, since dAn → dA in C(Uh (D)), dAn → dA in H 1 (Uh (D))-strong, and by Theorem 4.2 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1.

9. Compactness Theorems for Sets of Bounded Curvature

329

(iii) Properties (9.12). Consider the initial subsequence {dAn } which converges to dA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of dAn to dA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (A))N ×N ,   −→ −→ ∇dAn · div Φ dx = ∇dA · div Φ dx. lim n→∞

Uh (A)

Uh (A)

Each such Φ has compact support in Uh (A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ).

For n ≥ N (ε) consider the integral   −→ ∇dAn · div Φ dx = Uh (A)

 ⇒

Uh (A)

−→

Uh−2ε (A)

∇dAn



∇dAn · div Φ dx =

−→

∇dAn · div Φ dx U (A )

h n · div Φ dx ≤ D2 dAn M 1 (Uh (An )) ΦC(Uh (An )) ≤ c ΦC(Uh−2ε (A)) = c ΦC(Uh (A)) .

−→

By convergence of ∇dAn to ∇dA in L2 (D∪Uh (A))-weak, for all Φ ∈ D 1 (Uh (A))N ×N  −→ ∇dA · div Φ dx ≤ c ΦC(Uh (A)) ⇒ D2 dA M 1 (Uh (A)) ≤ c. Uh (A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof. Proof of Lemma 9.1. First check that Uh (A) = Uh (∂A) ∪x∈A−h B(x, h),

 def

A−h = x ∈ RN : B(x, h) ⊂ A .

It is sufficient to show that all points x of int A are contained in the right-hand side of the above expression. If d∂A (x) < h, then x ∈ Uh (∂A); if d∂A (x) ≥ h, then B(x, h) ⊂ A and x ∈ A−h . For all Φ ∈ D1 (Uh (A))N ×N , Φ has compact support and there exists ε > 0, 0 < 2ε < h, such that supp Φ ⊂ Uh−2ε (A). But Uh−ε (A) ⊂ Uh (A) and there exists {xj ∈ A−h : 1 ≤ j ≤ m} such that Uh−ε (A) ⊂ Uh (∂A)∪m j=1 B(xj , h). Let ψ0 ∈ D(Uh (∂A)) and let ψj ∈ D(B(xj , h)) be a partition of unity such that m ψj = 1 on Uh−2ε (A). 0 ≤ ψj ≤ 1, j=0

330

Chapter 6. Metrics via Distance Functions

Consider the integral   −→ ∇dA · div Φ dx = Uh (A)

 ∇dA · div Φ dx = −→

Uh−2ε (A)

 =

−→   ∇dA · div ψ0 Φ dx + Uh−2ε (A)

  m ∇dA · div  ψj Φ dx −→

Uh−2ε (A)

m



j=1

j=0

−→   ∇dA · div ψj Φ dx. Uh−2ε (A)

By construction for ΦC(Uh (A)) ≤ 1 ψ0 Φ ∈ D1 (Uh−2ε (A) ∩ Uh (∂A))N ×N , ψj Φ ∈ D1 (Uh−2ε (A) ∩ B(xj , h))N ×N ,

ψ0 ΦC(Uh (∂A)) ≤ 1,

ψj ΦC(B(xj ,h)) ≤ 1,

1 ≤ j ≤ m.

But for j, 1 ≤ j ≤ m, B(xj , h) ⊂ int A, where both dA and ∇dA are identically zero. As a result the above integral reduces to    −→ −→ −→ ∇dA · div Φ dx = ∇dA · div (ψ0 Φ) dx = ∇dA· div (ψ0 Φ) dx Uh (A)

 ⇒

Uh−2ε (A)

U (∂A)

h ∇dA · div Φ dx ≤ D2 dA M 1 (Uh (∂A)) ψ0 ΦC(Uh (∂A))

−→

Uh (A)

⇒ D2 dA M 1 (Uh (A)) ≤ D 2 dA M 1 (Uh (∂A)) , and this completes the proof. Theorem 9.4. Let D, ∅ = D ⊂ RN , be a bounded open holdall. Let {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets and assume that ∃h > 0 and ∃c > 0 such that

∀n,

D2 d
Ωn M 1 (Uh (∂Ωn )) ≤ c.

(9.15)

Then there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇d
Ω ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, d
Ωn → d
Ω in W01,p (D)-strong. k

(9.16)

Moreover for all ϕ ∈ D0 (Uh (Ω)) lim ∂ij d
Ωn , ϕ = ∂ij d
Ω , ϕ, 1 ≤ i, j ≤ N,

k→∞

k

D2 d
Ω M 1 (Uh (
Ω)) ≤ c, (9.17)

and χΩ belongs to BVloc (RN ). Proof. First note that ∂Ωn = ∅ since Ωn is bounded and nonempty. By Lemma 9.1, Uh (∂Ωn ) can be replaced by Uh (Ωn ) in condition (9.15). Moreover since Ωn can be unbounded, we shall work with the bounded neighborhoods Uh (D Ωn ) of D Ωn rather than with Uh (Ωn ). Lemma 9.2. Let h > 0 and let ∅ = Ω ⊂ D ⊂ RN be bounded open. Then D2 d
Ω M 1 (Uh (


D

Ω))

= D2 d
Ω M 1 (Uh (
Ω)) .

If ∂Ω = ∅, D 2 d
Ω M 1 (Uh (∂Ω)) = D2 d
Ω M 1 (Uh (
Ω)) .

9. Compactness Theorems for Sets of Bounded Curvature

331

The proof of this lemma will be given after the proof of the theorem. (i) By Theorem 2.4 (ii) since D is bounded, there exist a subsequence, still denoted by {d
Ωn }, and an open subset Ω of D such that d
Ωn → d
Ω

in C0 (D) and Cc (D ∪ Uh (Ω))

and another subsequence, still denoted by {d
Ωn }, such that d
Ωn → d
Ω

in H01 (D)-weak and H01 (D ∪ Uh (Ω))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N d
Ωn (x) ≤ d
Ω (x) + ε,

d
Ω (x) ≤ d
Ωn (x) + ε.

Clearly, since Ω ⊂ D and Ωn ⊂ D and D is open, D Ωn = ∅ and D Ω = ∅. Furthermore, D Ωn ⊂ Uh−2 ε (D Ωn ) ⊂ Uh−ε (D Ω) ⊂ Uh (D Ωn ), D Uh−ε (D Ω) ⊂ D Uh−2 ε (D Ωn ) ⊂ Ωn .

(9.18) (9.19)

From (9.15) and (9.18) ∀n ≥ N,

D 2 d
Ωn M 1 (Uh−ε (


D

Ω))

≤ c.

(9.20)

As in part (i) of the proof of Theorem 9.3, we can construct a bounded open Lipschitzian set UB between Uh−2ε (D Ω) and Uh−ε (D Ω) such that Uh−2ε (D Ω) ⊂ UB ⊂ Uh−ε (D Ω). Since UB is bounded and Lipschitzian, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {d
Ωn }, and f ∈ BV(UB )N such that ∇d
Ωn → f in L1 (UB )N . Since D ∪ UB is bounded, Cdc (D ∪ UB ) is compact in C0 (D ∪ UB ), and there exist another subsequence, still denoted by {d
Ωn }, and an open subset Ω of D such that d
Ωn → d
Ω in C0 (D ∪ UB ) and, a fortiori, in L1 (D ∪ UB ). Therefore d
Ωn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇d
Ω on BD and d
Ωn converges to d
Ω in W 1,1 (UB ). By Definition 3.3 and Theorem 5.1, ∇d
Ω and ∇d
Ωn all belong to BVloc (RN )N since they are BV in tubular neighborhoods Uh (D Ω) and Uh (D Ωn ) of their respective boundaries ∂Ω and ∂Ωn . Moreover, by Theorem 5.2 (ii), χ
Ω ∈ BVloc (RN ) and, a fortiori, χΩ since Ω is open. The above conclusions also hold for the subset Uh−2ε (D Ω) of UB . (ii) Convergence in W01,p (D). Consider the integral  |∇d
Ωn − ∇d
Ω |2 dx  D |∇d
Ωn − ∇d
Ω |2 dx + |∇d
Ωn − ∇d
Ω |2 dx. = D∩Uh−2ε (
D Ω)

D\Uh−2ε (
D Ω)

332

Chapter 6. Metrics via Distance Functions

From part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of D Uh−2ε (D Ω). From the relations (9.19) for all n ≥ N , |∇d
Ωn (x)| = 1 a.e. in Ωn ⊃ D Uh−3ε (D Ωn ) ⊃ D Uh−2ε (D Ω), |∇dΩ (x)| = 1 a.e. in Ω ⊃ D Uh−2ε (D Ω). The second integral reduces to   |∇d
Ωn − ∇d
Ω |2 dx = D\Uh−2ε (
D Ω)

2 (1 − ∇d
Ωn · ∇d
Ω ) dx,

D\Uh−2ε (
D Ω)

which converges to zero by weak convergence of ∇d
Ωn to ∇d
Ω in L2 (D)N and the fact that |∇d
Ω | = 1 almost everywhere in D\Uh−2ε (D Ω). Therefore, since d
Ωn → d
Ω in C0 (D) d
Ωn → d
Ω in H01 (D)-strong, and by Theorem 4.2 (i) the convergence is true in W01,p (D) for all p ≥ 1. (iii) Properties (9.17). Consider the initial subsequence {d
Ωn }, which converges to d
Ω in H01 (D ∪ Uh (Ω))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of d
Ωn to d
Ω in H01 (D ∪ Uh (Ω))-weak for each Φ ∈ D 1 (Uh (Ω))N ×N ,   −→ −→ ∇d
Ωn · div Φ dx = ∇d
Ω · div Φ dx. lim n→∞

Uh (
Ω)

Uh (
Ω)

Now each Φ ∈ D1 (Uh (Ω))N ×N has compact support in Uh (Ω) and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (Ω). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (Ωn ) ⊂ Uh−ε (Ω) ⊂ Uh (Ωn ).

For n ≥ N (ε) consider the integral   −→ ∇d
Ωn · div Φ dx = Uh (
Ω)

 ⇒

Uh (
Ω)

∇d
Ωn

Uh−2ε (
Ω)

∇d
Ωn

 · div Φ dx = −→

−→

∇dΩn · div Φ dx

U (
Ω )

h n · div Φ dx ≤ D 2 d
Ωn M 1 (Uh (
D Ωn )) ΦC(Uh (
D Ωn )) ≤ c ΦC(Uh−2ε (
D Ω)) = c ΦC(Uh (
Ω)) .

−→

By convergence of ∇d
Ωn to ∇d
Ω in L2 (D∪Uh (Ω))-weak, for all Φ ∈ D1 (Uh (Ω))N ×N  −→ ∇d
Ω · div Φ dx ≤ c ΦC(Uh (
Ω)) ⇒ D2 d
Ω M 1 (Uh (
Ω)) ≤ c. Uh (
Ω)

Finally the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

9. Compactness Theorems for Sets of Bounded Curvature Proof of Lemma 9.2. The proof is similar to that of Lemma 9.1 with Uh (Ω) = Uh (D Ω) ∪ int D = Uh (D Ω)∪x∈(
D)−h B(x, h),  def

(D)−h = x ∈ RN : B(x, h) ⊂ D , since both d
Ω and ∇d
Ω are identically zero on D.

333

Chapter 7

Metrics via Oriented Distance Functions 1

Introduction

In this chapter we study oriented distance functions and their role in the description of the geometric properties and smoothness of domains and their boundary. They constitute a special family of the so-called algebraic or signed distance functions. Signed distance functions make sense only when some appropriate orientation of the underlying set is available to assign a sign, while oriented distance functions are always well-defined. Our choice of terminology emphasizes the fact that, for a smooth open domain, the associated oriented distance function specifies the orientation of the normal to the boundary of the set. They enjoy many interesting properties. For instance, they retain the nice properties of the distance functions but also generate the classical geometric properties associated with sets and their boundary. The smoothness of the oriented distance function in a neighborhood of the boundary of the set is equivalent to the local smoothness of its boundary. Similarly, the convexity of the function is equivalent to the convexity of the closure of the set. In addition, their respective gradient and Hessian matrix respectively coincide with the unit outward normal and the second fundamental form on the boundary of the set. Finally, they provide a framework for the classification of domains and sets according to their degree of smoothness, much like Sobolev spaces and spaces of continuous and H¨ olderian functions do for functions. The first part of the chapter deals with the basic definitions and constructions and the main results. The second part specializes to specific subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. In the first part, section 2 presents the basic properties, introduces the uniform metric topology, and shows its connection with the Hausdorff and complementary Hausdorff topologies of Chapter 6. Section 3 is devoted to the differentiability properties and the associated set of projections onto the boundary, and it completes the treatment of skeletons and cracks. Section 8 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function in a neighborhood of its boundary for sets of class C 1,1 or better. 335

336

Chapter 7. Metrics via Oriented Distance Functions

When the domain is sufficiently smooth the trace of the Hessian matrix of secondorder partial derivatives on the boundary is the classical second fundamental form of geometry. Section 4 deals with the W 1,p -topology on the set of oriented distance functions and the closed subfamily of sets for which the volume of the boundary is zero. In the second part of the chapter, section 5 studies the subfamily of sets for which the gradient of the oriented distance function is a vector of functions of bounded variation: the sets with global or local bounded curvature. There are large classes for which general compactness theorems will be proved in section 11. Examples are given to illustrate the behavior of the norms in tubular neighborhoods as the thickness of the neighborhood goes to zero in section 5.1. Section 9 introduces Sobolev or W s,p -domains which provide a framework for the classification of sets according to their degree of smoothness. For smooth sets this classification intertwines with the classical classification of C k - and H¨ olderian domains. Section 10 extends the characterization of closed convex sets by the distance function to the oriented distance function and introduces the notion of semiconvex sets. The set of equivalence classes of convex subsets of a compact holdall D is again compact, as it was for all the topologies considered in Chapters 5 and 6. Section 7 shows that sets of positive reach introduced in Chapter 6 are sets of locally bounded curvature and that the boundary of their closure has zero volume. A compactness theorem is also given. Section 12 introduces the notion of γ-density perimeter, which is a relaxation of the (N − 1)-dimensional upper Minkowski content, and a compactness theorem for a family of subsets of a bounded holdall that have a uniformly bounded γ-density perimeter. In the last part of the chapter, section 11 gives compactness theorems for sets of global and local bounded curvatures from a uniform bound in tubular neighborhoods of their boundary. They are the analogues of the theorems of Chapter 6. Section 13 gives the compactness theorem for the family of subsets of a bounded holdall that have the uniform fat segment property. Equivalent conditions are given on the local graph functions of the family in section 13.2. The results are specialized to the compactness under the uniform cusp/cone property. In particular we recover the compactness of Theorem 6.11 of section 6.4 in Chapter 5 for the associated family of characteristic functions in Lp (D). Section 14 combines the compactness under the uniform fat segment property with a bound on the De Giorgi perimeter of Caccioppoli sets in section 14.1 and the γ-density perimeter in section 14.2. Section 15 introduces the families of cracked sets and uses them in section 16 to provide an original solution to a variation of the image segmentation problem of D. Mumford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks, and sets that can be made up of components of different codimensions. The Hausdorff (N − 1) measure of their boundary is not necessarily finite. Yet compact families (in the W 1,p -topology) of such sets can be constructed.

2. Uniform Metric Topology

2

337

Uniform Metric Topology

2.1

The Family of Oriented Distance Functions Cb (D)

The distance function dA provides a good description of a domain A from the “outside.” However, its gradient undergoes a jump discontinuity at the boundary of A which prevents the extraction of information about the smoothness of A from the smoothness of dA in a neighborhood of ∂A. To get around this, it is natural to take into account the negative of the distance function of its complement A that cancels the jump discontinuity at the boundary of smooth sets. Definition 2.1. Given A ⊂ RN , the oriented distance function from x ∈ RN to A is defined as def

bA (x) = dA (x) − d
A (x).

(2.1)

This function gives a level set description of a set whose boundary coincides with the zero level set. From Chapter 6 the function bA is finite in RN if and only if ∅ = A = RN , since bA = dA = +∞ when A = ∅ and bA = −d
A = −∞ when A = ∅. This condition is completely equivalent to ∂A = ∅, in which case bA coincides with the algebraic distance function to the boundary of A:   x ∈ int A, dA (x) = d∂A (x), bA (x) = 0, x ∈ ∂A,   −d
A (x) = −d∂A (x), x ∈ int A.

(2.2)

Noting that b
A = −bA , it means that we have implicitly chosen the negative sign for the interior of A and the positive sign for the interior of its complement. As bA is an increasing function from its interior to its exterior, we shall see later that for sets with a smooth boundary, the restriction of the gradient of bA to ∂A coincides with the outward unit normal to ∂A. Changing the sign of bA gives the inward orientation to the gradient and the normal. Associate with a nonempty subset D of RN the family  def

Cb (D) = bA : A ⊂ D and ∂A = ∅ .

(2.3)

Theorem 2.1. Let A be a subset of RN . Then the following hold: (i) A = ∅ and A = ∅ ⇐⇒ ∂A = ∅. (ii) Given bA and bB in Cb (D), A ⊃ B ⇒ b A ≤ bB

and

A = B ⇒ b A = bB ,

bA ≤ bB in D ⇐⇒ B ⊂ A and A ⊂ B, bA = bB in D ⇐⇒ B = A and A = B ⇐⇒ B = A and ∂A = ∂B.

338

Chapter 7. Metrics via Oriented Distance Functions In particular, bint A ≤ bA ≤ bA and bA = bA ⇐⇒ ∂A = ∂A

and

bint A = bA ⇐⇒ ∂ int A = ∂A.

(2.4)

(iii) |bA | = dA + d
A = max{dA , d
A } = d∂A and ∂A = {x ∈ RN : bA (x) = 0}. (iv) bA ≥ 0 ⇐⇒ A ⊃ ∂A ⊃ A ⇐⇒ ∂A = A. (v) bA = 0 ⇐⇒ A = ∂A = A ⇐⇒ ∂A = RN . (vi) If ∂A = ∅, the function bA is uniformly Lipschitz continuous in RN and |bA (y) − bA (x)| ≤ |y − x|.

∀x, y ∈ RN ,

(2.5)

Moreover, bA is (Fr´echet) differentiable almost everywhere and |∇bA (x)| ≤ 1

a.e. in RN .

(2.6)

Proof. (i) The proof is obvious. (ii) By assumption + − − dA = b+ A ≤ bB = dB and d
A = bA ≥ bB = d
B in D.

Since A ⊂ D and B ⊂ D, then A ⊃ D and B ⊃ D and d
A = d
B = 0 in D

⇒ d
A ≥ d
B in RN

⇒ A ⊂ B.

Also since B ⊂ D for all x ∈ B, dA (x) ≤ dB (x) = 0 and x ∈ A. Therefore, A ⊂ B and B ⊂ A. Conversely, B⊂A

⇒ dA ≤ dB in RN

and

A ⊂ B

⇒ d
B ≤ d
A in RN

and, a fortiori, in D. The equality case follows from the fact that bA = bB if and only if bA ≥ bB and bA ≤ bB . By definition int A = A ⊂ A ⊂ A and bint A ≤ bA ≤ bA . (iii) For x in A bA (x) = −d
A (x) =⇒ |bA (x)| = d
A (x) ≤ d∂A (x) since A ⊃ ∂A and the inf over A is smaller than the inf over its subset ∂A. Similarly, for x in A bA (x) = dA (x) =⇒ |bA (x)| = dA (x) ≤ d∂A (x), and finally |bA (x)| ≤ max{d
A (x), dA (x)} ≤ d∂A (x). Conversely, for each x in A, the set of projections ΠA (x) ⊂ A ∩ A = ∂A is not empty. Hence |bA (x)| = dA (x) =

min |x − y| ≥ inf |x − y| = d∂A (x),

y∈ΠA (x)

y∈∂A

2. Uniform Metric Topology

339

and similarly for all x in A, |bA (x)| = d
A (x) ≥ d∂A (x). Therefore |bA (x)| ≥ d∂A (x). (iv) If bA = dA −d
A ≥ 0, then dA ≥ d
A and A ⊂ A, and necessarily A ⊂ ∂A and A = ∂A. Conversely, if x ∈ ∂A, then by definition bA (x) = 0. If x ∈ / ∂A, then x ∈ ∂A = A = int A and bA (x) = dA (x) ≥ 0. (v) bA = 0 is equivalent to bA ≥ 0 and b
A = −bA ≥ 0. Then we apply (v) twice. But A = ∂A = A ⇐⇒ int A = ∅ = int A ⇐⇒ ∂A = RN . (vi) Clearly, ∀x, y ∈ A, ∀x, y ∈ A,

|bA (y) − bA (x)| = |d
A (y) − d
A (x)| ≤ |y − x|, |bA (y) − bA (x)| = |dA (y) − dA (x)| ≤ |y − x|.

For x ∈ A and y ∈ int A = A, dA (y) > 0 and bA (y) − bA (x) = dA (y) + d
A (x) > 0

⇒ |bA (y) − bA (x)| = dA (y) + d
A (x).

By assumption B(y, dA (y)) ⊂ int A. Define the point dA (y) x=y+ (x − y) ∈ A |x − y|   dA (y) ⇒ d
A (x) ≤ |x − x| = 1 − (y − x) = |x − y| − dA (y) |x − y| ⇒ |bA (y) − bA (x)| = dA (y) + d
A (x) ≤ |x − y|. The argument is similar for x ∈ int A and y ∈ A. The differentiability follows from Theorem 2.1 (vii) in Chapter 6.

2.2

Uniform Metric Topology

From Theorem 2.1 (i) the function bA is finite at each point when ∂A = ∅. This excludes A = ∅ and A = RN . The zero function bA (x) = 0 for all x in RN corresponds to the equivalence class of sets A such that A = ∂A = A or ∂A = RN . This class of sets is not empty. For instance, choose the subset of points of RN with rational coordinates or the set of all lines parallel to one of the coordinate axes with rational coordinates. Let D be a nonempty subset of RN and associate with each subset A of D, ∂A = ∅, the equivalence class  def

[A]b = B : ∀B, B ⊂ D, B = A and ∂A = ∂B

340

Chapter 7. Metrics via Oriented Distance Functions

and the family of equivalence classes  def

Fb (D) = [A]b : ∀A, A ⊂ D and ∂A = ∅ . The equivalence classes induced by bA are finer than those induced by dA since both the closures and the boundaries of the respective sets must coincide. For Cd (D), we have an invariant closed representative A in the equivalence class [A]d and for Cdc (D) an invariant open representative int A for [A]d . For Cb (D), the deficiencies of Cd (D) and Cdc (D) combine dA = dA ≤ dint A

and d
int A = d
A = d
A ≤ d
A ,

and, in general, there is neither a closed nor an open representative, since bA ≤ bA ≤ bint A . One invariant in the class is ∂A, but it does not completely describes the class without one of the other two A or A. We shall see that for sets verifying a uniform segment property bA = bA = bint A and for convex sets that bA = bA . As in the case of dA we identify Fb (D) with the family Cb (D) through the embedding [A]b → bA : Fb (D) → Cb (D) ⊂ C(D). When D is bounded, the space C(D) endowed with the norm f C(D) is a Banach space. Moreover, for each A ⊂ D, bA is bounded, uniformly continuous on D, and bA ∈ C(D). This will induce the following complete metric: def

ρ([A]b , [B]b ) = bA − bB C(D)

(2.7)

on Fb (D). When D is open but not necessarily bounded, we use the space Cloc (D) defined in section 2.2 of Chapter 6 endowed with the complete metric ρδ defined in (2.9) for the family of seminorms {qK } defined in (2.8). It will be shown below that Cb (D) is a closed subset of Cloc (D) and that this induces the following complete metric on Fb (D): def

ρδ ([A]b , [B]b ) =

∞ 1 qKk (bA − bB ) . 2k 1 + qKk (bA − bB )

(2.8)

k=1

The subfamilies def

Fb0 (D) = {[A]b : ∀A, A ⊂ D, ∂A = ∅, m(∂A) = 0} , def

Cb0 (D) = {bA ∈ Cb (D) : m(∂A) = 0} of Fb (D) and Cb (D) will be important. We now have the equivalent of Theorem 2.2 in Chapter 6.

2. Uniform Metric Topology

341

Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cb (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) defines a complete metric topology on Fb (D). (ii) For a bounded open subset D of RN , the set Cb (D) is compact in C(D). (iii) For a bounded open subset D of RN , the map − 3 bA → (b+ A , bA , |bA |) = (dA , d
A , d∂A ) : Cb (D) ⊂ C(D) → C(D)

is continuous: for all bA and bB in Cb (D),

 max dB − dA C(D) , d
B − d
A C(D) , d∂B − d∂A C(D) ≤ bB − bA C(D) .

(2.9)

Proof. (i) It is sufficient to consider the case for which D is bounded. Consider a sequence {An } ⊂ D, ∂An = ∅, such that bAn → f in C(D) for some f in C(D). Associate with each g in C(D) its positive and negative parts g + (x) = max{g(x), 0},

g − (x) = max{−g(x), 0}.

Then by continuity of this operation + − and d
An = b− in C(D), d A n = b+ An → f An → f

d∂An = |bAn | → |f | in C(D). By Theorem 2.2 (i) of Chapter 6, there exists a closed subset F , ∅ = F ⊂ D, such that f + = dF in RN

and f + > 0 in D.

Moreover F = RN since D is bounded. By Theorem 2.4 and the remark at the beginning of section 2.3 of Chapter 6, there exists an open subset G ⊂ D, G = RN , such that f − = d
G in RN and d
G ∈ C0 (D). Therefore, f = f + − f − = dF − d
G , f = f + − f − = dF in D and f > 0 in D. Define the sets 

 def

A+ = x ∈ RN : f (x) > 0 = x ∈ D : f (x) > 0 ∪ D, 

 def

A− = x ∈ RN : f (x) < 0 = x ∈ D : f (x) < 0 , 

 def

A0 = x ∈ RN : f (x) = 0 = x ∈ D : f (x) = 0 .

342

Chapter 7. Metrics via Oriented Distance Functions

They form a partition of RN , RN = A− ∪ A0 ∪ A+ , and RN = A0 ∪ A− = F = ∅,

A0 ∪ A+ = G = ∅

⇒ A0 = F ∩ G = ∅,

D ⊂ A+ and F = A0 ∪ A− ⊂ D, f = dA0 ∪A− − dA0 ∪A+ . If A0 were empty, RN could be partitioned into two disjoint nonempty closed subsets. Since A0 is closed ∂A0 = A0 ∩ A0 = A0 ∩ A+ ∪ A− = A0 ∩ [A+ ∪ A− ] = A0 ∩ [A+ ∪ A− ∪ ∂A− ∪ ∂A+ ] = A0 ∩ [∂A− ∪ ∂A+ ]. Moreover since A− and A+ are open, A− ⊂ A− ∪ A0 , A+ ⊂ A+ ∪ A0 , ∂A− = A− ∩ A− ⊂ [A0 ∪ A+ ] ∩ [A0 ∪ A− ] = A0 , ∂A+ = A+ ∩ A+ ⊂ [A0 ∪ A− ] ∩ [A0 ∪ A+ ] = A0 ⇒ ∂A− ∪ ∂A+ = [∂A− ∪ ∂A+ ] ∩ A0 = ∂A0 ⇒ ∂A− ∪ ∂A+ = ∂A0

⇒ ∂A0 = ∂A− ∪ (∂A+ \∂A− ).

Let Q be the subset of points in RN with rational coordinates. Define def

B 0 = int A0 ,

def

0 B+ = B 0 ∩ Q,

def

0 B− = B 0 ∩ Q,

and notice that by density of Q and Q in RN , 0 = B0 = B0 . B+ −

Consider the following new partition of RN : 0 0 RN = A+ ∪ (∂A+ \∂A− ) ∪ A− ∪ ∂A− ∪ B+ ∪ B− .

Define 0 A = A− ∪ (∂A+ \∂A− ) ∪ B+ def

0 ⇒ A = A+ ∪ ∂A− ∪ B−

and 0 = A− ∪ ∂A− ∪ (∂A+ \∂A− ) ∪ B 0 , A = A− ∪ (∂A+ \∂A− ) ∪ B+ 0 = A+ ∪ ∂A+ ∪ ∂A− ∪ B 0 . A = A+ ∪ ∂A− ∪ B−

But ∂A− ∪ ∂A+ = ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ ∂A+ ,

2. Uniform Metric Topology

343

and since ∂A− ∪ ∂A+ = ∂A0

A = A− ∪ ∂A0 ∪ int A0 = A− ∪ A0 A = A+ ∪ ∂A0 ∪ int A0 = A+ ∪ A0

⇒ ∂A = A0 = ∅

⇒ bA = dA − d
A = bA − d
A = dF − d
G = f. (ii) For the compactness, consider any sequence {bAn } ⊂ Cb (D) ⊂ C 0,1 (D). Since D is bounded, bAn and ∇bAn are both pointwise uniformly bounded in D. From Theorem 2.5 in Chapter 2, the injection of C 0,1 (D) into C(D) is compact and there exist f ∈ C(D) and a subsequence {bAnk } such that bAnk → f in C(D). From the proof of the closure in part (i), there exists A ⊂ D, ∂A = ∅, such that f = bA . (iii) For all bA and bB in Cb (D) and x in D, |bB (x)| ≤ |bA (x)| + |bB (x) − bA (x)| ⇒ | |bB (x)| − |bA (x)| | ≤ |bB (x) − bA (x)| ⇒ d∂B − d∂A C(D) =  |bB | − |bA | C(D) ≤ bB − bA C(D) . − Moreover, dA = b+ A = (|bA | + bA )/2 and d
A = bA = (|bA | − bA )/2, and necessarily ± b± B − bA C(D) ≤ bB − bA C(D) .

By combining the above three inequalities we get (2.9). We have the analogue of Theorem 2.3 of Chapter 6 and its corollary for A, A, and ∂A. In the last case it takes the following form (to be compared with T. J. Richardson [1, Lem. 3.2, p. 44] and S. R. Kulkarni, S. K. Mitter, and T. J. Richardson [1] for an application to image segmentation). Corollary 1. Let D be a nonempty open (resp., bounded open) holdall in RN . Define for a subset S of RN the sets def

Hb (S) = {bA ∈ Cb (D) : S ⊂ ∂A} , def

Ib (S) = {bA ∈ Cb (D) : ∂A ⊂ S} , def

Jb (S) = {bA ∈ Cb (D) : ∂A ∩ S = ∅} . (i) Let S be a subset of RN . Then Hb (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then Ib (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then Jb (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then Jb (S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then Ib (S) is open in Cloc (D) (resp., C(D)). (iv) For D bounded, associate with an equivalent class [A]b the number # ∂A = number of connected components of ∂A.

344

Chapter 7. Metrics via Oriented Distance Functions Then the map  def

[A]b → # ∂A : F#b (D) = [A]b : ∀A ⊂ D, ∂A = ∅, and # ∂A < ∞ → R is lower semicontinuous. For a fixed number c > 0, the subset {bA ∈ Cb (D) : # ∂A ≤ c} of Cb (D) is compact in C(D). In particular, the subset {bA ∈ Cb (D) : ∂A is connected} of Cb (D) is compact in C(D).

Proof. The proof is the same as for Theorem 2.3 of Chapter 6 using the Lipschitz continuity of the map bA → d∂A = |bA | from C(D) to C(D).

3

Projection, Skeleton, Crack, and Differentiability

This section is the analogue of section 3 in Chapter 6: connection between the gradient of bA and the projection onto ∂A and the characteristic functions associated with ∂A, singularities of the gradients, and the notions of skeleton1 and cracks. Let A ⊂ RN , ∅ = ∂A, and recall the notation  def

(3.1) Sing (∇bA ) = x ∈ RN : ∇bA (x)  for the set of singularities of the gradient of bA , def

Π∂A (x) = {z ∈ ∂A : |z − x| = d∂A (x)}

(3.2)

for the set of projections of x onto ∂A (when Π∂A (x) is a singleton, the unique element is denoted by p∂A (x)), and 

 def

Sk (∂A) = x ∈ RN : Π∂A (x) is not a singleton = x ∈ RN : ∇d2∂A (x)  (3.3) for the skeleton of ∂A. Since Π∂A (x) is a singleton for all x ∈ ∂A we have Sk (∂A) ⊂ RN \∂A. In general, the functions bA and d∂A are different and Sing ∇bA is usually smaller than Sing (∇d∂A ) = Sk (∂A) ∪ Ck (∂A). As a result, using the oriented distance function bA rather than d∂A requires a new definition of the set of cracks with respect to bA that will be justified in Theorem 3.1 (iv). Definition 3.1. Given A ⊂ RN , ∅ = ∂A, the set of b-cracks of A is defined as def

Ck b (A) = Sing ∇bA \Sk (∂A).

(3.4)

Sk (∂A), Ck b (A), and Sing (∇bA ) have zero N -dimensional Lebesgue measure, since bA is Lipschitz continuous and hence differentiable almost everywhere. 1 Our definition of a skeleton does not exactly coincide with the one used in morphological `re [1]). mathematics (cf., for instance, G. Matheron [1] or A. Rivie

3. Projection, Skeleton, Crack, and Differentiability

345

Remark 3.1. (1) In general Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A), but Ck b (A) can be strictly smaller than Ck (∂A). Consider the open ball of unit radius A = B1 (0):   x 1 x i xj bA (x) = |x| − 1, ∇bA (x) = , D2 bA (x)ij = δij − , |x| |x| |x| |x| Sing (∇bA ) = {0},

Sk (∂A) = {0},

and

Ck b (A) = ∅.

It is easy to check that for all x ∈ ∂A, ∇bA (x) is the outward unit normal, and D 2 bA (x) is the second fundamental form with eigenvalues 0 for the space spanned by the normal x and 1 for the tangent space to ∂A in x orthogonal to x. But d∂A (x) = ||x| − 1| , Sing (∇d∂A ) = {0} ∪ ∂B1 (0),

Sk (∂A) = {0},

and

Ck (∂A) = ∂B1 (0).

Using bA rather than d∂A or dA removes artificial singularities. (2) In general Sk (∂A) ⊂ RN \∂A, but if ∂A has positive reach h > 0 in the sense of Definition 6.1 in section 6 of Chapter 6 (there exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)), the skeleton will remain at a distance h from ∂A. The function bA enjoys properties similar to the ones of dA and d
A since |bA | = d∂A = max{dA , d
A } and we have the analogues of Theorems 3.2 and 3.3 in Chapter 6. Recall the definition of def

f∂A (x) =

 1 2 |x| − d2∂A (x) . 2

Theorem 3.1. Let A be a subset of RN such that ∅ = ∂A, and let x ∈ RN . (i) For all x ∈ RN , d2∂A (x) = b2A (x), f∂A (x) =

 1 2 |x| − b2A (x) 2

is convex and continuous, the function x →

|x|2 k∂A,h (x) = − |bA |(x) : RN \Uh (∂A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (∂A), ∇f∂A and ∇b2A belong to  N BVloc (RN )N , and ∇d∂A ∈ BVloc RN \∂(∂A) . (ii) The set Π∂A (x) is nonempty and compact, and ∀x ∈ / ∂A

Π∂A (x) ⊂ ∂(∂A)

and

∀x ∈ ∂A

Π∂A (x) = {x}.

/ A, Π∂A (x) = Π
A (x) ⊂ ∂A. For x ∈ / A, Π∂A (x) = ΠA (x) ⊂ ∂A and for x ∈ Moreover, ∂(∂A) = ∂A ∪ ∂A.

346

Chapter 7. Metrics via Oriented Distance Functions

(iii) For all x and v in RN , the Hadamard semiderivative2 of d2A always exists, b2A (x + tv) − b2A (x) = min 2(x − z) · v, t0 t z∈Π∂A (x) f∂A (x + tv) − f∂A (x) dH f∂A (x; v) = lim = σΠ∂A (x) (v) = σco Π∂A (x) (v), t0 t dH b2A (x; v) = lim

where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular,

 Sk (∂A) = x ∈ RN : ∇b2A (x)  ⊂ RN \∂A.

(3.5)

Given v ∈ RN , for all x ∈ RN \∂(∂A) the Hadamard semiderivative of bA exists, dH bA (x; v) =

1 min (x − p) · v, bA (x) p∈ΠA (x)

(3.6)

and for x ∈ ∂(∂A), dH bA (x; v) exists if and only if lim

t0

bA (x + tv) t

exists.

(3.7)

(iv) The following statements are equivalent: (a) b2A (x) is (Fr´echet) differentiable at x. (b) b2A (x) is Gateaux differentiable at x. (c) Π∂A (x) is a singleton. (v) ∇f∂A (x) exists if and only if Π∂A (x) = {p∂A (x)} is a singleton. In that case 1 p∂A (x) = ∇f∂A (x) = x − ∇b2A (x). (3.8) 2 For all x and y in RN ,  1 2  1 2 |y| − b2A (y) ≥ |x| − b2A (x) + p(x) · (y − x), (3.9) ∀p(x) ∈ Π∂A (x), 2 2 or equivalently, ∀p(x) ∈ Π∂A (x),

b2A (y) − b2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.10)

For all x and y in RN , ∀p(x) ∈ Π∂A (x), ∀p(y) ∈ Π∂A (y), 2A

(p(y) − p(x)) · (y − x) ≥ 0.

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Definition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.11)

3. Projection, Skeleton, Crack, and Differentiability

347

(vi) The functions p∂A : RN \Sk (∂A) → RN

and

∇b2A : RN \Sk (∂A) → RN

are continuous. For all x ∈ RN \Sk (∂A) 1 p∂A (x) = x − ∇b2A (x) = ∇f∂A (x). 2 In particular, for all x ∈ ∂A, Π∂A (x) = {x}, ∇b2A (x) = 0, 

Sk (∂A) = x ∈ RN : ∇b2A (x)  and ∇bA (x)  ⊂ RN \∂A,

 Ck b (A) = x ∈ RN : ∇b2A (x) ∃ and ∇bA (x)  ⊂ ∂(∂A),

(3.12)

(3.13)

and Sing (∇bA ) = Sk (∂A) ∪ Ck b (A). (vii) The function ∇bA : RN \ (Sk (∂A) ∪ ∂(∂A)) → RN is continuous. For all x ∈ int ∂A, ∇bA (x) = 0; for all x ∈ RN \ (Sk (∂A) ∪ ∂A),   x − p∂A (x)   N   \A, , x ∈ R  ∇b2A (x)  |x − p∂A (x)| = , |∇bA (x)| = 1. ∇bA (x) =  2 bA (x)  x − p∂A (x)  −  , x ∈ RN \A, |x − p∂A (x)| (3.14) If ∇d∂A (x) exists for some x ∈ ∂A, then ∇bA (x) = ∇d∂A (x) = 0. In particular, ∇bA (x) = 0 almost everywhere in ∂A. (viii) The functions b2A and bA are differentiable almost everywhere and m(Sk (∂A)) = m(Ck b (A)) = m(Sing (∇bA )) = 0. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.15)

If ∂A = ∅, then either A = RN or A = ∅. (ix) Given A ⊂ RN , ∂A = ∅, χ∂A (x) = 1 − |∇bA (x)| in RN \Sing (∇bA ),

(3.16)

the identity holds for almost all x in RN , and the set def

∂b A = {x ∈ ∂A : ∇bA (x) exists and ∇bA (x) = 0} ⊂ ∂(∂A) has zero N -dimensional Lebesgue measure.

(3.17)

348

Chapter 7. Metrics via Oriented Distance Functions

(x) Let µ be a measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1] and D ⊂ RN be bounded open such that µ(D) < ∞. The map bA → µ(∂A) : Cb (D) → R

(3.18)

is upper semicontinuous with respect to the topology of uniform convergence on Cb (D). Moreover, if bAn → bA in C(D) for bAn and bA in Cb (D), then µ(int A) ≤ lim inf µ(int An ) ≤ lim sup µ(An ) ≤ µ(A).

(3.19)

If µ(∂A) = 0, the map bA → µ(A) : Cb (D) → R is continuous in bA . def

(xi) Given x ∈ RN , α ∈ [0, 1], p ∈ Π∂A (x), and xα = p + α (x − p), then bA (xα ) = α bA (x)

∀α ∈ [0, 1] ,

and

Π∂A (xα ) ⊂ Π∂A (x).

In particular, if Π∂A (x) is a singleton, then Π∂A (xα ) is a singleton and ∂A, then for ∇b2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = all 0 < α ≤ 1 ∇bA (xα ) exists and ∇bA (xα ) = ∇bA (x). Proof. (i) to (v) follow from Theorems 3.2 and 3.3 of Chapter 6 using the identity b2A = d2∂A . (v) To complete the proof of identities (3.13), recall that the projection of all points of ∂A onto ∂A being a singleton, Sk (∂A ⊂ RN \∂A. Therefore, for any x ∈ Sk (∂A), we have bA (x) = 0 and if ∇b2A (x) does not exist, then ∇bA (x) cannot exist. This sharpens the characterization (3.5) of Sk (∂A). From this we readily get the characterization of Ck b (A) = Sing (∇bA )\Sk (∂A) and Ck b (A) ⊂ ∂A. This last property can be improved to Ck b (A) ⊂ ∂(∂A) by noticing that for x ∈ int ∂A, bA (x) = 0 and ∇b2A (x) = ∇bA (x) = 0. (vi) From Theorem 3.3 (iv) of Chapter 6. (vii) In RN \A, bA = dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in N R \A, bA = −d
A and ∇bA is continuous in RN \(Sk (A) ∪ A); in int ∂A, bA = dA = d
A = 0 and ∇bA = 0. So, in view of part (v) it is continuous in RN \(Sk (A) ∪ A) ∪ RN \(Sk (A) ∪ A) ∪ int ∂A = RN \ (Sk (∂A) ∪ ∂(∂A)) . If x ∈ RN \A, then bA = −d
A and repeat the proof of Theorem 3.3 (vi) in Chapter 6 to obtain −∇bA (x) = ∇d
A (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| −bA (x)

and p∂A (x) ∈ ∂A is unique. Similarly, for x ∈ RN \A, bA = dA , ∇bA (x) = ∇dA (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| bA (x)

and p∂A (x) ∈ ∂A is unique. In both cases |∇bA (x) = 1.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

349

Finally, from Theorem 3.3 (i) in Chapter 6, whenever d∂A is differentiable at a point x ∈ ∂A, ∇d∂A (x) = 0. But for all v in RN and t > 0 bA (x + tv) − bA (x) |bA (x + tv)| d∂A (x + tv) − d∂A (x) = = t t t since |bA | = d∂A . Therefore, if ∇d∂A (x) exists, ∇bA (x) exists and ∇bA (x) = ∇d∂A (x) = 0. Finally, since ∇d∂A (x) exists almost everywhere in RN , then for almost all x ∈ ∂A, ∇bA (x) = 0. (viii) From Theorem 3.3 (vii) in Chapter 6 and the fact that Ck b (A) ⊂ C(∂A). Indeed, for x ∈ ∂A, if ∇dA (x) and ∇d
A (x) exist, they are 0 and for all t > 0 and v ∈ RN bA (x + tv) − bA (x) dA (x + tv) − dA (x) d
A (x + tv) − d
A (x) = →0 − t t t as t  0. Hence ∇bA (x) = 0. As a consequence, if ∇bA (x) does not exist, then either ∇dA (x) or ∇d
A (x) does not exist and Ck b (A) ⊂ C(A) ∪ C(A) = C(∂A) ⊂ ∂(∂A) from Theorem 3.3 (vii) in Chapter 6. (ix) Since bA is a Lipschitzian function, it is differentiable almost everywhere in RN , and in view of the previous considerations, when it is differentiable |∇bA (x)| =

1, x ∈ / ∂A, 0 a.e. in ∂A

⇒ χ∂A (x) = 1 − |∇bA (x)|

a.e. in RN .

Therefore, the set ∂b A has at most a zero measure. (x) From Theorem 4.1 in Chapter 6 applied to A, A, and ∂A. (xi) Same proof as the one of Theorem 3.3 (ix) in Chapter 6. Remark 3.2. In general ∇dA (x) and ∇d
A (x) do not exist for x ∈ ∂A. This is readily seen by constructing the directional derivatives for the half-space A = {(x1 , x2 ) ∈ R2 : x1 ≤ 0}

(3.20)

at the point (0, 0). Nevertheless ∇bA (0, 0) exists and is equal to (1, 0), which is the outward unit normal at (0, 0) ∈ ∂A to A. Note also that for all x ∈ ∂A, |∇bA (x)| = 1. This is possible since mN (∂A) = 0. If ∇bA (x) exists, then it is easy to check that |∇bA (x)| ≤ 1. For sufficiently smooth domains, the subset ∂b A of the boundary ∂A coincides with the reduced boundary of finite perimeter sets.

4 4.1

W 1,p (D)-Metric Topology and the Family Cb0 (D) Motivations and Main Properties

1,p From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to Wloc (RN ) for all p ≥ 1. So the previous constructions for dA and d
A can be repeated with the space

350

Chapter 7. Metrics via Oriented Distance Functions

1,p (RN ) in place of Cloc (D) to generate new W 1,p metric topologies on the family Wloc Cb (D). Moreover the other distance functions can be recovered from the map − bA → (b+ A , bA , |bA |) = (dA , d
A , d∂A )

and the characteristic functions from the maps bA → b− A = d
A → χint A = |∇d
A |,

(4.1)

bA → = dA → χint
A = |∇dA |, bA → χ∂A = 1 − |∇bA |.

(4.2) (4.3)

b+ A

One of the advantages of the function bA is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of int A, int A, and ∂A, that is, continuity of the volume of these sets. It will be useful to introduce the following set and terminology. Definition 4.1. (i) A set A ⊂ RN , ∂A = ∅, is said to have a thin 3 boundary if mN (∂A) = 0. (ii) Denote by def

Cb0 (D) = {bA : ∀A, A ⊂ D, ∂A = ∅ and m(∂A) = 0}

(4.4)

the subset of oriented distance functions with thin boundaries. In this section we give the analogues of Theorems 4.2 and 4.3 in Chapter 6. Theorem 4.1. Let D be an open (resp., bounded open) subset of RN . 1,p (i) The topologies induced by Wloc (D) (resp., W 1,p (D)) on Cb (D) are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cb (D) is closed in Wloc def

ρD ([A2 ]b , [A1 ]b ) =

∞ bA2 − bA1 W 1.p (B(0,n)) 1 n 2 1 + bA2 − bA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ]b , [A1 ]b ) = bA2 − bA1 W 1,p (D) ) defines a complete metric structure on Fb (D). (iii) For p, 1 ≤ p < ∞, the map 1,p bA → χ∂A = 1 − |∇bA | : Cb (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and bA1 and bA2 in Cb (D), χ∂A2 − χ∂A1 Lp (K) ≤ ∇bA2 − ∇bA1 Lp (K) ≤ bA2 − bA1 W 1,p (K) . In particular, Cb0 (D) is a closed subset of Cb (D) for the W 1,p -topology. 3 This

terminology should not be confused with the notion of thin set in the capacity sense.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

351

(iv) Let D be a bounded open subset of RN . For each bA ∈ Cb (D), bA W 1,p (D) =  |bA | W 1,p (D) = d∂A W 1,p (D) , dA W 1,p (D) ≤ bA W 1,p (D) ,

d
A W 1,p (D) ≤ bA W 1,p (D) .

(4.5)

The map − 1.p (D) → W 1.p (D)3 (4.6) bA → (b+ A , bA , |bA |) = (dA , d
A , d∂A ) : Cb (D) ⊂ W

is continuous. (v) Let D be an open (resp., bounded open) subset of RN . For all p, 1 ≤ p < ∞, the map bA → (χ∂A , χint A , χint
A ) 1,p (D) → Lploc (D)3 (resp.,W 1,p (D) → Lp (D)3 ) : Wloc

(4.7)

is continuous. Proof. The proofs of (i) and (ii) are essentially the same as the proof of Theorem 4.2 of Chapter 6 using the properties established for Cb (D) in C(D) of Theorem 2.2. (iii) From Theorem 3.1 (ix) for any two subsets A1 and A2 of D with nonempty boundaries and for any open U in D |∇bA2 | ≤ |∇bA1 | + |∇bA2 − ∇bA1 | + |∇bA2 − ∇bA1 | ⇒ |χ∂A1 − χ∂A2 | ≤ |∇bA2 − ∇bA1 |

⇒ χ∂A1 ≤ χ∂A2  |χ∂A2 − χ∂A1 |p dx ≤ ∇bA2 − ∇bA1 pLp (U ) ≤ bA2 − bA1 pW 1,p (U ) ⇒ U

for p, 1 ≤ p < ∞, and with the ess sup norm for p = ∞. The closure of Cb0 (D) follows from the continuity of the maps  bA → m(∂A ∩ U ) = χ∂A dx : W 1,p (U ) → R ∂A∩U

for all bounded open subsets U of D. (iv) First observe that |bA (x)| = dA (x) + d
A (x) and ∇|bA (x)| = ∇dA (x) + ∇d
A (x), bA (x) = dA (x) − d
A (x) and ∇bA (x) = ∇dA (x) − ∇d
A (x) and since ∇bA = ∇dA = ∇d
A = 0 almost everywhere on ∂A = A ∩ A, then |∇bA (x)| = |∇dA (x)| + |∇d
A (x)| = |∇|bA (x)| | for almost all x in RN . From this we readily get (4.5). However, this is not sufficient to prove the continuity since the map (4.6) is not linear. To get around this consider a sequence {bAn } ⊂ Cb (D) converging to bA in W 1,p (D). From Theorem 2.2 (iii), |bAn | = d∂An , b+ An = + − dAn , and b− = d converge to |b | = d , b = d , and b = d in C(D) A ∂A A
An
A An A A

352

Chapter 7. Metrics via Oriented Distance Functions

and hence in W 1,p (D)-weak. To prove that the convergence is strong, consider the L2 -norm  |∇dAn − ∇dA |2 + |∇d
An − ∇d
A |2 dx D  = |∇dAn |2 + |∇dA |2 + |∇d
An |2 + |∇d
A |2 D − 2 ∇dAn · ∇dA − 2 ∇d
An · ∇d
A dx  |∇bAn |2 + |∇bA |2 − 2 ∇dAn · ∇dA − 2 ∇d
An · ∇d
A dx = D   2 2 2 → 2 |∇bA | − 2 |∇dA | − 2 |∇d
A | dx = 2 |∇bA |2 − 2 |∇bA |2 dx = 0 D

D

by weak L -convergence of ∇dAn and ∇d
An to ∇dA and ∇d
A . So both sequences dAn and d
An converge in W 1,2 (D)-strong, and hence from part (i) in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (v) It is sufficient to prove the result for D bounded open. From part (iii) it is true for χ∂A , and from part (iv) and Theorem 4.2 (ii) and (iii) of Chapter 6 for the other two. 2

4.2

Weak W 1,p -Topology

Theorem 4.2. Let D be a bounded open domain in RN . (i) If {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {bAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely, if {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cb (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Proof. (i) Recall that for D bounded there exists a constant c > 0 such that for all bA ∈ Cb (D) |bA (x)| ≤ c and |∇bA (x)| ≤ 1 a.e. in D. If {bAn } weakly converges in W 1,p (D) for some p ≥ 1, then {bAn } weakly converges in Lp (D), {∇bAn } weakly converges in Lp (D)N . By Lemma 3.1 (i) in Chapter 5 both sequences weakly converge for all p ≥ 1 and hence {bAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {bAn } converges in C(D), then by Theorem 2.2 (ii) there exists bA ∈ Cb (D) such that bAn → bA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N     ∇bAn · ϕ dx = − bAn div ϕ dx → − bA div ϕ dx = ∇bA · ϕ dx. D

D

D

D

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

353

By density of D(D) in L2 (D), ∇bAn → ∇bA in L2 (D)N -weak and hence bAn → bA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D), there exist a subsequence, still indexed by n, and bA such that bAn → bA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit, bA = f . Therefore, all convergent subsequences in C(D) converge to the same limit. So the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {bAn } in Cd (D). From Theorem 2.2 (ii) Cb (D) is compact and there exist a subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cb (D) is compact in W 1,p (D)-weak. For sets with thin boundaries the strong and weak W 1,p (D)-convergences of elements of Cb0 (D) to an element of Cb0 (D) are equivalent as in the case of Theorem 3.2 in section 3.2 of Chapter 5 for characteristic functions in Lp (D)-strong. Theorem 4.3. Let ∅ = D ⊂ RN be bounded open. (i) Let {An } be a sequence of subsets of D such that ∂An = ∅ and m(∂An ) = 0 ¯ be such that ∂A = ∅ and m(∂A) = 0. Then and A ⊂ D ⇒ bAn → bA in W 1,2 (D)-strong,

bAn  bA in W 1,2 (D)-weak

and hence in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (ii) The map bA → mN (int A) = mN (A) = mN (A) : Cb0 (D) → R

(4.8)

is continuous with respect to the W 1,p -topology on Cb0 (D) for all p, 1 ≤ p < ∞. Proof. (i) From Theorem 4.1 (i) on the equivalence of the W 1,p -topologies, it is sufficient to prove the result for p = 2. By Theorem 4.2 (ii), the weak L2 -convergence of {bAn } to bA implies the strong convergence in C(D) and hence in L2 (D)-strong. Since, for all n ≥ 1, m(∂An ) = 0 = m(∂A), |∇bA | = 1 = |∇bAn | almost everywhere in D by Theorem 3.1 (vii). As a result   |∇bAn − ∇bA |2 dx = |∇bAn |2 + |∇bA |2 − 2∇bAn · ∇bA dx D D    2 χ∂A dx = 0. = 2 (1 − ∇bAn · ∇bA ) dx → 2 (1 − |∇bA | ) dx = 2 D

D

D

Therefore ∇bAn → ∇bA in L2 (D)N -strong and bAn → bA in W 1,2 (D)-strong, since the convergence bAn → bA in L2 (D)-strong follows from the weak convergence in W 1,2 (D). The convergence in W 1,p (D)-strong follows from the equivalence of the topologies on Cb (D) (cf. Theorem 4.1 (i)). (ii) From Theorem 3.1 (x) since mN (∂A) = 0 for bA ∈ Cb0 (D) and Cb0 (D) is closed in the W 1,p -topology.

354

5

Chapter 7. Metrics via Oriented Distance Functions

Boundary of Bounded and Locally Bounded Curvature

We introduce the family of sets with bounded or locally bounded curvature which are the analogues for bA of the sets of exterior and interior bounded curvature associated with dA and d
A in section 5 of Chapter 6. They include C 1,1 -domains, convex sets, and the sets of positive reach of H. Federer [3]. They lead to compactness theorems for subfamilies of Cb (D) in W 1,p (D). Definition 5.1. (i) Given a bounded open nonempty holdall D in RN and a subset A of D, ∂A = ∅, its boundary ∂A is said to be of bounded curvature with respect to D if ∇bA ∈ BV(D)N .

(5.1)

This family of sets will be denoted as follows:  def

BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N . (ii) Given a subset A of RN , ∂A = ∅, its boundary ∂A is said to be of locally bounded curvature if ∇bA belongs to BVloc (RN )N . The family of sets whose boundary is of locally bounded curvature will be denoted by  def

BCb = bA ∈ Cb (RN ) : ∇bA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, and in Theorem 5.1 of Chapter 6 for sets with a locally bounded curvature, it is sufficient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. Let A, ∂A = ∅, be a subset of RN . Then ∂A is of locally bounded curvature if and only ∀x ∈ ∂(∂A), ∃ρ > 0 such that ∇bA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Definition 5.1 and Theorem 3.1 (i). Theorem 5.2. (i) Let D be a nonempty bounded open Lipschitzian holdall in RN and let A ⊂ D, ∂A = ∅. If ∇bA ∈ BV(D)N , then ∇χ∂A M 1 (D) ≤ 2 D 2 bA M 1 (D) ,

χ∂A ∈ BV(D),

and ∂A has finite perimeter with respect to D. (ii) For any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N and ∂A has locally finite perimeter.

⇒

χ∂A ∈ BVloc (RN ),

5. Boundary of Bounded and Locally Bounded Curvature

355

Proof. Given ∇bA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇bA in L1 (D)N

and Duk M 1 (D) → D2 bA M 1 (D)

as k goes to infinity, and since |∇bA (x)| ≤ 1, this sequence can be chosen in such a way that ∀k ≥ 1, |uk (x)| ≤ 1. This follows from the use of mollifiers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N 



−

 (|∇bA | − 1) div V dx =

|∇bA |2 div V dx.

2

χ∂A div V dx = D

D

D

For each uk    |uk |2 div V dx = −2 [ ∗Duk ]uk · V dx = −2 uk · [Duk ]V dx, D

D

D

where ∗Duk is the transpose of the Jacobian matrix Duk and   |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore, as k goes to infinity,   χ∂A div V dx = |∇bA |2 div V dx ≤ 2D2 bA M 1 V C(D) . D

D

Therefore, ∇χ∂A ∈ M 1 (D)N .

5.1

Examples and Limit of Tubular Norms as h Goes to Zero

It is informative to compute the Laplacian of bA for a few examples. The first three examples are illustrated in Figure 7.1. Example 5.1 (Half-plane in R2 ). (cf. Example 5.1 in Chapter 6). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that bA (x1 , x2 ) = x1 ,

∇bA (x1 , x2 ) = (1, 0),

∆bA (x) = 0,

356

Chapter 7. Metrics via Oriented Distance Functions

and

 b∂A (x1 , x2 ) = |x1 |,

∇b∂A (x1 , x2 ) =  ∆b∂A , ϕ = 2 ϕ dx.

 x1 ,0 , |x1 |

∂A

Figure 7.1. ∇bA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). (cf. Example 5.2 in Chapter 6). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly, x bA (x) = |x| − R, ∇bA (x) = , |x|  1 ∆bA , ϕ = ϕ dx. 2 |x| R Also b∂A (x) = |x| − R ,

∇b∂A (x) =



 ϕ ds −

∆b∂A , ϕ = 2 ∂A

A

|x| > R,

x , |x| x − |x| ,

1 ϕ dx + |x|

|x| < R,


A

1 ϕ dx. |x|

Again ∆b∂A contains twice the boundary measure on ∂A. Example 5.3 (Unit square in R2 ). (cf. Example 5.3 in Chapter 6). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1} . Since A is symmetrical with respect to both axes, it is sufficient to specify bA in the first quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in

5. Boundary of Bounded and Locally Bounded Curvature

357

the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence for 1 ≤ i ≤ 4   |x2 | − 1, bA (x) = |x − ci |,   |x1 | − 1,

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

and for the whole plane ∆bA , ϕ =

4  i=1

D3 ∩Qi

  (0, 1), x−ci , ∇bA (x) = |x−c i|   (1, 0),

√ 1 ϕ dx + 2 |x − ci |

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

 ϕ dx, D1 ∩D2

where D1 ∩ D2 is made up of the two diagonals of the square where ∇bA has a singularity (that is, the skeleton). Moreover, for 1 ≤ i ≤ 4   x ∈ D2 ∩ Qi , ||x2 | − 1| , b∂A (x) = |x − ci |, x ∈ D3 ∩ Qi ,   ||x1 | − 1| , x ∈ D1 ∩ Qi , and ∆b∂A , ϕ =

4  i=1

D3 ∩Qi

√ 1 ϕ dx − 2 |x − ci |



 ϕ dx + 2 D1 ∩D2

ϕ dx. ∂A

Notice that the structures of the Laplacian are similar to the ones observed in the previous examples except for the presence of a singular term along the two diagonals of the square. All C 2 -domains with a compact boundary belong to all the categories of Definition 5.1 and of Definition 5.1 of Chapter 6. The h-dependent norms D2 bA M 1 (Uh (∂A)) , D 2 d
A M 1 (Uh (
A)) , and D2 dA M 1 (Uh (A)) are all decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A. Example 5.4. If A ⊂ RN is of class C 2 with compact boundary, then lim D2 bA M 1 (Uh (∂A)) = 0.

h0

Example 5.5. Let A = {xi }Ii=1 be I distinct points in RN . Then ∂A = A, and lim D 2 bA M 1 (Uh (∂A)) =

h0

2 I − 1, N = 1, 0, N ≥ 2.

358

Chapter 7. Metrics via Oriented Distance Functions

Example 5.6. Let A be a closed line in RN of length L > 0. Then ∂A = A, and lim D2 bA M 1 (Uh (∂A)) =

h0

2 L, N = 2, 0, N ≥ 3.

Example 5.7. Let N = 2. For the finite square and the ball of finite radius lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorff measure (cf. Examples 5.2 and 5.3 in Chapter 6). Also, looking at ∆bA in Uh (A) as h goes to zero provides information about Sk (A). Example 5.8. Let A be the unit square in R2 . Then lim ∆bA M 1 (Uh (A))

h0

√ 2 = H1 (Sk (A)), 2

where H1 is the one-dimensional Hausdorff measure and Sk(A) = Skint (A) is the skeleton of A made up of the two interior diagonals (cf. Example 5.3). It would seem that in general  lim ∆bA M 1 (Uh (A)) = | [∇bA ] · n| dH1 , h0

Sk(A)

where [∇bA ] is the jump in ∇bA and n is the unit normal to Sk(A) (if it exists!).

6

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 7.1 of Chapter 6 that dA can be approximated by the distance function dAr of the dilated set Ar = {x ∈ RN : dA (x) ≤ r}, r > 0, of A in the W 1,p -topology. The approximation can also be achieved with the oriented distance function: bAr (x) → bA (x) uniformly in RN . But it requires that the measure of the boundary ∂A be zero to get the approximation in the W 1,p -topology. Lemma 6.1. Given A ⊂ RN , ∂ A¯ = ∅ if and only if A = ∅ and there exists h > 0 such that Ah = RN . Proof. If ∂ A¯ = ∅, then A = ∅ and A = ∅, A = ∅ and A = ∅, and RN = A = ∩h>0 Ah

⇒ ∃h > 0 such that Ah = RN .

Conversely, A ⊃ A = ∅ and, for some h > 0, A = ∩k>0 Ak ⊂ Ah = RN . Therefore A = ∅ and A = ∅ imply that ∂A = ∅.

6. Approximation by Dilated Sets/Tubular Neighborhoods

359

Theorem 6.1. Given A ⊂ RN such that ∂A = ∅, there exists h > 0 such that for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) For all r, 0 < r ≤ h, A ⊃ Ar = ∅, ∂Ar = ∅, bUr (A) = bAr , and Ur (A) and Ar are invariant open and closed representatives in the equivalence class [Ar ]b . Moreover, d
Ur (A) (x) = d
Ar (x) = 0,

if dA (x) ≥ r,

d
Ur (A) (x) = d
Ar (x) ≥ r − dA (x),

if 0 < dA (x) < r,

(6.1)

d
Ur (A) (x) = d
Ar (x) ≥ d
A (x) + r,

if dA (x) = 0,

(6.2)

∀x ∈ R , N

0 ≤ bA (x) − bAr (x) ≤ r,

(6.3)

and, as r → 0, bUr (A) (x) = bAr (x) → bA (x) uniformly in RN , χAr → χA in Lploc (RN ),

∀p, 1 ≤ p < ∞.

(ii) The following conditions are equivalent: (a) m(∂A) = 0. 1,p (RN ) for some p, 1 ≤ p < ∞. (b) As r → 0, bUr (A) = bAr → bA in Wloc 1,p (RN ) for some p, 1 ≤ p < ∞. (c) As r → 0, d
Ur (A) = d
Ar → d
A in Wloc

If (a), (b), or (c) is verified, then (b) and (c) are verified for all p, 1 ≤ p < ∞. In particular, (a) is verified for all bounded sets A = ∅ in RN such that m(∂A) = 0. Proof. The existence of the h > 0 such that Ah = RN and A = ∅ is a consequence of Lemma 6.1. From this for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) From the initial remarks, RN = A ⊃ A¯ ⊃ Ar ⊃ Ah = ∅ and ∂Ar = ∅ and ∂A = ∅, and the functions d
A , d
Ar , bA , and bAr are well-defined. From Theorem 3.1 of Chapter 6, Ar = Ur (A), and Ar = Ur (A), and ∂Ar = d−1 A {r}. Hence, dAr = dUr (A) . Hence, dAr = dUr (A) , d
Ar = d
Ur (A) , and bAr = bUr (A) . For x ∈ RN such that dA (x) ≥ r, x ∈ Ur (A) and d
Ur (A) (x) = 0. Inequality (6.1). For all y ∈ Ur (A) and all p ∈ A, r < dA (y) ≤ |y − p| ≤ |y − x| + |x − p|, r ≤ inf |y − x| + inf |x − p| = d
Ur (A) (x) + dA (x). y∈
Ur (A)

p∈A

Inequality (6.2). Since A ∩ Ur (A) = ∅, for x ∈ A, there exists pr ∈ ∂Ur (A) = d−1 A {r} such that d
Ur (A) (x) = |pr − x| > 0. Therefore dA (pr ) = r. Either d
A (x) = 0 or d
A (x) > 0. If d
A (x) = 0, then d
Ur (A) (x) = |pr − x| ≥ dA (pr ) = r = d
A (x) + r.

360

Chapter 7. Metrics via Oriented Distance Functions

def If d
A (x) > 0, then x ∈ / A and x ∈ int A¯ = A\∂A, Bx = B(x, d
A (x)) ⊂ int A, and Bx ⊂ A. Define pr − x def p = x + d
A (x) ∈ Bx ⇒ |p − x| = d
A (x) and p ∈ Bx ⊂ A. |pr − x|

By construction d
Ur (A) (x) = |x − pr | = |x − p| + |p − pr | = d
A (x) + |p − pr |, |p − pr | ≥ inf |p − pr | ≥ inf |p − pr | = dA (pr ) = r p ∈Bx

p ∈A

and d
Ur (A) (x) ≥ d
A (x) + r. Inequality (6.3). From Theorem 2.1 (ii), A¯ ⊂ Ar implies that bA¯ (x) ≥ bAr (x). As for the second part of inequality (6.3), we use identity (7.2) from Theorem 7.1 in Chapter 6 for dUr (A) = dAr combined with (6.2) for x ∈ A and with (6.1) for x ∈ RN \A: r ≤ dA (x)

⇒ x ∈ Ur (A) ⊂ A

0 < dA (x) < r

⇒ bUr (A) (x) = dUr (A) (x) = dA (x) − r = bA (x) − r, ⇒ dUr (A) (x) = 0 and d
A (x) = 0 ⇒ bUr (A) (x) = −d
Ur (A) (x) ≤ dA (x) − r = bA (x) − r,

⇒ d
Ur (A) (x) = d
A (x) = 0

⇒ x ∈ A and dUr (A) (x) = 0 ⇒ bUr (A) (x) = −d
Ur (A) (x) ≤ −d
A (x) − r = bA (x) − r.

dA (x) = 0

(ii) If m(∂A) = 0, from part (i), d
Ur (A) (x) → d
A (x) implies that ∇d
Ur (A)  ∇d
A in L2 (D)N -weak for all bounded open sets. Consider the estimate     |∇d
Ur (A) − ∇d
A |2 dx = |∇d
Ur (A) |2 + |∇d
A |2 − 2 ∇d
Ur (A) · ∇d
A dx D D    1 − χ
Ur (A) + 1 − χ − 2 ∇d
Ur (A) · ∇d
A dx =
A

D

from the identities of Theorem 3.3 (viii) in Chapter 6. As r → 0, {Ur (A)} is a family of increasing closed sets and Ur (A)  ∪r>0 Ur (A) =  ∩r>0 Ur (A) = A

⇒ m(D ∩ Ur (A)) → m(D ∩ A).

Going to the limit    1 − χ
Ur (A) + 1 − χ − 2 ∇d
Ur (A) · ∇d
A dx
A    D    2 − χ
A − χ − 2 |∇d
A |2 dx = χ − χ
A dx = → χ∂A dx = 0, D


A

since |∇d
A | = 1 − χ


A

D


A

D

, ∂A = A\A, and m(∂A) = 0. Therefore d
Ur (A) → d
A

1,p in W (D) and hence in Wloc (RN ) for all p, 1 ≤ p < ∞. So, from Theorem 7.1 1,p in Chapter 6, bUr (A) = dUr (A) − d
Ur (A) → dA − d
A = bA in Wloc (RN ) for all p, 1 ≤ p < ∞. 1,2

7. Federer’s Sets of Positive Reach

7

361

Federer’s Sets of Positive Reach

7.1

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 6.1 that the oriented distance function bA can be approximated uniformly in RN by the oriented distance function bAr of its dilated set Ar , r > 0, and that the W 1,p -convergence can be achieved if and only if m(∂A) = 0. It turns out that the W 1,p -convergence can be obtained for sets with positive reach such that ∂A = ∅ without assuming a priori that m(∂A) = 0. Theorem 7.1. Let A ⊂ RN such that ∂A = ∅ and reach (A) > 0. (i) A = ∅ and there exists h such that 0 < h ≤ reach (A) and Ah = RN . For all r, 0 < r < h, the sets Ur (A) are of class C 1,1 , bUr (A) = bAr , and bAr (x) = bA (x) − r in RN .

(7.1)

As r goes to zero, bAr (x) → bA (x) uniformly in RN , 1,p bAr → bA in Wloc (RN ),

dAr → d A ,

d
Ar → d
A ,

d∂Ar → d∂A in

(7.2) 1,p Wloc (RN ),

(7.3)

m(∂A) = 0, and χAr → χA = χint A in Lploc (RN ) for all p, 1 ≤ p < ∞. (ii) Furthermore, ∂A is of locally bounded curvature and of locally finite perimeter: ∇bA ∈ BVloc (RN )N

and

χ∂A ∈ BVloc (RN ).

Moreover, for all r, 0 < r < h, /  1 1. 2 bA (x) = f∂Ar (x) − f∂A (x) + r = bA (x) − b2Ar (x) + r, r r ∀x ∈ RN , ∀v ∈ RN , dH bA (x; v) exists.

(7.4)

(7.5) (7.6)

Remark 7.1. Recall from Theorem 3.1 (iii) that the Hadamard semiderivative of bA always exists at points off the boundary. The last statement in part (ii) says that for a closed set with nonempty boundary and positive reach, the Hadamard semiderivative exists in all points of its boundary and there is an explicit expression for it. In particular, this is true for all closed convex sets. Proof. (i) The uniform convergence of bAr to bA in RN was proved in Theorem 6.1 (i), and it remains to prove the W 1,p -convergence without the a priori assumption that m(∂A) = 0 that will come as a consequence rather than as a hypothesis. To do that 1,p (RN ). we first prove the convergence d
Ur (A) → d
A in Wloc From Theorem 6.1 (i), it sufficient to show that inequalities (6.1) and (6.2) become equalities for x ∈ Uh (A): for 0 < r < h d
Ur (A) (x) = 0,

if dA (x) ≥ r,

d
Ur (A) (x) ≥ r − dA (x),

if 0 < dA (x) < r,

d
Ur (A) (x) ≥ d
A (x) + r, if dA (x) = 0.

(7.7) (7.8)

362

Chapter 7. Metrics via Oriented Distance Functions

For x ∈ RN such that dA (x) ≥ r, d
A (x) = d
Ur (A) (x) = 0 and, from identity (7.2) of Chapter 6 in Theorem 6.1, bAr (x) = dAr (x) = dA (x) − r = bA (x) − r. For x ∈ RN such that 0 < dA (x) < r, d
A (x) = 0 = dAr (x). From Theorem 6.2 (iii) of Chapter 6 for all r, 0 < dA (x) < r < h ≤ reach (A), def

x(r) = pA (x) + r

x − pA (x) dA (x)

⇒ dA (x(r)) = r and pA (x(r)) = pA (x)

/ Ur (A), d
Ur (A) (x) = inf p∈∂Ar |x − p| and and x(r) ∈ ∂Ar = ∂Ur (A). Since x ∈ d
Ur (A) (x) = d
Ar (x) = d∂Ar (x) ≤ |x(r) − x| = r − dA (x). Combining this with inequality (7.7), bAr (x) = bA (x) − r. Finally, for x ∈ RN such that 0 = dA (x), x ∈ A, and since ∂A = ∅, there exists p¯ ∈ ∂A such that p − x|. There exists a sequence {yn } ⊂ Ur (A)\A such that yn → p and d
A (x) = |¯ cr such that p)| ≤ cr |yn − p¯| |pA (yn ) − pA (¯

⇒ pA (yn ) → pA (¯ p) = p¯.

Associate with each yn the points def

qn = pA (yn ) + r ∇dA (yn ). Since 0 < dA (qn ) ≤ r < h ≤ reach (A), by Theorem 6.2 (iii) of Chapter 6 pA (qn ) = pA (yn ),

dA (qn ) = r,

qn ∈ ∂Ar .

The sequence {qn } is bounded since p)| |qn − p¯| ≤ |qn − pA (qn )| + |pA (qn ) − p¯| = |qn − pA (qn )| + |pA (yn ) − pA (¯ ≤ r + cr |yn − p¯|. There exist q and a subsequence, still indexed by n, such that qn → q ⇒ r = dA (qn ) → dA (q) ⇒ pA (yn ) = pA (qn ) → pA (q) ⇒ dA (q) = r and pA (q) = p¯, r = dA (q) = |q − pA (q)| = |q − p¯|. Finally, since x ∈ A, A ∩ Ar = ∅, and ∂Ar = ∂Ar = d−1 A {r} d
Ar (x) =

inf dA (pr )=r

|x − pr | ≤ |x − q| ≤ |x − p¯| + |¯ p − q| = d
A (x) + r.

Hence, from inequality (7.8), d
Ar (x) = d
A (x) + r, and since dA (x) = 0 = dAr (x), bAr (x) = bA (x) − r. Since ∇bAr = ∇bA , for all bounded open subsets D of RN as r → 0, bAr − bA W 1,p (D) = bAr − bA Lp (D) ≤ r m(D)1/p → 0.

7. Federer’s Sets of Positive Reach

363

From Theorem 7.1 of Chapter 6 1,p (RN ) dAr → dA in Wloc

and χAr → χA in Lploc (RN )

for all p, 1 ≤ p < ∞. For all 0 < r < h ≤ reach (A), d
Ar = dAr − bAr → dA − bA = 1,p 1,p (RN ) and d∂Ar = dAr + d
Ar → dA + d
A = d∂A in Wloc (RN ). Finally, d
A in Wloc 1,p N since bAr → bA in Wloc (R ), 0 = χ∂Ar = 1 − |∇bAr | → 1 − |∇bA | = χ∂A in Lploc (RN ) and χ∂A = 0 and m(∂A) = 0. (ii) For 0 < r < h < reach (A), the open set Ur (A) is defined via the r-level N set d−1 A {r} = {x ∈ R : dA (x) = r} of the function dA . From Theorem 6.2 (vi) in Chapter 6, for each x ∈ d−1 A {r} = ∂Ur (A) there exists a neighborhood Bρ (x), ρ = min{r, h − r}/2 > 0, of x in Uh (A)\A such that dA ∈ C 1,1 (Bρ (x)). From part (i) for all 0 < r < h and y ∈ Bρ (x) ⊂ Uh (A)\A, d
A (y) = 0, bAr (y) = bA (y)−r = dA (y)−r. Therefore, bAr ∈ C 1,1 (Bρ (x)), and, a fortiori, for all x ∈ ∂Ar , there exists Bρ (x) such that ∇bAr = ∇dA ∈ BV(Bρ (x)). By Theorem 5.1, ∂Ar is of locally bounded curvature. By Definition 5.1, ∇bAr ∈ BVloc (RN )N and, since bAr = bA − r in RN , we also have ∇bA = ∇bAr ∈ BVloc (RN )N . By Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). Finally, since bAr (x) = bA (x) − r in RN from part (i) |x|2 − b2Ar (x) = |x|2 − b2A (x) − r2 + 2r bA (x), f∂Ar (x) − f∂A (x) + r2 = r bA (x) and bA is the difference of two convex continuous functions for which the Hadamard semiderivative exists in all points x ∈ RN and for all directions v ∈ RN .

7.2

Boundaries with Positive Reach

The next theorem is a specialization of Theorem 6.2 of Chapter 6 to the oriented distance function. Theorem 7.2. Let A ⊂ RN , ∂A = ∅. (i) Associate with a ∈ ∂A the sets  def

P (a) = v ∈ RN : Π∂A (a + v) = {a} ,

 def

Q(a) = v ∈ RN : d∂A (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta ∂A)∗ . (ii) Given a ∈ ∂A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : Π∂A (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), Π∂A (a + tv) = {a} and d∂A (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (∂A). (iii) If there exists h such that 0 < h ≤ reach (∂A), then for all x ∈ Uh (∂A)\∂A and 0 < |t| < h, Π∂A (x(t)) = {p∂A (x)} and bA (x(t)) = t, where def

x(t) = p∂A (x) + t ∇bA (x) = p∂A (x) + t

x − p∂A (x) . bA (x)

(7.9)

364

Chapter 7. Metrics via Oriented Distance Functions

(iv) Given b ∈ ∂A, x ∈ RN \Sk (∂A), a = p∂A (x), such that reach (∂A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (∂A, a)

(7.10)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (∂A) such that |bA (x)| ≤ r and reach (∂A, p∂A (x)) ≥ q, y ∈ RN \Sk (∂A) such that |bA (y)| ≤ r and reach (∂A, p∂A (y)) ≥ q, then |p∂A (y) − p∂A (x)| ≤

q |y − x|. q−r

(7.11)

(vi) If 0 < reach (∂A) < +∞, then for all h, 0 < h < reach (∂A), and all x, y ∈ (∂A)h = {x ∈ RN : |bA (x)| ≤ h} reach (∂A) |p∂A (y) − p∂A (x)| ≤ |y − x|, reach (∂A) − h   2 ∇bA (y) − ∇b2A (x) ≤ 2 1 + reach (∂A) |y − x|; (7.12) reach (∂A) − h

 for all s, 0 < s < h < reach (∂A), x, y ∈ ∂As,h = x ∈ RN : s ≤ |bA (x)| ≤ h ,   reach (∂A) 1 2+ |y − x| . |∇bA (y) − ∇bA (x)| ≤ s reach (∂A) − h If reach (∂A) = +∞, ∀x, y ∈ RN , |p∂A (y) − p∂A (x)| ≤ |y − x|, 2 ∇bA (y) − ∇b2A (x) ≤ 4 |y − x|;

(7.13)

for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ d∂A (x)} |∇bA (y) − ∇bA (x)| ≤

3 |y − x| . s

(vii) If reach (∂A) > 0, then, for all 0 < r < reach (∂A), Ur (∂A) is a set of class C 1,1 , Ur (∂A) = (∂A)r , ∂Ur (∂A) = {x ∈ RN : |bA (x)| = r}, and (∂A)r = Ur (∂A) = {x ∈ RN : |bA (x)| ≥ r}. (viii) For all a ∈ ∂A,

 Ta ∂A =

 |bA (a + th)| =0 . h ∈ R : lim inf t0 t N

In particular, if a ∈ int ∂A, Ta ∂A = RN .

8. Boundary Smoothness and Smoothness of bA

365

The next theorem is a specialization of Theorem 6.3 of Chapter 6 to the oriented distance function. Theorem 7.3. Given A ⊂ RN such that ∂A = ∅, the following conditions are equivalent: (i) ∂A has positive reach, that is, reach (∂A) > 0. (ii) There exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)). (iii) There exists h > 0 such that for all s, 0 < s < h, bA ∈ C 1,1 (Us,h (∂A)). (iv) There exists h > 0 such that bA ∈ C 1 (Uh (∂A)\∂A). (v) There exists h > 0 such that for all x ∈ Uh (∂A), Π∂A (x) is a singleton. Recall from Theorem 3.1 (viii) and (3.15) the following properties. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A). If ∂A has positive reach, then both A and A have positive reach.

8

Boundary Smoothness and Smoothness of bA

In this section the smoothness of the boundary ∂A is related to the smoothness of the function bA in a neighborhood of ∂A. The next two theorems give a complete characterization of sets A such that m(∂A) = 0 and bA ∈ C k,
in a neighborhood of ∂A. We first present two equivalence theorems that will follow from the more detailed Theorems 8.3 and 8.4. It is useful to recall that for a set that is locally of class C k,
in a point of ∂A, ∂A is locally a C k,
-submanifold of dimension N − 1 in that point, but the converse is not true (cf. Definitions 3.1 and 3.2 in Chapter 2)4 . Theorem 8.1 (Local version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤  ≤ 1 be a real number, and let x ∈ ∂A. 4 It is important to note that Theorem 8.1 is not true when b is replaced by d A ∂A since its gradient ∇d∂A is discontinuous across ∂A. Part (iii) of Theorem 8.1 in the direction (⇒) was asserted by J. Serrin [1] in 1969 for N = 3 and proved in 1977 by D. Gilbarg and N. S. Trudinger [1] for k ≥ 2 (provided that d∂A is replaced by bA in Lemma 1, p. 382 in [1], and Lemma 14.16, p. 355 in [2]) with a different proof by S. G. Krantz and H. R. Parks [1] in 1981. Another proof with the ´sio [17], who extended the result function bA was given in 1994 by M. C. Delfour and J.-P. Zole from (iii) to (ii) in the direction (⇒) down to the C 1,1 case and established the equivalence (⇔) in the whole range from C ∞ to C 1,1 . The counterexample of Example 6.2 for domains of class C 1,1− is the same as the one provided earlier in S. G. Krantz and H. R. Parks [1], where they observe only that the domain is C 2−ε , leaving the reader under the misleading impression that part (ii) of Theorem 8.1 would not be true for domains ranging from class C 1,1 to C 2 . Part (i) of Theorem 8.1 in the direction (⇒) was proved in the C 1 case by S. G. Krantz and H. R. Parks [1] in 1981. The equivalence (⇔) here for C 1,λ , 0 ≤ λ < 1, was given in 2004 by M. C. Delfour and ´sio [40]. J.-P. Zole

366

Chapter 7. Metrics via Oriented Distance Functions

(i) (k = 1, 0 ≤  < 1). A is a set of class C 1,
in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅ if and only if ∃ρ > 0 such that bA ∈ C 1,
(Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.1)

(ii) (k = 1,  = 1). A is locally a set of class C 1,1 at x if and only if ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.2)

(iii) (k ≥ 2, 0 ≤  ≤ 1). A is locally a set of class C k,
at x if and only if ∃ρ > 0 such that bA ∈ C k,
(Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.3)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is locally a C k,
-submanifold of dimension N − 1 at x. Remark 8.1. The condition Sk (∂A) ∩ V (x) = ∅ in part (i) is really necessary. Example 6.2 in Chapter 6 gives a family of two-dimensional sets of class C 1,λ , 0 ≤ λ < 1, for which bA ∈ C 1,λ and d2∂A = b2A ∈ / C 1 in any neighborhood of the point (0, 0) of its boundary ∂A. From Theorem 8.1 we get the global version that has had and still has an interesting history of inaccurate statements or incomplete proofs Theorem 8.2 (Global version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, and let 0 ≤  ≤ 1 be a real number. (i) (k = 1, 0 ≤  < 1). A is a set of class C 1,
and ∂A ∩ Sk (∂A) = ∅ if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1,
(Bρ (x)).

(8.4)

(ii) (k = 1,  = 1). A is a set of class C 1,1 if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x)).

(8.5)

(iii) (k ≥ 2, 0 ≤  ≤ 1). A is a set of class C k,
if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C k,
(Bρ (x)).

(8.6)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is a C k,
-submanifold of dimension N − 1. Remark 8.2. Theorems 8.1 and 8.2 deal only with sets of class C 1,
whose boundary is a C 1,
submanifold of dimension (N − 1). For arbitrary closed submanifolds M of RN of codimension greater than or equal to 1, ∇bM generally does not exist on M . In that case the smoothness of M is related to the existence and the smoothness of

8. Boundary Smoothness and Smoothness of bA

367

∇d2M in a neighborhood of M that implies that M is locally of positive reach. The reader is referred to Theorem 6.5 in section 6.2 of Chapter 6. The above two equivalence theorems follow from the following two local theorems in each direction. Theorem 8.3. Let A ⊂ RN be such that ∂A = ∅. Given x ∈ ∂A, assume that ∃ a bounded open neighborhood V (x) of x such that bA ∈ C k,
(V (x))

(8.7)

for some integer k ≥ 1 and some real , 0 ≤  ≤ 1, and that m(∂A ∩ V (x)) = 0. (i) Then A is locally a set of class C k,
in V (x) and bA = bint A = bA in V (x). (ii) Π∂A (y) = {p∂A (y)} is a singleton for all y ∈ V (x) and 1,1 k,
(V (x)) ∩ Cloc (V (x)) b2A ∈ Cloc

and

0,1 k−1,
p∂A ∈ Cloc (V (x))N ∩ Cloc (V (x))N .

Remark 8.3. If condition (8.7) is verified, the condition m(∂A) = 0 is equivalent to ∂A = RN . The condition bA ∈ Cb0 (RN ) is necessary to rule out the case ∅ = ∂A = RN that yields bA = 0 and A = A = RN that trivially satisfies condition (8.7). Proof. (i) Since ∇bA exists and is continuous everywhere in V (x), |∇bA | is continuous in V (x). But |∇bA (x)| = 1 in V (x)\∂A and |∇bA (x)| = 0 almost everywhere in V (x) ∩ ∂A. By continuity, either |∇bA | = 1 or |∇bA | = 0 in V (x). If ∇bA = 0 in V (x), then bA is constant in V (x) and equal to bA (x) = 0. Therefore, V (x) ⊂ ∂A and this yields the contradiction 0 < m(V (x)) ≤ m(∂A ∩ V (x)) = 0. Consider def the set Ω = {x ∈ RN : −bA (x) > 0}. By Theorem 4.2 in Chapter 2, Ω is a set of class C k,
since bA ∈ C k,
(V (x)) and ∇bA = 0 on b−1 A {0} ∩ V (x). In particular, ∂Ω = b−1 {0} = ∂A, int A = Ω, and int A = int Ω in V (x). Finally A bA = bΩ = bint A = bA . k,
k,
(ii) By assumption, bA ∈ Cloc (V (x)) and hence b2A belongs to Cloc (V (x)). By Theorem 3.1 (iv), Π∂A (y) = {p∂A (y)} is a singleton in V (x). By Theorem 7.3, 1,1 k,
0,1 (V (x)) and, a fortiori, in Cloc (V (x)). This yields p∂A ∈ Cloc (V (x)) ∩ b2A ∈ Cloc k−1,
Cloc (V (x)). In the other direction the smoothness of the domain implies the smoothness of bA in a neighborhood of ∂A when the domain is at least of class C 1,1 . A counterexample will be given in dimension 2. Theorem 8.4. Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤  ≤ 1 be a real number, and let x ∈ ∂A if it is not. (i) (k = 1, 0 ≤  < 1). If A is locally a set of class C 1,
in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1,
(Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.8)

368

Chapter 7. Metrics via Oriented Distance Functions

(ii) (k = 1,  = 1). If A is locally a set of class C 1,1 in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.9)

(iii) (k ≥ 2, 0 ≤  ≤ 1). If A is locally a set of class C k,
in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C k,
(Bρ (x))

and

p∂A ∈ C k−1,
(Bρ (x)).

(8.10)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A. Proof. (i) From Definition 3.2 of Chapter 2 of a locally a C 1,
-submanifold of dimension N − 1, m(∂A ∩ V (x)) = 0 in a bounded open neighborhood V (x) and the set A can be locally described by the level sets of the C 1,
-function def

f (y) = gx (y) · eN ,

(8.11)

since by definition int A ∩ V (x) = {y ∈ V (x) : f (y) > 0} , ∂A ∩ V (x) = {y ∈ V (x) : f (y) = 0} . The boundary ∂A in V (x) is the zero level set of f and the gradient ∇f (y) = Dgx (y)∗ eN = 0 in V (x) is normal to that level set. Thus the outward unit normal n to A on V (x) ∩ ∂A is given by n(y) = −

∇f (y) Dgx (y)∗ eN Dhx (gx (y))−∗ eN =− = − . |∇f (y)| |Dgx (y)∗ eN | |Dhx (gx (y))−∗ eN |

(8.12)

Since V (x) ∩ Sk (∂A) = ∅, for all y ∈ V (x), ∃ unique p∂A (y) ∈ ∂A,

d∂A (y) = inf |y − p| = |y − p∂A (y)|, p∈∂A

1 1 p∂A (y) = y − ∇d2∂A (y) = y − ∇b2A (y). 2 2 Choose ρ > 0 such that B3ρ (x) ⊂ V (x). Then for all y ∈ Bρ (x) |p∂A (y) − x| ≤ |p∂A (y) − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p∂A (y) ∈ B2ρ (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}.

(8.13)

8. Boundary Smoothness and Smoothness of bA

369

Therefore, for each y ∈ Bρ (x), there exists a unique p∂A (y) ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |z − y|2 + λ f (z) ˆ ∈ R such that there exists a λ ˆ ∇f (p∂A (y)) = 0 and f (p∂A (y)) = 0 2(p∂A (y) − y) + λ ˆ |∇f (p∂A (y))| ⇒ 2d∂A (y) = |λ| ⇒ p∂A (y) = y + bA (y)

∇f (p∂A (y)) = y − bA (y) n(p∂A (y)) |∇f (p∂A (y))|

since ∇f (p∂A (y)) = 0 on ∂A ∩ V (x). Combining this with (8.13) 1 y − ∇b2A (y) = p∂A (y) = y − bA (y) n(p∂A (y)) 2 ⇒ ∀y ∈ Br (x)\∂A, ∇bA (y) = n(p∂A (y))

∀y ∈ Br (x),

(8.14)

since V (x)∩Sk (∂A) = ∅ implies that ∇bA (y) exists in V (x)\∂A by Theorem 3.1 (vii). Consider the normal n(z) = −∇f (z)/|∇f (z)| in V (x)\∂A. For z2 and z1 in ∂A ∩ B2ρ (x) the difference ∇f (z2 ) ∇f (z1 ) − |n(z2 ) − n(z1 )| = |∇f (z2 )| |∇f (z1 )|   1 1 1 = (∇f (z2 ) − ∇f (z1 )) + − ∇f (z1 ) |∇f (z2 )| |∇f (z2 )| |∇f (z1 )| 1 ∇f (z2 ) − ∇f (z1 ) + (|∇f (z1 )| − |∇f (z2 )|) ∇f (z1 ) = |∇f (z2 )| |∇f (z1 )| 1 1 |∇f (z2 ) − ∇f (z1 )| ≤ 2 |∇f (z2 ) − ∇f (z1 )| . ≤2 |∇f (z2 )| inf z∈∂A∩B2ρ (x) |∇f (z)| But f ∈ C 1,
(B2ρ (x)) and since |∇f | = 0 is continuous on V (x), |∇f (z2 ) − ∇f (z1 )| ≤ cx |z2 − z1 |


def

and α =

inf

|∇f (z)| > 0

z∈∂A∩B2ρ (x)

and the outward unit normal is -H¨olderian, ∀z2 , z1 ∈ B2ρ (x) ∩ ∂A,

|n(z2 ) − n(z1 )| ≤

2 cx |z2 − z1 |
, α

where cx is the Lipschitz constant of f in B2ρ (x). By construction, B3ρ (x) ∩ Sk (∂A) ⊂ V (x) ∩ Sk (∂A) = ∅ and for all z ∈ Bρ (x), B2ρ (z) ∩ Sk (∂A) = ∅

370

Chapter 7. Metrics via Oriented Distance Functions

implies that reach (∂A, z) ≥ 2r > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = ρ < q = 2ρ, ∀z1 , z2 ∈ Bρ (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.15)

and hence p∂A ∈ C 0,1 (Bρ (x))N . So the composition n◦p∂A belongs to C 0,
(Bρ (x))N since we have already shown that n ∈ C 0,
(B2ρ (x) ∩ ∂A)N . Finally, ∇bA ∈ L∞ (Bρ (x))N , n(p∂A ) ∈ C 0,
(Bρ (x))N , m(∂A) = 0, and from (8.14) ∇bA (y) = n(p∂A (y))

a.e. in Bρ (x)

imply that ∇bA ∈ C 0,
(Bρ (x))N , bA ∈ C 1,
(Bρ (x)), and ∇bA = n ◦ p∂A in Bρ (x). (ii) The first steps are the same as in part (i) up to (8.12) with C 1,1 in place 1,
of C . For each y ∈ V (x), Π∂A (y) is compact and nonempty and ∀p ∈ Π∂A (y),

d2∂A (y) = inf |a − y|2 = |p − y|2 . a∈∂A

Choose r > 0 such that B3r (x) ⊂ V (x). Then for all y ∈ Br (x) and p ∈ Π∂A (y) |p − x| ≤ |p − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p ∈ B2r (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}. Therefore, for each y ∈ Br (x) and p ∈ Π∂A (y), p ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |a − y|2 + λ f (a) ˆ ∈ R and p ∈ Π∂A (y) such that there exist λ ˆ ∇f (p) = 0 and f (p) = 0 2(p − y) + λ ⇒ p = y + bA (y)

ˆ |∇f (p)| ⇒ 2d∂A (y) = |λ|

∇f (p) = y − bA (y) n(p) |∇f (p)|

⇒ ∀y ∈ Br (x), ∀p ∈ Π∂A (y),

p = y − bA (y) n(p),

(8.16)

since ∇f (p) = 0 on ∂A ∩ V (x). By the same proof as in part (i) with  = 1 2 ∀z2 , z1 ∈ B2r (x) ∩ ∂A, |n(z2 ) − n(z1 )| ≤ cx |z2 − z1 |, α where cx is the Lipschitz constant associated with f in B2r (x). Therefore, n ∈ C 0,1 (B2r (x) ∩ ∂A). From (8.16) for y ∈ Br (x) and for all p1 , p2 ∈ Π∂A (y), |p2 − p1 | ≤ |bA (y)| |n(p2 ) − n(p1 )| ≤ |bA (y)|

2 cx |p2 − p1 |. α

8. Boundary Smoothness and Smoothness of bA

371

By choosing h such that 0 < h < min{r, α/(2 cx )}, h 2 cx /α < 1, for y ∈ Bh (x) ⊂ Br (x) ∩ {z ∈ RN : |bA (z)| ≤ h} and all p1 , p2 ∈ Π∂A (y), p2 = p1 , Π∂A (y) is a singleton, and p∂A (y) = y − bA (y) n(p∂A (y)). By construction, Bh (x)∩Sk (∂A) = ∅ and for all z ∈ Bh/3 (x), B2h/3 (z)∩Sk (∂A) = ∅ implies that reach (∂A, z) ≥ 2h/3 > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = h/3 < q = 2h/3 ∀z1 , z2 ∈ Bh/3 (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.17)

and hence p∂A ∈ C 0,1 (Bh/3 (x))N . Thus n ◦ p∂A belongs to C 0,1 (Bh/3 (x))N since we have already shown that n ∈ C 0,1 (B2r (x) ∩ ∂A)N . The remainder and the conclusion of the proof is the same as in part (i) with ρ = h/3. (iii) When A is of class C k,
in V (x) for an integer k, k ≥ 2, and a real number , 0 ≤  ≤ 1, it is C 1,1 and the previous constructions and conclusions of (ii) are verified. There exists ρ > 0 such that for each y ∈ Bρ (x) there exists a unique p∂A (y) ∈ B2ρ (x) such that p∂A (y) = y − bA (y)

∇f (p∂A (y)) . |∇f (p∂A (y))|

Since ∇f is continuous and ∇f = 0 on ∂A ∩ V (x), there exists r, 0 < 2r < ρ, such that inf

|∇f (z)| > 0

z∈B2r (x)

and defining N (z) = ∇f (z)/|∇f (z)|,  ∇f (z) 1 D2 f (z) − ∇N (z) = |∇f (z)| |∇f (z)|

∗



∇f (z) D f (z) |∇f (z)|



2

belongs to C k−1,
(B2r (x)), where ∗v denotes the transpose of a vector v ∈ RN , so that (u ∗v)ij = ui vj is the tensor product. Consider the map def

(y, z) → G(y, z) = z − y − bA (y)

∇f (z) : Br (x) × B2r (x) → RN . |∇f (z)|

From part (ii), there exists a unique p(y) = p∂A (y) ∈ B2r (x) ∩ ∂A such that p∂A ∈ C 0,1 (B2r (x))N , bA ∈ C 1,1 (Br (x)), and G(y, p(y)) = 0 where DGz (y, z) = I −

⇒ DGy (y, p(y)) + DGz (y, p(y)) Dp(y) = 0,

 bA (y) ∇f (z) D2 f (z) − |∇f (z)| |∇f (z)|

∗

 D2 f (z)

DGy (y, z)) = −I − N (z) ∗∇bA (y).

∇f (z) |∇f (z)|

 ,

372

Chapter 7. Metrics via Oriented Distance Functions

DGz (y, z) is at least C 1,1 in (y, z) since bA ∈ C 1,1 and N is at least C 1,1 for k ≥ 2. Since f is at least C 2 in V (x), f and its first and second derivatives are bounded in some smaller neighborhood Br (x) of x. By reducing the upper bound on |bA (y)|, there exists a new smaller neighborhood Br (x) of x such that Dz G(y, z) is invertible for all (y, z) ∈ Br (x) × B2r (x). So the conditions of the implicit function theorem are met. There exist a neighborhood W (x) ⊂ Br (x) of x and a unique C 1 -mapping p : W (x) → RN such that ∀y ∈ W (x),

G(y, p(y)) = 0

that coincides with p∂A (cf., for instance, L. Schwartz [3, Thm. 26, p. 283]). Therefore Dp∂A (y) = (DGz (y, p∂A (y)))−1 [I + N (p∂A (y)) ∗∇bA (y)] and Dp∂A is C 0 and p∂A is C 1 . Thus ∇bA = N ◦ p∂A ∈ C 1 and bA is now C 2 . Since N ∈ C k−1,
and D2 f ∈ C k−2,
, we can repeat this argument until we get Dp∂A in C k−2,
, and finally p∂A ∈ C k−1,
, ∇bA = N ◦ p∂A ∈ C k−1,
, and bA ∈ C k,
. From Theorem 8.1 (ii) additional information is available on the flow of ∇bΩ . def

Theorem 8.5. Let Ω ⊂ RN be of class C 1,1 . Denote its boundary by Γ = ∂Ω. (i) For each x ∈ Γ there exists a bounded open neighborhood W (x) of x such that bΩ ∈ C 1,1 (W (x)),

∇bΩ = n ◦ pΓ = ∇bΩ ◦ pΓ ,

|∇bΩ | = 1 in W (x), (8.18)

where n is the unit exterior normal to Ω on Γ. (ii) Associate with x ∈ Γ and y in W (x) the flow Tt (y) of ∇bΩ in W (x), that is, dx (t) = ∇bΩ (x(t)), dt

x(0) = y,

def

Tt (y) = x(t).

(8.19)

Then for t in a neighborhood of 0 we have the following properties: pΓ ◦ Tt = pΓ ,

∇bΩ ◦ Tt = ∇bΩ ,

Tt (y) = y + t ∇bΩ (y).

(8.20)

(iii) Almost everywhere in W (x) d DTt = D 2 bΩ dt

−1 D2 bΩ ◦ Tt = D2 bΩ I + tD2 bΩ ,

DTt = I + t D 2 bΩ , ⇒ D2 bΩ ◦ Tt DTt = D 2 bΩ ,

(DTt )−1 ∇bΩ = ∇bΩ = ∗(DTt )−1 ∇bΩ .

(8.21) (8.22) (8.23)

Proof. For simplicity, we use the notation b = bΩ and p = pΓ . (i) The result follows from Theorem 8.4 (i). (ii) By assumption from part (i) the function b belongs to C 1,1 (W (x)) and hence p ∈ C 0,1 (W (x); RN ). Thus almost everywhere in W (x), Dp = I − ∇b ∗∇b − bD2 b

⇒ Dp∇b = 0.

9. Sobolev or W m,p Domains

373

Now dTt d (p ◦ Tt ) = Dp ◦ Tt = Dp ◦ Tt ∇b ◦ Tt = (Dp ∇b) ◦ Tt = 0 dt dt ⇒ p ◦ Tt = p in W (x). But since p and Tt belong to C 0,1 (W (x); RN ), the identity necessarily holds everywhere in W (x). Moreover, ∇b ◦ Tt = n ◦ p ◦ Tt = n ◦ p = ∇b in W (x). Finally dTt = ∇b ◦ Tt = ∇b dt

⇒ Tt (y) = y + t∇b(y) in W (x).

In particular, almost everywhere in W (x), d DTt = D2 b, dt ⇒ (DTt )−1 ∇b = ∇b = ∗(DTt )−1 ∇b.

DTt = I + t D2 b DTt ∇b = ∇b

9

⇒

Sobolev or W m,p Domains

The N -dimensional Lebesgue measure of the boundary of a Lipschitzian subset of RN is zero. However, this is generally not true for sets of locally bounded curvature, as can be seen from the following example. Example 9.1. Let B be the open unit ball centered in 0 of R2 and define A = {x ∈ B : x with rational coordinates}. Then ∂A = B, bA = dB , and for all h > 0 ∇bA ∈ BV(Uh (∂A))2 ,   1 ∆bA , ϕ = ϕ dx. ϕ dx + |x| ∂B
B The next natural question that comes to mind is whether the boundary or the skeleton has locally finite (N − 1)-Hausdorff measure. Then come questions about the mean curvature of the boundary. For the function bA , ∆bA = tr D2 bA is proportional to the mean curvature of the boundary, which is only a measure in RN for sets of bounded local curvature. This calls for the introduction of some classification of sets that would complement the classical H¨olderian terminology introduced in Chapter 2, would fill the gap in between, and would possibly say something about sets whose boundary is not even continuous. The following definition seems to have ´sio [32]. been first introduced in M. C. Delfour and J.-P. Zole

374

Chapter 7. Metrics via Oriented Distance Functions

Definition 9.1 (Sobolev domains). Given m > 1 and p ≥ 1, a subset A of RN is said to be an (m, p)-Sobolev domain or simply a W m,p -domain if ∂A = ∅ and there exists h > 0 such that m,p bA ∈ Wloc (Uh (∂A)).

This classification gives an analytical description of the smoothness of boundaries in terms of the derivatives of bA . The definition is obviously vacuous for m = 1 1,∞ since bA ∈ Wloc (RN ) for any set A such that ∂A = ∅. For 1 < m < 2 we shall see that it encompasses sets that are less smooth than sets of locally bounded curvature. For m = 2 the W 2,p -domains are of class C 1,1−N/p for p > N . For m > 2 they are intertwined with sets of class C k,
. Theorem 9.1. Given any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N 1 1+η,p ⇒ ∀p, 1 ≤ p < ∞, ∀η, 0 ≤ η < , bA ∈ Wloc (RN ). p N N Proof. Since |∇bA (x)| ≤ 1 almost everywhere, ∇bA ∈ BVloc (RN )N ∩ L∞ loc (R ) and the theorem follows directly from Theorem 6.9 in Chapter 5.

It is quite interesting that a domain of locally bounded curvature is a W 2−ε,1 domain for any arbitrary small ε > 0. So it is almost a W 2,1 -domain, and domains of class W 2−ε,1 seem to be a larger class than domains of locally bounded curvature. In addition, their boundary does not generally have zero measure. Now consider the “threshold case” m = 2. Theorem 9.2. Given an integer N ≥ 1, let A be a subset of RN such that ∂A = ∅. (i) If m(∂A) = 0 and there exist p > N and h > 0 such that 2,p bA ∈ Wloc (Uh (∂A)),

(9.1)

then reach (∂A) ≥ h > 0,

1,1−N/p

bA ∈ Cloc

(Uh (∂A)),

and A is a H¨ olderian set of class C 1,1−N/p . 2,p (Uh (∂A)) is equivalent to ∆bA ∈ (ii) In dimension N = 2 the condition bA ∈ Wloc p Lloc (Uh (∂A)).

Proof. (i) Consider the function |∇bA |2 . Since |∇bA | ≤ 1, then 2,p 1,∞ (Uh (∂A)) ∩ Wloc (Uh (∂A)) bA ∈ Wloc

2,p 1,∞ ⇒ b2A ∈ Wloc (Uh (∂A)) ∩ Wloc (Uh (∂A)).

In particular, for all x ∈ ∂A, bA , b2A ∈ W 2,p (B(x, h)) ∩ W 1,∞ (B(x, h)).

10. Characterization of Convex and Semiconvex Sets

375

For p > N , from R. A. Adams [1, Thm. 5.4, Part III, and Rem. 5.5 (3), p. 98], bA , b2A ∈ C 1,λ (B(x, h)), ⇒

bA , b2A

∈

1,λ Cloc (Uh (A)),

0 < λ ≤ 1 − N/p 0 < λ ≤ 1 − N/p.

From Theorem 8.2 (i), if m(∂A) = 0, the set A is a set of class C 1,1−N/p , ∂A ∩ Sk (∂A) = ∅, and reach (∂A) ≥ h. (ii) From part (i), |∇bA (x)|2 = 1 in Uh (∂A), and D2 bA (x) ∇bA (x) = 0 almost everywhere. Hence in dimension N = 2, 2 2 bA (x) = ∂21 bA (x) = −∂1 bA (x) ∂2 bA (x) ∆bA (x), ∂12 2 ∂11 bA (x) = (∂2 bA (x))2 ∆bA (x), 2 ∂22 bA (x) = (∂1 bA (x))2 ∆bA (x).

As can be seen, a certain amount of work would be necessary to characterize all the Sobolev domains and answer the many associated open questions.

10 10.1

Characterization of Convex and Semiconvex Sets Convex Sets and Convexity of bA

We have seen in Chapter 6 that the convexity of dA is equivalent to the convexity of A. This characterization remains true with bA in place of dA . However, if in both cases the convexity of dA or bA is sufficient to get the convexity of A, it is not the case for A as can be seen from the following example. Example 10.1. Let B be the open unit ball in RN and A be the set B minus all the points in B with rational coordinates. By definition, A¯ = B, A = RN , dA = dB = dA ,

∂A = B,

∂A = ∂B = ∂A,

d
A = d
A = dRN = 0

⇒ bA = dB = bB = bA . By Theorem 8.1 of Chapter 6 the functions dB and, a fortiori, bA are convex, but A is not convex and ∂A = ∂A. Theorem 10.1. (i) Let A be a convex subset of RN . Then bA = bA

and

[A]b = [A]b .

(10.1)

Hence, the equivalence class [A]b can be identified with the unique closed set A in the class.

376

Chapter 7. Metrics via Oriented Distance Functions

(ii) Let A be a convex subset of RN such that int A = ∅. Then bint A = bA = bA

and

[int A]b = [A]b = [A]b .

(10.2)

Hence, the equivalence class [A]b can be identified with either the unique closed set A or the unique open set int A = A = ∅ in the class. If, in addition, A = RN , then ∂A = ∅ and int A = ∅. (iii) Let A be a subset of RN such that ∂A = ∅.5 Then bA is convex

⇐⇒

A is convex.

(10.3)

Proof. (i) If A = ∅, then A = ∅, A = RN , and A = RN . Hence bA = bA . If A = RN , we use the identity bA = −b
A . If ∂A = ∅, then dA = dA and, by Theorem 3.5 (ii) in Chapter 5, A = A, and bA = bA . (ii) From Theorem 3.5 (ii) in Chapter 5, int A = A and int A = A = A. Hence bA = bint A . From part (i), bA = bA . If, in addition, A = RN , always from Theorem 3.5 (ii) in Chapter 5, ∂A = ∅ and int A = ∅. (iii) (⇒) Denote by xλ the convex combination λx + (1 − λ)y of two points x and y in A for some λ ∈ [0, 1]. By convexity of bA , bA (xλ ) ≤ λbA (x) + (1 − λ)bA (y) = − [λd
A (x) + (1 − λ)d
A (y)] ≤ 0 ⇒ dA (xλ ) = b+ A (xλ ) = max{bA (xλ ), 0} = 0

⇒ xλ ∈ A

and A is convex. (⇐) We first prove that for A closed the function bA is convex. From this we conclude that if A is convex, then bA is convex and from part (i) bA = bA is convex. So for A closed, consider three cases. The first one deserves a lemma. Lemma 10.1. Let A be a subset of RN such that ∂A = ∅ and let A be convex. Then bA = −d
A is convex in A. Proof. Again we can assume that A is closed. Associate with x and y in A the radii rx = d
A (x), ry = d
A (y), and rλ = λrx + (1 − λ)ry and the closed balls B x of center x and radius rx , B y of center y and radius ry , and B λ of center xλ and radius rλ . By the definition of d
A , B x ⊂ A and B y ⊂ A since A is closed. Associate with 5 In 1985 D. H. Armitage and U. ¨ Kura [1] showed that a proper closed domain Ω is convex if and only if −bA is superharmonic and that, for N = 2, the result still holds if −bA is superharmonic only on A. They also showed that, for A compact, d∂A is subharmonic on A if and only if A is convex. This work was pursued in 1987 by M. J. Parker [1] and in 1988 by M. J. Parker [2]. ´sio [17] that the property that A is convex In 1994 it was shown in M. C. Delfour and J.-P. Zole if and only if dA is convex remains true with bA in place of dA .

10. Characterization of Convex and Semiconvex Sets

377

each z ∈ B λ the points zx = x and zy = y if rλ = 0, and if rλ > 0 the points rx (z − xλ ) ⇒ zx ∈ B x , rλ ry def zy = y + (z − xλ ) ⇒ zy ∈ B y rλ λrx + (1 − λ)ry ⇒ λzx + (1 − λ)zy = xλ + (z − xλ ) = z. rλ def

zx = x +

Obviously z = xλ if rλ = 0. Therefore, B λ ⊂ λB x + (1 − λ)B y ⊂ A, since A is closed and convex. In particular d
A (xλ ) ≥ d
B λ (xλ ) ≥ rλ = λd
A (x) + (1 − λ)d
A (y), and this proves the lemma. For the second case consider x and y in A. By definition bA (x) = dA (x)

and bA (y) = dA (y).

By Theorem 8.1 of Chapter 6, dA is convex when A is convex and bA (xλ ) ≤ dA (xλ ) ≤ λdA (x) + (1 − λ)dA (y) = λbA (x) + (1 − λ)bA (y). The third and last case is the mixed one for x ∈ A and y ∈ A. Define def

xλ = λx + (1 − λ)y,

def

pλ = p∂A (xλ ).

Since A is convex, denote by H the tangent hyperplane to A through pλ , by H + the closed half-space containing A, and by H − the other closed subspace associated with H. By the definition of H, d∂A (xλ ) = dH (xλ ),

H ⊂ H ±,

and by convexity of A, A ⊂ H + and H − ⊂ A. The projection onto H is a linear operator and pλ = pH (xλ ) = λpH (x) + (1 − λ)pH (y), pλ − xλ = λ(pH (x) − x) + (1 − λ)(pH (y) − y). If y ∈ H + and x ∈ H + , dH (xλ ) = λdH (x) + (1 − λ)dH (y),

(10.4)

378

Chapter 7. Metrics via Oriented Distance Functions

and if y ∈ H + and x ∈ H − ,

dH (xλ ) =

λdH (x) − (1 − λ)dH (y), (1 − λ)dH (y) − λdH (x),

if xλ ∈ H − , if xλ ∈ H + .

(10.5)

By convexity any y ∈ A belongs to H + and since H − ⊂ A, we readily have dH (y) = dH − (y) ≥ d
A (y) = −bA (y). First consider the case xλ ∈ A. Then dA (xλ ) > 0, xλ ∈ H − , and necessarily x ∈ H − . From (10.5) bA (xλ ) = dA (xλ ) = dH (xλ ) = λdH (x) − (1 − λ)dH (y). But since x ∈ H − and A ⊂ H + dH (x) = dH + (x) ≤ dA (x) and bA (xλ ) = dH (xλ ) ≤ λdA (x) − (1 − λ)d
A (y) = λbA (x) + (1 − λ)bA (y). Next consider the case xλ ∈ A for which xλ ∈ H + and −bA (xλ ) = d
A (xλ ) = dH (xλ ). If x ∈ H − , then since A ⊂ H + , dH (x) = dH + (x) ≤ dA (x), and from (10.5) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) − λdH (x) ≥ (1 − λ)d
A (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. If, on the other hand, x ∈ H + , then x, y ∈ H + and xλ ∈ A ⊂ H + . So from (10.4) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) + λdH (x) ≥ (1 − λ)d
A (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. This covers all cases and concludes the proof.

10. Characterization of Convex and Semiconvex Sets

10.2

379

Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D)

We have seen in Theorem 10.1 (i) that for a convex subset A of D, bA = bA and that A is the unique closed representative in the equivalence class. To work with open convex subsets is more delicate. Always from Theorem 10.1 (ii), int A becomes the unique open representative in the equivalence class when the convex set A has a nonempty interior. In order to work with families of open rather than closed convex subsets, it will be necessary to add a constraint on the interior of the subsets to prevent the occurrence of subsets with an empty interior. Theorem 10.2. Let D be a nonempty open (resp., bounded open) subset of RN . (i) The family  def

Cb (D) = bA : A ⊂ D, ∂A = ∅, A convex

(10.6)

is closed in Cloc (D) (resp., compact in C(D)). It coincides with the family  bA : A ⊂ D, ∂A = ∅, A closed and convex .



(ii) Let E be a nonempty open subset of D. Then the families def

Cb (E; D) = {bΩ : E ⊂ Ω ⊂ D, ∂Ω = ∅, Ω open and convex} , def

Cb,loc (E; D) =

bΩ :

∃x ∈ RN , ∃A ∈ O(N) such that x + AE ⊂ Ω and Ω is open and convex ⊂ D

&

(10.7) (10.8)

are closed in Cloc (D) (resp., compact in C(D)). 1,p (D)- (resp., (iii) The conclusions of parts (i) and (ii) remain true with Wloc 1,p W (D)-) strong in place of Cloc (D) (resp., C(D)).

Proof. It is sufficient to prove the result for D bounded. Furthermore, by compactness of Cb (D) in C(D), it is sufficient to prove that Cb (D), Cb (E; D), and Cb,loc (E; D) are closed in C(D). (i) Let {bAn } be a Cauchy sequence in Cb (D). It converges to some bA ∈ Cb (D). The function bA is convex as the pointwise limit of a family of convex functions. By Theorem 10.1 (iii), A is convex, and by Theorem 10.1 (i) bA = bA is convex. Therefore the Cauchy sequence converges to bA in Cb (D) and Cb (D) is closed in C(D). (ii) From part (i), given a Cauchy sequence {bΩn } in Cb (E; D), there exists an open convex subset Ω of D such that bΩn → bΩ . Moreover, since d
Ωn → d
Ω , E ⊂ Ωn ⇒ E ⊃ Ωn ⇒ d
E ≤ d
Ωn ⇒ d
E ≤ d
Ω ⇒ E ⊃ Ω ⇒ E ⊂ Ω. c The proof for Cb,loc (E; D) is similar to the proof for Cd,loc (E; D) in Theorem 2.6 of Chapter 6. (iii) Let {bAn } ⊂ Cb (D) be a Cauchy sequence in W 1,p (D). From Theorem 4.2 (ii), it is also Cauchy in C(D). Hence from part (i) there exists a convex set

380

Chapter 7. Metrics via Oriented Distance Functions

A, ∂A = ∅, such that bAn → bA in C(D) and, a fortiori, in W 1,p (D). This shows that Cb (D) is closed in W 1,p (D). To show that it is compact consider a sequence {bAn } ⊂ Cb (D). From part (i) there exist a subsequence, still indexed by n, and a convex subset A, ∂A = ∅, of D such that bAn → bA in C(D). Since D is bounded, it also converges in Lp (D). Since |∇bAn | is pointwise bounded by one, there exists another subsequence of {bAn }, still indexed by n, such that ∇bAn converges to ∇bA in W 1.p (D)N -weak. But An and A are convex. Thus m(∂A) = 0 = m(∂An ) and |∇bAn | = 1 = |∇bA | almost everywhere in RN . As a result for p = 2   2 |∇bAn − ∇bA | dx = |∇bAn |2 + |∇bA |2 dx − 2∇bAn · ∇bA dx D D 2 − 2∇bAn · ∇bA dx → 0. = D

Hence we get the compactness in W 1,2 (D)-strong and by Theorem 4.1 (i) in W 1,p (D)strong, 1 ≤ p < ∞.

10.3

BV Character of bA and Semiconvex Sets

The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 10.3. For all convex sets A such that ∂A = ∅, (i) ∇bA ∈ BVloc (RN )N , ∇χ∂A ∈ BVloc (RN ), ∇dA ∈ BVloc (RN )N , ∇χA ∈ BVloc (RN ), and for all x ∈ RN and v ∈ RN , dH bA (x, v) exists. (ii) The Hessian matrix D2 bA of second-order derivatives is a matrix of signed Radon measures which are nonnegative on the diagonal. (iii) bA has a second-order derivative almost everywhere, and for almost all x and y in RN , bA (y) − bA (x) − ∇bA (x) · (y − x) − 1 (y − x) · D 2 bA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. Same argument as in the proof of Theorem 8.2 of Chapter 6. When A is convex and of class C 2 , then for any x0 ∈ ∂A, there exists a strictly convex neighborhood N (x0 ) of x0 such that bA ∈ C 2 (N (x0 )) and ∀x ∈ N (x0 ), ∀ξ ∈ RN ,

D 2 bA (x)ξ · ξ ≥ 0,

(10.9)

or, since D 2 bA (x) ∇bA (x) = 0 in N (x0 ), then for all x ∈ N (x0 ), ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D2 bA (x)ξ · ξ ≥ 0.

(10.10)

This is related to the notion of strong elliptic midsurface in the theory of shells: there exists c > 0 ∀x ∈ ∂A, ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D 2 bA (x) ξ · ξ ≥ c|ξ|2 .

All this motivates the introduction of the following notions.

11. Compactness and Sets of Bounded Curvature

381

Definition 10.1. Let A be a closed subset of RN such that ∂A = ∅. (i) The set A is locally convex (resp., locally strictly convex ) if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that bA is convex (resp., strictly convex) in N (x0 ). (ii) The set A is semiconvex if ∃α ≥ 0,

bA (x) + α|x|2 is convex in RN .

(iii) The set A is locally semiconvex if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 and ∃α ≥ 0,

bA (x) + α|x|2 is convex in N (x0 ).

Remark 10.1. When A is a C 2 domain with a compact boundary, D 2 bA is bounded in a bounded neighborhood of ∂A and A is necessarily locally semiconvex. When A is a locally semiconvex set, for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that ∇bA ∈ BV(N (x0 ))N . If, in addition, ∂A is compact, then there exists h > 0 such that ∇bA ∈ BV(Uh (∂A))N . Remark 10.2. Given a fixed constant β > 0, consider all the subsets of D that are semiconvex with constant 0 ≤ α ≤ β. Then this set is closed for the uniform and the W 1,p -topologies, 1 ≤ p < ∞.

11

Compactness and Sets of Bounded Curvature

For the family of sets with bounded curvature, the key result is the compactness of the embedding

 (11.1) BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N → W 1,p (D) for bounded open Lipschitzian subsets of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(11.2)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(11.3)

for bounded open Lipschitzian subsets of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As in the case of characteristic functions in Chapter 5, we give a first version involving global conditions on a fixed bounded open Lipschitzian holdall D. In the second version the sets are contained in a bounded open holdall D with local conditions in the tubular neighborhood of their boundary.

382

11.1

Chapter 7. Metrics via Oriented Distance Functions

Global Conditions on D

Theorem 11.1. Let D be a nonempty bounded open Lipschitzian subset of RN . The embedding (11.1) is compact. Thus for any sequence {An }, ∂An = ∅, of subsets of D such that ∃c > 0, ∀n ≥ 1, D2 bAn M 1 (D) ≤ c, (11.4) there exist a subsequence {Ank } and a set A, ∂A = ∅, such that ∇bA ∈ BV(D)N and bAnk → bA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

n→∞

1 ≤ i, j ≤ N,

D2 bA M 1 (D) ≤ c.

(11.5)

 def

Proof. Given c > 0 consider the set Sc = bA ∈ Cb (D) : D2 bA M 1 (D) ≤ c . By compactness of the embedding (11.3), given any sequence {bAn } there exist a subsequence, still denoted by {bAn }, and f ∈ BV(D)N such that ∇bAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cb (D) is compact in C(D) for bounded D and there exist another subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D) and, a fortiori, in L1 (D). Therefore, bAnk converges in W 1,1 (D) and also in L1 (D). By uniqueness of the limit, f = ∇bA and bAnk converges in W 1,1 (D) to bA . For Φ ∈ D1 (D)N ×N as k goes to infinity   −→ −→ ∇bAnk · div Φ dx → ∇bA · div Φ dx D D   −→ −→ ∇bA · div Φ dx = lim ∇bAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D bA M 1 (D) ≤ c, and ∇bA ∈ BV(D) . This proves the sequential compactness of the embedding (11.1) for p = 1 and properties (11.5). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cb (D) in Theorem 4.1 (i). 2

11.2

N

Local Conditions in Tubular Neighborhoods

The global condition (11.4) is now weakened to a local one in a neighborhood of each set of the sequence. Simultaneously, the Lipschitzian condition on D is removed since only the uniform boundedness of the sets of the sequence is required. ∅, be a Theorem 11.2. Let ∅ = D ⊂ RN be bounded open and {An }, ∂An = sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 bAn M 1 (Uh (∂An )) ≤ c.

(11.6)

Then, there exist a subsequence {Ank } and A ⊂ D, ∅ = ∂A, such that ∇bA ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, bAnk → bA in W 1,p (Uh (D))-strong.

(11.7)

11. Compactness and Sets of Bounded Curvature

383

Moreover, for all ϕ ∈ D0 (Uh (∂A)), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

1 ≤ i, j ≤ N,

k→∞

D2 bA M 1 (Uh (∂A)) ≤ c,

(11.8)

and χ∂A belongs to BVloc (RN ). Proof. (i) By assumption, An ⊂ D implies that Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a subset A of D, ∂A = ∅, such that bAn → bA

in C(Uh (D))-strong

and another subsequence, still indexed by n, such that bAn → bA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and all x ∈ Uh (D), d∂An (x) ≤ d∂A (x) + ε,

d∂A (x) ≤ d∂An (x) + ε.

(11.9)

Therefore, ∂An ⊂ Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ), Uh−ε (∂A) ⊂ Uh−2 ε (∂An ) ⊂ ∂An .

(11.10) (11.11)

From (11.6) and (11.10) ∀n ≥ N,

D 2 bAn M 1 (Uh−ε (∂A)) ≤ c.

In order to use the compactness of the embedding (11.3) as in the proof of Theorem 11.1, we would need Uh−ε (∂A) to be Lipschitzian. To get around this we construct a bounded Lipschitzian set between Uh−2ε (∂A) and Uh−ε (∂A). Indeed by definition, Uh−ε (∂A) = ∪x∈∂A B(x, h − ε)

and

Uh−2ε (∂A) ⊂ Uh−ε (∂A),

and by compactness there exists a finite sequence of points {xi }ni=1 in ∂A such that def

Uh−2ε (∂A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a finite number of balls, it now follows by compactness of the embedding (11.3) for UB that there exist a subsequence, still denoted by {bAn }, and f ∈ BV(UB )N such that ∇bAn → f in L1 (UB )N . Since Uh (D) is bounded, Cb (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {bAn }, and A ⊂ D, ∂A = ∅, such that bAn → bA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, bAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇bA on UB and bAn converges to bA in W 1,1 (UB ). By Definition 5.1 and Theorem 3.1 (i), ∇bA and ∇bAn all belong

384

Chapter 7. Metrics via Oriented Distance Functions

to BVloc (RN )N since they are of bounded variation in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (∂A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral  |∇bAn − ∇bA |2 dx Uh (D)





|∇bAn − ∇bA |2 dx +

=

|∇bAn − ∇bA |2 dx.

Uh−2ε (∂A)

Uh (D)\Uh−2ε (∂A)

From part (i), the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of Uh−2ε (∂A). From (11.11) for all n ≥ N, |∇bAn (x)| = 1 a.e. in ∂An ⊃ Uh−3ε (∂An ) ⊃ Uh−2ε (∂A), |∇bA (x)| = 1 a.e. in ∂A ⊃ Uh−2ε (∂A). The second integral reduces to 

 |∇bAn − ∇bA |2 dx =

Uh (D)\Uh−2ε (∂A)

2 (1 − ∇bAn · ∇bA ) dx,

Uh (D)\Uh−2ε (∂A)

which converges to zero by weak convergence of ∇bAn to ∇bA in the space L2 (Uh (D))N in part (i) and the fact that |∇bA | = 1 almost everywhere in Uh (D)\ Uh−2ε (∂A). Therefore, since bAn → bA in C(Uh (D)) bAn → bA in H 1 (Uh (D))-strong, and by Theorem 4.1 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1. (iii) Properties (11.8). Consider the initial subsequence {bAn } which converges to bA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of bAn to bA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (∂A))N ×N ,  lim

n→∞

Uh (∂A)

−→



∇bAn · div Φ dx =

−→

∇bA · div Φ dx. Uh (∂A)

Each such Φ has compact support in Uh (∂A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (∂A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ).

12. Finite Density Perimeter and Compactness For n ≥ N (ε) consider the integral   −→ ∇bAn · div Φ dx = Uh (∂A)

∇bAn

385

 · div Φ dx = −→

Uh−2ε (∂A)

 ⇒

∇bAn

Uh (∂A)

−→

∇bAn · div Φ dx

U (∂A )

n h · div Φ dx ≤ D2 bAn M 1 (Uh (∂An )) ΦC(Uh (∂An )) ≤ c ΦC(Uh−2ε (∂A)) = c ΦC(Uh (∂A)) .

−→

By convergence of ∇bAn to ∇bA in the space L2 (Uh (D))-weak, then for all Φ ∈ D1 (Uh (∂A))N ×N  −→ ∇bA · div Φ dx ≤ c ΦC(Uh (∂A)) ⇒ D2 bA M 1 (Uh (∂A)) ≤ c. Uh (∂A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

12

Finite Density Perimeter and Compactness

´sio [14] is a reThe density perimeter introduced by D. Bucur and J.-P. Zole laxation of the (N − 1)-dimensional upper Minkowski content which leads to the compactness Theorem 12.1. Definition 12.1. Let h > 0 be a fixed real number and let ∅ = A ⊂ RN , ∂A = ∅. Consider the quotient mN (Uk (∂A)) def Ph (∂A) = sup , (12.1) 2k 0
dmN ≤ c 2k

⇒ mN (∂A) = 0.

Uk (∂A)

´sio [14] has been revisited The compactness result of D. Bucur and J.-P. Zole ´sio [38], and established in the W 1,p -topology by M. C. Delfour and J.-P. Zole from which convergence in all other topologies of Theorem 4.1 follows. Theorem 12.1. Let D = ∅ be a bounded open subset of RN and {An }, ∂An = ∅, be a sequence of subsets of D. Assume that ∃h > 0 and c > 0 such that ∀n,

Ph (∂An ) ≤ c.

(12.2)

Then there exist a subsequence {Ank } and A ⊂ D, ∂A = ∅, such that Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c, n→∞

∀p, 1 ≤ p < ∞,

bAnk → bA in W

1,p

(Uh (D))-strong.

(12.3) (12.4)

386

Chapter 7. Metrics via Oriented Distance Functions

Proof. The proof essentially rests on Theorem 4.3 since Ph (∂A) ≤ c implies mN (∂A) = 0. Since D is bounded, the family of oriented distance functions Cb (D) is compact in C(D) and W 1,p (D)-weak for all p, 1 ≤ p < ∞ (cf. Theorems 2.2 (ii) and 4.2 (iii)). So there exist bA ∈ Cb (D) and a subsequence, still indexed by n, such that bAn → bA in the above topologies. Moreover, for all k, 0 < k < h, and all ε, 0 < ε < h − k, ∃N (ε) > 0 such that ∀n ≥ N (ε),

Uk−ε (∂An ) ⊂ Uk (∂A) ⊂ Uk+ε (∂An )

(cf. proof of part (i) of Theorem 11.2). As a result for all n ≥ N (ε), mN (Uk−ε (∂An )) k − ε mN (Uk (∂A)) mN (Uk+ε (∂An )) k + ε k+ε ≤ ≤ ≤c 2(k − ε) k 2k 2(k + ε) k k mN (Uk+ε (∂An )) k + ε mN (Uk (∂A)) k+ε ≤ ≤ Ph (∂An ) ⇒ 2k 2(k + ε) k k mN (Uk (∂A)) k+ε ≤ lim inf Ph (∂An ) . (12.5) ⇒ n→∞ 2k k Going to the limit as ε goes to zero in the second and fourth terms mN (Uk (∂A)) ≤ lim inf Ph (∂An ) ≤ c n→∞ 2k ⇒ Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c ⇒ mN (∂A) = 0.

∀k, 0 < k < h,

n→∞

The theorem now follows from the fact that mN (∂A) = 0 and Theorem 4.3. Corollary 1. Let D = ∅ be a bounded open subset of RN and A, ∂A = ∅, be a subset of D such that ∃h > 0 and c > 0 such that Ph (∂A) ≤ c.

(12.6)

Then the mapping bA → Ph (∂A ) : Cb (D) → R ∪{+∞} is lower semicontinuous in A for the W 1,p (D)-topology. Proof. Since we have a metric topology, it is sufficient to prove the property for W 1,p (D)-converging sequences {bAn } to bA . From that point on the argument is the same as the one used to get (12.5) in the proof of Theorem 12.1 after the extraction of the subsequence. Remark 12.1. It is important to notice that even if {An } is a W 1,p -convergent sequence of bounded open subsets of RN with a uniformly bounded perimeter, the limit set A need not be an open set or have a nonempty interior int A such that bA = bint A . It would be tempting to say that bAn → bA implies d
An → d
A and use the open set int A = A for which d
A = d
int A to conclude that bA = bint A . This is incorrect as can be seen in the following example shown in Figure 7.2. Consider a family {An } of open rectangles in R2 of width equal to 1 and height 1/n, n ≥ 1, an integer. Their density perimeter is bounded by 4, and the bAn ’s converge to bA for A equal to the line of length 1 which has an empty interior.

13. Compactness and Uniform Fat Segment Property

An

1 n

1

387

A

1

Figure 7.2. W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R with uniformly bounded density perimeter to a set with empty interior. 2

13 13.1

Compactness and Uniform Fat Segment Property Main Theorem

In Theorem 6.11 of section 6.4 of Chapter 5, we have seen a compactness theorem for the family of subsets of a bounded holdall D satisfying the uniform cone property. This compactness theorem is no longer true when the uniform cone property is replaced by a uniform segment property. This is readily seen by considering the following example. Example 13.1. Given an integer n ≥ 1, consider the following sequence of open domains in R2 :   def Ωn = (x, y) ∈ R2 : |x| < 1 and |x|1/n < y < 2 . They satisfy the uniform segment property of Definition 6.1 (ii) of Chapter 2 by choosing λ = r = 1/4. The sequence {Ωn } converges to the closed set  def

A = (x, y) ∈ R2 : |x| ≤ 1 and 1 ≤ y ≤ 2 ∪ L,  def

L = (0, y) ∈ R2 : 0 ≤ y ≤ 1 in the uniform topologies associated with dΩn and bΩn or in the Lp -topologies associated with χΩn and χ
Ωn . However, the segment property is not satisfied along the line L and the corresponding family of subsets of the holdall D = B(0, 4) satisfying the uniform segment property with r = λ = 1/4 is not closed and, a fortiori, not compact. This example shows that a uniform segment property is too meager to make the corresponding family compact. Looking back at the proof of Theorem 6.11 of section 6.4 in Chapter 5, the fact that the cone is an open set around the segment is used critically. That was the motivation to introduce the more general uniform fat segment property (Definition 6.1 (iii) in section 6 of Chapter 2) that includes as special cases both the uniform cone and the uniform cusp properties (cf. Definitions 6.3 and 6.4 in section 6.4 of Chapter 2). However, the proof of Theorem 6.11 in Chapter 5 was based on the fact that the perimeter of all the sets in the family

388

Chapter 7. Metrics via Oriented Distance Functions

were uniformly bounded. There is no hope of getting that property for general H¨ olderian domains, and a completely different proof is necessary. Given a bounded open subset D of RN consider the family & Ω satisfies the uniform fat def L(D, r, O, λ) = Ω ⊂ D : . (13.1) segment property for (r, O, λ) We first give a general proof of the compactness of L(D, r, O, λ) in the C(D)- and W 1,p (D)-topologies associated with the oriented distance function bΩ . As a consequence, we get the compactness for the C(D)- and W 1,p (D)-topologies associated with dΩ and d
Ω and the Lp (D)-topologies associated with χΩ and χ
Ω . We recover the compactness Theorem 6.11 of Chapter 5 under the uniform cone property for χΩ in Lp (D) as a special case. In the last part of this section, we give several other ways to specify the compact families by using the equivalent conditions of Chapter 2 of the uniform fat segment property in terms of conditions on the local epigraphs via the dominating function and equicontinuity of the local graph functions. Theorem 13.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ) is not empty for r > 0, λ > 0, and an open / O. Then the family subset O of RN such that (0, eN ) ⊂ O and 0 ∈ def

B(D, r, O, λ) = {bΩ : ∀Ω ∈ L(D, r, O, λ)} is compact in C(D) and W 1,p (D). As a consequence the families def

Bd (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)}, def Bdc (D, r, O, λ) = {d
Ω : ∀Ω ∈ L(D, r, O, λ)}, def

Bd∂ (D, r, O, λ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in C(D) and W 1,p (D), and the families def

X(D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)}, def X c (D, r, O, λ) = {χ
Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in Lp (D). From Lemma 6.1 of Chapter 2, Theorem 13.1 can be specialized to structured open sets O. Corollary 1. The conclusions of Theorem 13.1 hold when O is specialized to open regions of the form 

(13.2) O(h, ρ, λ) = ξ ∈ RN : ξ  ∈ BH (0, ρ) and h(|ξ  |) < ξN < λ for dominating functions h ∈ H. Theorem 13.1 can be further specialized to Lipschitzian and H¨ olderian sets.

13. Compactness and Uniform Fat Segment Property

389

Corollary 2. The conclusions of Theorem 13.1 hold under the uniform cone property of Definition 6.3 (ii) in section 6.4 of Chapter 2 with O = O(h, ρ, λ) with h(θ) = θ/ tan ω and ρ = λ tan ω,

0 < ω < π/2,

(13.3)

and the uniform cusp property of Definition 6.4 (ii) of Chapter 2 with


O = O(h
, ρ, λ) with h
(θ) = λ (θ/ρ) ,

0 <  < 1.

The proof of Theorem 13.1 will require Theorem 4.3 and the following lemma. Lemma 13.1. Given a sequence {bΩn } ⊂ Cb (D) such that bΩn → bΩ in C(D) for some bΩ ∈ Cb (D), we have the following properties: ∀x ∈ Ω, ∀R > 0,

∃N (x, R) > 0, ∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅,

and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

(13.4)

Moreover, ∀x ∈ ∂Ω, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ Ωn ) = ∅, and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ Ω,

∃R > 0,

∀N > 0,

∃n ≥ N,

B(x, R) ∩ Ωn = ∅.

So there exists a subsequence {Ωnk }, nk → ∞, such that bΩnk (x) = dΩnk (x) ≥ R = 0 = dΩ (x) ≥ bΩ (x), which contradicts the fact that bΩnk → bΩ . Of course, the same assertion is true

for the complements and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

When x ∈ ∂Ω, x ∈ Ω ∩ Ω and we combine the two results. For the last result use the fact that the open ball cannot be partitioned into two nonempty disjoint open subsets. Proof of Theorem 13.1. (i) Compactness in C(D). Since for all Ω in L(D, r, O, λ), Ω is locally a C 0 -epigraph, ∂Ω = ∅, bΩ ∈ Cb (D), and mN (∂Ω) = 0. Consider an arbitrary sequence {Ωn } in L(D, r, O, λ). For D compact, Cb (D) is compact in C(D) and there exist Ω ⊂ D, ∂Ω = ∅, and a subsequence {Ωnk } such that bΩnk → bΩ in C(D). It remains to prove that Ω ∈ L(D, r, O, λ), that is, ∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + λAO ⊂ int Ω.

390

Chapter 7. Metrics via Oriented Distance Functions

From Lemma 13.1, for each x ∈ ∂Ω, for all k ≥ 1, there exists nk ≥ k such that   B x, r/2k ∩ ∂Ωnk = ∅. Denote by xk an element of that intersection:   ∀k ≥ 1, xk ∈ B x, r/2k ∩ ∂Ωnk . By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the first part of the lemma, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that for all k ≥ 1, B(y, r/2k ) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of that intersection. By construction yk ∈ Ωnk → y ∈ Ω ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this, note that y ∈ B(x, r) and that ∃ρ > 0,

B(y, ρ) ⊂ B(x, r)

and |y − x| + ρ/2 < r.

Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r ρ ρ/ r r ≤ k + r − + k ≤ r + k−1 − < r. 2 2 2 2 2 Since r/ρ > 1 the result is true for r 2k−1

−

ρ <0 2

⇒ k > 2 + log

  r . ρ

So we have constructed a subsequence {Ωnk } such that for k ≥ K xk ∈ ∂Ωnk → x ∈ ∂Ω

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ Ω ∩ B(x, r).

For all k, there exists Ak ∈ O(N), Ak ∗Ak = ∗Ak Ak = I, such that yk + λAk O ⊂ int Ωnk . Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N), A ∗A = ∗A A = I,

Ak → A.

Now consider z ∈ y + λAO. Since y + λAO is open there exists ρ > 0 such that B(z, ρ) ⊂ y + λAO, and there exists K  ≥ K such that ∀k ≥ K  ,

B(z, ρ/2) ⊂ yk + λAk O ⊂ int Ωnk = Ωnk

⇒ B(z, ρ/2) ⊃ Ωnk ⇒ 0 < ρ/2 = d
B(z,ρ/2) (z) ≤ d
Ωn (z) → d
Ω (z) k

⇒ 0 < ρ/2 ≤ d
Ω (z)

⇒ z ∈ Ω = int Ω

⇒ y + λAO ⊂ int Ω.

Hence Ω ⊂ D satisfies the uniform fat segment property.

13. Compactness and Uniform Fat Segment Property

391

(ii) W 1,p (D)-compactness. From Theorem 4.1 (i), it is sufficient to prove the result for p = 2. Consider the subsequence {Ωnk } ⊂ L(D, λ, O, r) and let Ω ∈ L(D, λ, O, r) be the set previously constructed such as bΩnk → bΩ in C(D). Hence bΩnk → bΩ in L2 (D). Since D is compact   |bΩnk |2 dx ≤ diam (D)2 dx ≤ diam (D)2 mN (D), ∀Ω ⊂ D, D D   2 |∇bΩnk | dx ≤ dx = mN (D), D

D

and there exists a subsequence, still denoted by {bΩnk }, which converges weakly to bΩ . Since all the sets in L(D, r, O, λ) verify a uniform segment property, they are locally C 0 -epigraphs, mN (∂Ωnk ) = 0 = mN (∂Ω) and, by Theorem 4.3, bΩnk → bΩ in W 1,p (D)-strong, 1 ≤ p < ∞. (iii) The other compactness follows by continuity of the maps (4.6) and (4.7) in Theorem 4.1 and the fact that mN (∂Ω) = 0 implies χint Ω = χΩ and χint
Ω = χ
Ω almost everywhere for Ω ∈ L(D, λ, O, r).

13.2

Equivalent Conditions on the Local Graph Functions

We have shown in Theorem 6.2 of Chapter 2 that the uniform fat segment property is equivalent to the equi-C 0 epigraph property. This means that we can equivalently characterize the compact family L(D, r, O, λ) in terms of conditions on the local graph functions such as the ones introduced in Theorem 5.1 of Chapter 2. Recall the following equivalent conditions: (i) Ω verifies the fat segment property. (ii) Ω is an equi-C 0 -epigraph; that is, Ω is a C 0 -epigraph and the local graph functions are uniformly bounded and equicontinuous. (iii) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ  , ξ  ∈ BH (0, ρ) such that |ξ  − ζ  | < ρ,

|ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |). (13.5)

(iv) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ  ∈ BH (0, ρ),

ax (ξ  ) ≤ h(|ξ  |).

(13.6)

Recall also from Definition 5.1 (ii) in Chapter 2 that Ω is said to be a C 0 epigraph if it is locally a C 0 -epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

392

Chapter 7. Metrics via Oriented Distance Functions

By applying each of the last three characterizations to all members of a family of subsets of a bounded holdall D, we get the analogue of the compactness Theorem 13.1 under the uniform fat segment property. Theorem 13.2. Let D be a bounded open nonempty subset of RN and consider the family L(D, ρ, U ) of all subsets Ω of D that verify the C 0 -epigraph property: there exist ρ > 0 and a bounded open neighborhood U of 0 such that 

def (13.7) U ⊂ ζ ∈ RN : PH (ζ) ∈ BH (0, ρ) , V = BH (0, ρ), and for each Ω ∈ L(D, ρ, U ) and each x ∈ ∂Ω, there exists AΩ x ∈ O(N) such that def

U Ω (x) = x + AΩ xU

(13.8)

Ω and a C 0 -graph function aΩ x : V → R such that ax (0) = 0 and

  U Ω (x) ∩ ∂Ω = U Ω (x) ∩ x + Ax (ζ  + ζN eN ) : ζ  ∈ V, ζN = aΩ x (ζ ) ,

  U Ω (x) ∩ int Ω = U Ω (x) ∩ x + Ax (ζ  + ζN eN ) : ζ  ∈ V, ζN > aΩ x (ζ ) .

Then, for all h ∈ H, the subfamilies   {aΩ  x : Ω ∈ L(D, ρ, U ), x ∈ ∂Ω}   def L0 (D, ρ, U ) = Ω ∈ L(D, ρ, U ) : is uniformly bounded ,     and equicontinuous  def

  L1 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : ∀x ∈ ∂Ω, ∀ζ  ∈ V, aΩ x (ζ ) ≤ h(|ζ |) ,   ∀x ∈ ∂Ω, ∀ζ  , ξ  ∈ V     def   L2 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : such that |ξ − ζ | < ρ     |ax (ξ  ) − ax (ζ  )| ≤ h(|ξ  − ζ  |)

(13.9)

(13.10)

(13.11) (13.12)

are compact (possibly empty) in C(D) and W 1,p (D), 1 ≤ p < ∞, and the conclusions of Theorem 13.1 hold. Remark 13.1. The geometric fat segment property of Corollary 1 was introduced in the form (13.2) ´sio [37] under the name uniform in the book of M. C. Delfour and J.-P. Zole cusp property in 2001, where the condition was specialized to (13.3) for Lipschitzian and H¨ olderian domains as stated in Corollary 2. Condition (13.10) is a streamlined version of the sufficient condition given in an unpublished report by D. Tiba [2], announced without proof in the note ¨ ki, and D. Tiba [1] in 2000, and later documented by W. Liu, P. Neittaanma in D. Tiba [3] in 2003. He gets a compactness result by introducing the following conditions on the local C 0 -epigraphs: the boundaries of the domains belong (after a rotation and a translation) to a family F of uniformly equicontinuous functions defined in a fixed neighborhood V of 0 in RN −1 . This means that there exists a modulus of continuity µ(ε) > 0 such that ∀ε > 0, ∀f ∈ F, ∀x, y ∈ V,

|x − y| < µ(ε)

⇒ |f (y) − f (x)| < ε.

Then the compactness follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2.

14. Compactness under Uniform Fat Segment Property and Bound on a Perimeter 393 The connection between that condition and our geometric condition was ´sio [1, 2] in first established in M. C. Delfour, N. Doyon, and J.-P. Zole 2005. Condition (13.11) that relaxes the equicontinuity condition (13.10) to the simpler dominating function condition and condition (13.12) were both introduced ´sio [43] in 2007 along with the general form in M. C. Delfour and J.-P. Zole of the uniform fat segment property that involves only the unparametrized open set O.

14

Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter

Domains that are locally Lipschitzian epigraphs or equivalently satisfy the local uniform cone property enjoy the additional property that for any bounded open set D the (N − 1)-Hausdorff measure of D ∩ ∂Ω is finite. In general this is no longer true for domains verifying a uniform cusp property for some function h(θ) = |θ/ρ|α , 0 < α < 1 (cf., for instance, Example 6.2 given6 in section 6.5 of Chapter 2 of a bounded domain Ω in RN for which HN −1 (∂Ω) = +∞ and the Hausdorff dimension of ∂Ω is exactly N − α). Yet, there are many domains with cusps whose boundary has finite HN −1 measure.

14.1

De Giorgi Perimeter of Caccioppoli Sets

One of the classical notions of perimeter is the one introduced in Definition 6.2 in section 6 of Chapter 5 in the context of the problem of minimal surfaces for Caccioppoli sets. Theorem 13.1 extends to the following subfamily of L(D, r, O, λ) for a bounded open subset D of RN and a constant c :   Ω satisfies the uniform fat     def L(D, r, O, λ, c ) = Ω ⊂ D : segment property for (r, O, λ) . (14.1)      and PD (Ω) ≤ c Theorem 14.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ, c ) is not empty for r > 0, λ > 0, c > 0, and a / O. Then the family bounded open subset O of RN such that (0, eN ) ⊂ O and 0 ∈ B(D, r, O, λ, c ) = {bΩ : ∀Ω ∈ L(D, r, O, λ, c )} def

is compact in C(D) and W 1,p (D). As a consequence the families Bd (D, r, O, λ, c ) = {dΩ : ∀Ω ∈ L(D, r, O, λ, c )}, def Bdc (D, r, O, λ, c ) = {d
Ω : ∀Ω ∈ L(D, r, O, λ, c )}, def Bd∂ (D, r, O, λ, c ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ, c )} def

6 This

´sio [1]. was originally in M. C. Delfour, N. Doyon, and J.-P. Zole

394

Chapter 7. Metrics via Oriented Distance Functions

are compact in C(D) and W 1,p (D), and the families X(D, r, O, λ, c ) = {χΩ : ∀Ω ∈ L(D, r, O, λ, c )} , def X c (D, r, O, λ, c ) = {χ
Ω : ∀Ω ∈ L(D, r, O, λ, c )} def

are compact in Lp (D).

14.2

Finite Density Perimeter

It is also possible to use the density perimeter introduced in Definition 12.1 in place of the De Giorgi perimeter. The proof of the next result combines Theorem 14.1, which says that the family L(D, r, O, λ, c ) is compact, with Theorem 12.1, which says that the family of sets verifying (12.2) is compact in W 1,p (D). The intersection of the two families of oriented distance functions is compact in W 1,p (D). Theorem 14.2. For fixed h > 0 and an open subset Ω of RN , Theorem 14.1 remains true when PD (Ω) is replaced by the density perimeter Ph (∂Ω).

15

The Families of Cracked Sets

In this section we introduce new families of thin sets, that is, sets A such that mN (∂A) = 0, by introducing conditions on the semiderivative of the distance function at points of the boundary. They have been used in M. C. Delfour and ´sio [38] in the context of the image segmentation problem of D. MumJ.-P. Zole ford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks and sets that can be made up of components of different codimensions. The Hausdorff (N − 1) measure of their boundary is not necessarily finite. Yet, compact families (in the W 1,p -topology) of such sets can be constructed. First recall lower and upper semiderivatives7 of Dini type for the differential quotient of a function f : V (x) ⊂ RN → R defined in a neighborhood V (x) of a point x ∈ RN in the direction d ∈ RN : f (x + td) − f (x) def f (x + td) − f (x) , = lim inf δ0 0
lim inf t0 w→d

f (x + tw) − f (x) def = lim δ0 t

inf

0
f (x + tw) − f (x) , t

7 Note that the (t, d), 0 < t < δ, is allowed in the inf and the sup of the last two definitions and the constraint (t, w) = (0, v) can be removed.

15. The Families of Cracked Sets lim sup t0 w→d

f (x + tw) − f (x) def = lim δ0 t

395 sup 0
f (x + tw) − f (x) . t

Definition 15.1. def Let Ω ⊂ RN be such that Γ = ∂Ω = ∅. (i) Ω is said to be weakly cracked if ∀x ∈ Γ, ∃d ∈ RN , |d| = 1, such that lim sup t0 w→d

dΓ (x + tw) > 0. t

(15.1)

dΓ (x + td) > 0. t

(15.2)

dΓ (x + td) > 0. |t|

(15.3)

(ii) Ω is said to be cracked if ∀x ∈ Γ, ∃d ∈ RN , |d| = 1, such that lim inf t0

(iii) Ω is said to be strongly cracked 8 if ∀x ∈ Γ, ∃d ∈ RN , |d| = 1, such that lim inf t→0

(iv) A set ∅ = A ⊂ RN is said to be thin if mN (A) = 0. Strongly cracked implies cracked, and cracked implies weakly cracked since lim inf t0 w→d

dΓ (x + tw) dΓ (x + td) ≤ lim inf t0 t t dΓ (x + td) dΓ (x + tw) ≤ lim sup ≤ lim sup . t t t0 t0 w→d

The special terminology of Definition 15.1 is introduced to provide an intuitive description of the sets. It is motivated by the fact that the boundary of such a set has zero N -dimensional Lebesgue measure (cf. Lemma 15.1) and can be made up of internal cracks, external hairs, cusps, or points, as shown in Figure 7.3. The weakly cracked property is verified in any point of the boundary where the gradient of dΓ does not exist; in boundary points where the gradient exists it is not identically 0. This is a very large family of sets that includes domains which are locally the epigraph of a continuous function. There are obvious variations of the above definitions and the forthcoming compactness Theorem 15.1 by replacing dΓ by dΩ , d
Ω , or bΩ . 8 Here

the definition is lim inf t→0

f (x + td) − f (x) def f (x + td) − f (x) = lim inf . |t| δ0 0<|t|<δ t

396

Chapter 7. Metrics via Oriented Distance Functions

Figure 7.3. Example of a two-dimensional strongly cracked set. Lemma 15.1. Let Ω ⊂ RN , let Γ = ∅, let x ∈ Γ, and let d, |d| = 1, be a direction in RN . If the semiderivative ddΓ (x; d) does not exist, then lim sup t0 w→d

dΓ (x + tw) > 0. t

Proof. Since the function dΓ is Lipschitzian, the limit of the quotient def

ddΓ (x; d) = lim

t0

dΓ (x + td) − dΓ (x) t

exists if and only if the limit of the quotient def

dH dΓ (x; d) = lim

t0 w→d

dΓ (x + tw) − dΓ (x) t

exists (cf. Theorem 2.4 (i) in Chapter 9). Moreover, by the Lipschitz continuity, dΓ (x + tw) − dΓ (x) tw ≤ = |w| → |d| t t and hence the liminf and the limsup of the quotient exist and are finite. For x ∈ Γ, dΓ (x) = 0. Therefore ddΓ (x; d) does not exist if and only if lim sup t0 w→d

dΓ (x + tw) dΓ (x + tw) > lim inf ≥0 t0 t t w→d

since the last term is nonnegative. This completes the proof. Theorem 15.1. Let ∅ = Ω ⊂ RN . (i) Given a weakly cracked set Ω, mN (Γ) = 0

and

Γ ⊂ int Ω ∪ int Ω.

Moreover in any point x ∈ Γ either ∇bΩ (x) exists and is different from zero or ∇bΩ (x) does not exists (set of cracks).

15. The Families of Cracked Sets

397

(ii) Given a cracked set Ω, for each x ∈ Γ def

∃ a direction d ∈ RN , |d| = 1, such that 1 ≥ (x) = lim inf t0

dΓ (x + td) >0 t

and for all ε, 0 < ε < (x), there exists δ > 0 such that ∀t, 0 < t < δ,

dΓ (x + td) ≥ ((x) − ε)t

⇒ x + C(δ cos ω, ω, d) ⊂ R \Γ, N

def

sin ω = (x) − ε, 0 < ω ≤ π/2

(C(λ, ω, d) is the open cone in 0 of direction d, height λ, and aperture ω). (iii) Given a strongly cracked set Ω, for each x ∈ Γ there exists a direction d ∈ RN , |d| = 1, such that def

1 ≥ (x) = lim inf t→0

dΓ (x + td) >0 |t|

and for all ε, 0 < ε < (x), there exists δ > 0 such that ∀t, 0 < |t| < δ,

dΓ (x + td) ≥ ((x) − ε)|t|

⇒ x ± C(δ cos ω, ω, d) ⊂ RN \Γ,

sin ω = (x) − ε, 0 < ω ≤ π/2.

This means that in each point of the boundary of a cracked set, it is possible to place an open cone that does not intersect the boundary. If, in addition, the set is strongly cracked, this cone can be replaced by a double cone. Proof. (i) We already know that ∇dΓ exists almost everywhere in RN and that, whenever it exists, |∇dΓ (x)| =

0, 1,

if x ∈ Γ, if x ∈ /Γ

(cf. Theorem 3.3 (vi) in Chapter 6). Therefore, if ∇dΓ (x) exists in a point x ∈ Γ, ∇dΓ (x) = 0 and for all d, |d| = 1, lim sup t0 w→d

dΓ (x + tw) dΓ (x + tw) − dΓ (x) = lim sup = ∇dΓ (x) · d = 0, t t t0 w→d

which contradicts the weakly cracked property. Hence the points of Γ are points where ∇dΓ (x) does not exist, which is itself a set of zero measure. Moreover, if ∇bΩ (x) exists in a point x ∈ ∂Ω, then bΩ (x) = 0 and for all d, |d| = 1, lim

t0 w→d

bΩ (x + tw) bΩ (x + tw) − bΩ (x) = lim = ∇bΩ (x) · d. t0 t t w→d

398

Chapter 7. Metrics via Oriented Distance Functions

If ∇bΩ (x) = 0, then for all d bΩ (x + tw) bΩ (x + tw) − bΩ (x) = lim = ∇bΩ (x) · d = 0 t0 t0 t t w→d w→d bΩ (x + tw) dΓ (x + tw) =0 ⇒ lim = lim t0 t0 t t lim

w→d

w→d

and ∇d∂Ω (x) = 0, which contradicts our assumption. Therefore if ∇bΩ (x) exists in a point x ∈ ∂Ω, ∇bΩ (x) = 0. For any x ∈ Γ introduce the notation dΓ (x + tw) ¯ def . (x) = lim sup t t0 w→d

¯ > 0 and for all δ > 0 By assumption (x) ¯ ≤ (x)

sup 0
dΓ (x + tw) . t

Hence there exist sequences {tn }, tn → 0, and {wn }, wn → d, such that dΓ (x + tn wn ) ¯ 0 < (x)/2 ≤ t

def

⇒ RN \Γ  xn = x + tn wn → x

and necessarily Γ ⊂ int Ω ∪ int Ω ⊂ int Ω ∪ int Ω. (ii) Given a cracked set Ω, for each x ∈ Γ there exists a direction d ∈ RN , |d| = 1, such that def

(x) = lim inf t0

dΓ (x + td) >0 t

and for all ε, 0 < ε < (x), there exists δ > 0 such that ∀t, 0 < t < δ,

dΓ (x + td) ≥ ((x) − ε)t.

Recall that since dΓ is Lipschitzian of constant 1, we necessarily have 1 ≥ (x) for |d| = 1. Therefore x + C(δ cos ω, ω, d) ⊂ RN \Γ,

sin ω = (x) − ε, 0 < ω ≤ π/2.

(iii) This is similar to the proof of part (ii). Theorem 15.2. Let D be a bounded open subset of RN and α, 0 < α ≤ 1, and h > 0 be real numbers.9 Consider the families   Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,   def ¯ : F(D, h, α) = Ω ⊂ D , dΓ (x + td)  ≥ α such that inf (15.4) 0
Cbh,α (D) = {bΩ : Ω ∈ F(D, h, α)} , 9 In view of the fact that the distance function d is Lipschitzian with constant 1, we necessarily Γ have 0 < α ≤ 1.

15. The Families of Cracked Sets

399

  Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,   def ¯ : Fs (D, h, α) = Ω ⊂ D , dΓ (x + td)  ≥ α such that inf 0<|t|
(15.5)

def

(Cbh,α )s (D) = {bΩ : Ω ∈ Fs (D, h, α)} . Then Cbh,α (D) and (Cbh,α )s (D) are compact in W 1,p (D), 1 ≤ p < ∞. Proof. (i) The family Cbh,α (D) is contained in Cb (D), which is compact in the uniform topology of C(D) and in the weak topology of W 1,p (D), 1 ≤ p < ∞ (cf. Theorem 2.2 (ii) and Theorem 4.2 (iii) in Chapter 6). Therefore, given a sequence {bΩn } in Cbh,α (D), there exist a subsequence, still denoted by {bΩn }, and bΩ ∈ Cb (D) such that bΩn → bΩ in W 1,p (D)-weak and C(D). In addition, by definition of the elements of Cbh,α (D), condition (15.2) is verified and each Ωn has thin boundary. We want to show that bΩ ∈ Cbh,α (D). Once this is proved, from Theorem 15.1 (i), Ω has a thin boundary. Hence the weak W 1,p (D) convergence implies the strong W 1,p (D) convergence by Theorem 4.3. In view of the continuity of the map bΩ → dΓ = |bΩ | : C(D) → C(D), dΓn → dΓ in C(D). From Lemma 13.1, given x ∈ Γ, there exists a subsequence of {bΩn }, still denoted by {bΩn }, and for each n ≥ 1 points xn ∈ Γn ∩ B(x, 1/n). Hence xn → x. By assumption ∀n ≥ 1, ∃dn ∈ RN , |dn | = 1, such that

inf

0
dΓn (xn + tdn ) ≥ α. t

Since the dn ’s have norm 1, there exist a subsequence, still denoted by {dn }, and d, |d| = 1, such that dn → d. Fix t, 0 < t < δ. Given ε > 0, there exists N such that for all n ≥ N |xn − x| < εt,

|dn − d| < δn < ε,

dΓn − dΓ C(D) < εt.

Fix n = N and consider the following estimates: dΓ (x + td) t dΓn (xn + tdn ) |dΓn (x + tdn ) − dΓn (xn + td)| − ≥ t t |dΓ (x + td) − dΓ (xn + td)| |dΓ (xn + td) − dΓn (xn + td)| − − t t |x − xn | dΓ − dΓn CD) ≥ α − ε − |d − dn | − − ≥ α − 4ε t t dΓ (x + td) dΓ (x + td) ≥ α − 4ε ⇒ inf ≥ α. ⇒ ∀ε > 0, inf 0
400

16 16.1

Chapter 7. Metrics via Oriented Distance Functions

A Variation of the Image Segmentation Problem of Mumford and Shah Problem Formulation

In this section we specialize to the segmentation of (N -dimensional) images where the segmentation can be composed of objects of codimension greater than or equal to one. To represent the set of Figure 7.3 as the boundary of an open set one can use the unbounded plane R2 minus all the lines and the points in Figure 7.3. If it is important that the open set be bounded, a fixed open frame D is introduced. The open set Ω in Figure 7.4 is then defined as the interior of the bounded open frame D minus all the lines and points used to draw the picture.

open frame D

Figure 7.4. The two-dimensional strongly cracked set of Figure 7.3 in an open frame D.

Definition 16.1. Let D be a bounded open subset of RN with Lipschitzian boundary. (i) An image in the frame D is specified by a function f ∈ L2 (D). (ii) We say that {Ωi }i∈I is an open partition of D if {Ωi }i∈I is a family of disjoint connected open subsets of D such that mN (∪i∈I Ωi ) = mN (D) and mN (∂ ∪i∈I Ωi ) = 0. Denote by P(D) the family of all such open partitions of D. Given an open partition {Ωi }i∈I of D, associate with each i ∈ I a function ϕi ∈ H 1 (Ωi ). In its intuitive form the problem formulated by D. Mumford and J. Shah [2] aims at finding an open partition P = {Ωi }i∈I in P(D) solution of the following minimization problem:  inf1 ε |∇ϕi |2 + |ϕi − f |2 dx (16.1) inf P ∈P(D)

i∈I

ϕi ∈H (Ωi )

Ωi

16. A Variation of the Image Segmentation Problem of Mumford and Shah

401

for some fixed constant ε > 0. The term in ε times the norm of the gradient can be seen as a Tikhonov regularization on each Ωi . Observe that without the condition mN (∪i∈I Ωi ) = mN (D) the empty set would be a solution of the problem or a phenomenon of the type discussed in Remark 12.1 and the phenomenon of Figure 7.2 could occur. The question of existence requires a more specific family of open partitions or a penalization term which preserves the “length” of the interfaces in some appropriate sense:  inf inf1 ε |∇ϕi |2 + |ϕi − f |2 dx + c HN −1 (∂ ∪i∈I Ωi ) (16.2) P ∈P(D)

i∈I

ϕi ∈H (Ωi )

Ωi

for some c > 0. The choice of the relaxation of the (N − 1)-Hausdorff measure HN −1 is critical as discussed in section 9 of Chapter 1. Another way of looking at the problem would be to minimize the number I of open subsets of the open partition, but this seems more difficult to formalize.

16.2

Cracked Sets without the Perimeter

In this section we specialize the compact family of Theorem 15.2 to get the existence of a solution to the following minimization problem:  inf

ε |∇ϕ|2 + |ϕ − f |2 dx.

inf

Ω∈F (D,h,α) ϕ∈H 1 (Ω) Ω open ⊂D, mN (Ω)=mN (D)

(16.3)

Ω

This allows for open sets Ω with HN −1 (Γ) = ∞. The pair (h, α) are the control parameters of the segmentation. Recall the characterization of Theorem 15.1 (ii), which says that in each point of the boundary there exists a small open cone of uniform height and aperture which does not intersect the boundary. Theorem 16.1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundary and real numbers h, α, and ε > 0, there exist an open subset Ω∗ of D in F(D, h, α) such that mN (Ω∗ ) = mN (D) and a function y ∈ H 1 (Ω∗ ) that are solutions of problem (16.3). 16.2.1

Technical Lemmas

The proof of the existence theorems will require the following technical results. Lemma 16.1. Given a subset A of RN with nonempty boundary ∂A, ∃ an open subset Ω ⊂ RN such that bA = bΩ if and only if dA = dint A or equivalently A¯ = int A. Proof. By definition for Ω open bA = bΩ ⇐⇒ dA = dΩ and d
Ω = d
A ,

402

Chapter 7. Metrics via Oriented Distance Functions

which is also equivalent to ¯ and Ω = A ⇐⇒ A¯ = Ω ¯ and Ω = int A. A¯ = Ω ¯ Hence the necessary and sufficient condition finally reduces to int A = A. Lemma 16.2. Let D ⊂ RN be bounded open with Lipschitzian boundary. Then Ω ⊂ D, Γ = ∅, mN (Γ) = 0, and mN (Ω) = mN (D)

⇒ int Ω = Ω.

Proof. By contradiction. If int Ω  Ω, there exists x ∈ Ω such that dint Ω (x) = ρ > 0 and hence B(x, ρ) ⊂ Ω. Therefore Ω ∩ B(x, ρ) ⊂ Γ and mN (Ω ∩ B(x, ρ)) ≤ mN (Γ) = 0. By assumption mN (D) = mN (Ω) = mN (Ω) implies mN (D ∩ B(x, ρ)) = mN (Ω∩B(x, ρ)) = 0. Since Ω ⊂ D, there exists x ∈ D such that mN (D∩B(x, ρ)) = 0. But this is a contradiction since D is an open set with Lipschitzian boundary. 16.2.2

Another Compactness Theorem

The compactness of the following special family of cracked sets contained in a frame is a corollary to the compactness Theorem 15.2. Theorem 16.2. Let D be a bounded open subset of RN with Lipschitzian boundary and h > 0 and α > 0 be real numbers. Consider the family   Ω open ⊂ D and mN (Ω) = mN (D)        and ∀x ∈ Γ, ∃d, |d| = 1, def c ¯ , F (D, h, α) = Ω ⊂ D :   dΓ (x + td)   (16.4)    such that inf ≥α  0
Cbc,h,α (D) = {bΩ : Ω ∈ F c (D, h, α)} . Then Cbc,h,α (D) is compact in W 1,p (D), 1 ≤ p < ∞. Proof. By standard arguments and Lemma 16.2. The conclusion follows from Theorem 15.2 by adding the constraint mN (Ωn ) = mN (D) which will be verified for the limit set Ω for which a subsequence of {bΩn } converges to bΩ in W 1,1 (D) and hence {χΩn } converges to χΩ in L1 (D). 16.2.3

Proof of Theorem 16.1

Proof of Theorem 16.1. (i) For each open Ω ∈ F c (D, h, α), the problem  def ε|∇ϕ|2 + |ϕ − f |2 dx inf1 F (Ω, ϕ), F (Ω, ϕ) = ϕ∈H (Ω)

Ω

has a unique solution y in H 1 (Ω) since the objective function F (Ω, ϕ) is continuous and coercive on H 1 (Ω). Define  def inf inf ε |∇ϕ|2 + |ϕ − f |2 dx. m = c 1 Ω∈F (D,h,α) ϕ∈H (Ω)

Ω

16. A Variation of the Image Segmentation Problem of Mumford and Shah

403

(ii) The minimum is finite since the objective function is positive and   inf1 ε |∇ϕ|2 + |ϕ − f |2 dx ≤ |f |2 dx ∀ open Ω ∈ F c (D, h, α), ϕ∈H (Ω)

Ω

Ω

by choosing ϕ = 0. Let {Ωn } be a minimizing sequence of open subsets of D in F c (D, h, α) and for each n let yn ∈ H 1 (Ωn ) be the minimizing element of F (Ωn , ϕ) over H 1 (Ωn ). Therefore  ε |∇yn |2 + |yn − f |2 dx → m Ωn  ⇒ ∃c > 0 such that ∀n, ε |∇yn |2 + |yn − f |2 dx ≤ c. Ωn

By coercivity the sequence {yn } is uniformly bounded in H 1 (Ωn ); that is, there exists a constant c > 0 such that ∀n,

yn L2 (Ωn ) ≤ c and ∇yn L2 (Ωn ) ≤ c.

By Theorem 16.2, there exist a subsequence of {Ωn } and an open set Ω ∈ F c (D, h, α) such that bΩn → bΩ in H 1 (D) and C(D). In particular d
Ωn → d
Ω in C(D). By the compactivorous property of Theorem 2.4 (iii) in Chapter 6 ∀K compact ⊂ int Ω,

∃N such that ∀n ≥ N,

K ⊂ Ωn .

Moreover χΩn → χΩ ∈ L2 (D)

⇒ χΩn  χΩ ∈ L∞ (D)-weak∗

⇒ f χΩn  f χΩ ∈ L (D)-weak. 2

(16.5) (16.6)

Define the distributions  def ˜ yn , ϕ = yn ϕ dx, ∀ϕ ∈ D(D), Ω  n 9n , Φ def = ∇yn · Φ dx, ∀Φ ∈ D(D)N , ∇y Ωn  y˜n div Φ dx, ∀Φ ∈ D(D)N . ∇˜ yn , Φ = − D

9n with the extensions of yn and It is readily seen that we can identify y˜n and ∇y ∇yn by zero from Ωn to D. As a result there exist subsequences, still denoted by 9n }, and y˜ ∈ L2 (D) and Y ∈ L2 (D)N such that y˜n  y˜ in L2 (D)-weak {˜ yn } and {∇y 9 and ∇yn  Y in L2 (D)N -weak. By the compactivorous property, for all Φ ∈ D( Ω), there exists N such that ∀n > N,

supp Φ ⊂ Ωn

⇒ Φ ∈ D(Ωn ).

404

Chapter 7. Metrics via Oriented Distance Functions

Therefore for all n > N 9n , Φ = − ∇˜ yn − ∇y



 yn div Φ dx −

∇yn · Φ dx = 0

Ωn

Ωn

since yn ∈ H 1 (Ωn ). But D(Ωn ) ⊂ D(D) and by letting n go to infinity 



0=−



yn div Φ dx − Ωn

Ωn

⇒ ∀Φ ∈ D(Ω),



∇yn · Φ dx → − y˜ div Φ dx − D   y˜ div Φ dx + Y · Φ dx = 0. D

Y · Φ dx D

D

Define the new distribution def



y, ϕ =

∀ϕ ∈ D(Ω).

y˜ ϕ dx, Ω

It is easy to check that y ∈ L2 (Ω) and hence 

 ∀Φ ∈ D(Ω),

  Y · Φ dx = y˜ div Φ dx + Y · Φ dx D Ω Ω   y˜ div Φ dx = Y · Φ dx ∇y, Φ = −

y˜ div Φ dx +

0= D

⇒ ∀Φ ∈ D(Ω),

Ω

Ω

⇒ ∇y = Y |Ω ∈ L2 (Ω)

⇒ y ∈ H 1 (Ω).

(iv) Coming back to our objective function 

 inf 1

ϕ∈H (Ωn )

ε |∇ϕ|2 + |ϕ − f |2 dx = Ωn

ε |∇yn |2 + |yn − f |2 dx

Ωn 9n |2 + |˜ = ε |∇y yn − f χΩn |2 dx. D

By convexity and continuity of the objective function with respect to the pair (˜ yn − 9n ) in L2 (D) × L2 (D)N and the fact that, from (16.6), (˜ 9n )  f χΩn , ∇y yn − f χΩn , ∇y (˜ y − f χΩ , Y ) in L2 (D) × L2 (D)N -weak, 



9n |2 + |˜ ε |∇y yn − f χΩn |2 dx  ε |∇yn |2 + |yn − f |2 dx = m = lim inf n→∞ Ωn   2 2 ⇒ ε |∇y| + |y − f | dx = ε |∇y|2 + |y − f χΩ |2 dx ≤ m.

ε |Y |2 + |˜ y − f χΩ |2 dx ≤ lim inf D

n→∞

Ω

D

D

By definition of the minimum we have the equality and there exist an open set Ω ∈ F c (D, h, α) and y ∈ H 1 (Ω) solution of the segmentation problem.

16. A Variation of the Image Segmentation Problem of Mumford and Shah

16.3

405

Existence of a Cracked Set with Minimum Density Perimeter

Theorem 16.1 gives an existence result for the family F(D, h, α) of open sets Ω such that mN (Ω) = mN (D) without constraint on the “perimeter” of Ω. Denote by F ∗ (D, h, α) the set of solutions to problem (16.3). In general, the perimeter can be infinite as can be seen from the following example. Example 16.1. The function f : D → R is defined as follows: f (x) =

if x ∈ Ω1 , if x ∈ Ω2 ,

1, 0,

(16.7)

where D = {(x, y) : −2 < x < 3, −1 < y < 3}, Ω = Ω1 ∪ Ω2 , Ω2 = D\Ω1 , and the open set Ω1 is constructed below (see Figure 7.5). The set Ω with y = f is a solution of problem (16.3) with infinite perimeter. The set Ω1 is a two-dimensional example similar to Examples 6.1 and 6.2 of section 6.5 in Chapter 2 of an open domain satisfying the uniform cusp condition for the function h(θ) = θ α , 0 < α < 1. It can easily be generalized to an N -dimensional example. Consider the open domain Ω1 in R2 def

Ω1 = {(x, y) : −1 < x ≤ 0 and 0 < y < 2} ∩ {(x, y) : 0 < x < 1 and f (x) < y < 2} ∩ {(x, y) : 1 ≤ x < 2 and 0 < y < 2} , where f : [0, 1] → R is defined as follows: def

f (x) =

∞



fk (x),

k=0

 1 1 fk : 1 − k , 1 − k+1 → R . 2 2 α

Associate with 0 < α < 1 and k ≥ 0 the even integer ηk = 2 [(2k+1 ) 1−α ], where [β] is the smallest integer greater than or equal to β. Assume that for each k ≥ 0 def

fk =



ηk /2

gk,j

gk,j : [xk,j−1 , xk,j−1 + δk ] → R,

j=1 def

xk,j = 1 −

1 + (j − 1)2δk , 1 ≤ j ≤ ηk /2, 2k

def

δk =

1 ηk 2k+1

and that the function gk,j is given by the expression  0, 0 ≤ x < xk,j−1 ,    (x − x α ) , xk,j−1 ≤ x ≤ xk,j−1 + δk , def k,j−1 gk,j (x) = α  (2xk,j−1 − x) , xk,j−1 + δk ≤ x ≤ xk,j−1 + 2δk ,    0, x > xk,j−1 + 2δk .

406

Chapter 7. Metrics via Oriented Distance Functions

Ω2 f = 0

frame D

Ω1 f = 1

Figure 7.5. The two open components Ω1 and Ω2 of the open domain Ω for N = 2. Note that gk,j (xk,j−1 + δk ) = (δk )α is independent of j and is the maximum of the function gk,j . The uniform cusp property is verified for ρ = 1/8, λ = h(ρ), and h(θ) = θα . The boundary of Ω1 is made up of straight lines of total length 9 plus the length of the curve def

C = {(x, f (x)) : 0 ≤ x < 1} ,

C = ∪∞ k=0 Ck ,

η /2

k Ck = ∪j=1 Ck,j ,

def

Ck,j = {(x, f (x)) : xk,j−1 ≤ x < xk,j−1 + δk } . The length of the curve Ck,j is bounded below by HN −1 (Ck,j ) ≥ 2

'



ηk /2

(δk )2 + (δk )2α ≥ 2 (δk )α

ηk 2 (δk )α = ηk (δk )α = ηk HN −1 (Ck ) ≥ 2  HN −1 (Ck ) ≥ ηk 1−α

⇒ HN −1 (C) =

α

1 2k+1

∞

⇒ HN −1 (Ck ) ≥ 

1 ηk 2k+1

HN −1 (Ck,j ),

j=1

α



= ηk

1−α

1 2k+1

α ,

 α  1−α α 1 = 2[(2k+1 ) 1−α ] 2k+1  α 1 ≥ 21−α (2k+1 )α = 21−α 2k+1

HN −1 (Ck ) ≥ +∞ 21−α = +∞.

k=0

When at least one solution has a bounded perimeter, it is possible to show that there is one that minimizes the h-density perimeter.

16. A Variation of the Image Segmentation Problem of Mumford and Shah

407

Theorem 16.3. Assume that the assumptions of Theorem 16.1 are verified. There exists an Ω∗ in F ∗ (D, h, α) which minimizes the h-density perimeter. Proof. If for all Ω in F ∗ (D, h, α) the h-density perimeter is +∞, the theorem is true. If for some Ω ∈ F ∗ (D, h, α), Ph (Γ) ≤ c, then there exists a sequence {Ωn } in F ∗ (D, h, α) such that Ph (Γn ) →

inf

Ω∈F ∗ (D,h,α)

Ph (Γ).

By the compactness Theorems 12.1 and 16.2, there exist a subsequence and Ω∗ , Γ∗ = ∅, such that bΩn → bΩ∗ in W 1,p (D), Ω∗ ∈ F (D, h, α), mN (Ω) = mN (D), and Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c. n→∞

Finally by going back to the proof of Theorem 16.1 and using the fact that all the Ωn ’s are already minimizers in F ∗ (D, h, α), it can be shown that Ω∗ is indeed one of the minimizers in the set F ∗ (D, h, α).

16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter

To complete the results on the segmentation problem, we turn to the existence of a segmentation for a family of sets with a uniform bound or with a penalization term in the objective function on the h-density perimeter. Theorem 16.4. Given a bounded open frame ∅ = D ⊂ RN with a Lipschitzian boundary and real numbers h > 0 and c > 0,10 there exists an open subset Ω∗ of D, Γ∗ = ∅, with finite h-density perimeter such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.8)), and y ∈ H 1 (Ω∗ ) solutions of the respective problems  inf inf1 ε |∇ϕ|2 + |ϕ − f |2 dx, (16.8) Ω open ⊂D, Ph (Γ)≤c ϕ∈H (Ω) mN (Ω)=mN (D)

Ω



inf

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

inf

Ω open ⊂D ϕ∈H 1 (Ω) mN (Ω)=mN (D)

(16.9)

Ω

Proof. The proof for the objective function (16.8) is exactly the same as the one of Theorem 16.1. It uses Lemma 16.2 to show that the minimizing set has a thin boundary and the compactness Theorem 12.1. The proof for the objective function (16.9) uses the fact that there is a minimizing sequence for which the h-density perimeter is uniformly bounded and the lower semicontinuity of the density perimeter in the W 1,p -topology is given by Corollary 1. ´sio [8]. Problem (16.9) was originally considered in D. Bucur and J.-P. Zole The above two identification problems can be further specialized to the family of cracked sets F(D, h, α). 10 Note that the constant c must be large enough to take into account the contribution of the boundary of D.

408

Chapter 7. Metrics via Oriented Distance Functions

Corollary 1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundary and real numbers h > 0, α > 0, and c > 0, there exists an open subset Ω∗ of D in F(D, h, α) such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.10)), and y ∈ H 1 (Ω∗ ) solutions of the problem  inf

inf1

ε |∇ϕ|2 + |ϕ − f |2 dx,

(16.10)

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

(16.11)

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) Ph (Γ)≤c, mN (Ω)=mN (D)

Ω

 inf

inf1

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) mN (Ω)=mN (D)

Ω

Proof. Since the minimizing sequence {bΩn } constructed in the proof of Theorem 16.1 strongly converges to bΩ∗ in W 1,p (D) for all p, 1 ≤ p < ∞, from property (12.3) in Theorem 12.1, we have Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c n→∞

and the optimal Ω∗ constructed in the proof of the theorem satisfies the additional constraint on the density perimeter.

Chapter 8

Shape Continuity and Optimization 1

Introduction and Generic Examples

The underlying philosophy behind Chapters 5, 6, and 7 was to introduce metric topologies on sets associated with families of functions parametrized by sets rather than parametrize sets by functions. In the case of the characteristic function we have seen several examples where it appears explicitly in the modeling of the problem. The distance functions provide other topologies and constructions. For instance the oriented distance function gives a direct analytic access to geometric entities such as the normal and the fundamental forms on the boundary of a geometric domain without ad hoc local bases or Christoffel symbols. Many of the advantages of working in a truly intrinsic framework for shape derivatives will be illustrated in Chapters 9 and 10. Now that we have meaningful topologies on sets, we can consider the continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc. In this chapter we concentrate on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is defined as an infimum over a family of functions over a fixed domain or set such as eigenvalue and compliance problems. What is nice about them is that no adjoint system is necessary to characterize the minimizing function. Other problems have the structure of optimal control theory, that is, a state equation depending on the control that describes the evolution of the state and an objective functional that depends on the state and the control. In that case, the characterization of the optimal control involves an adjoint state equation coupled with the state equation. In the context of shape and geometric optimization the control will be the underlying geometry. The state equation will be a static or dynamical partial differential equation on the domain, and the objective functional will depend on the domain and the state that itself depends on the domain. As in control theory, we need continuity of the objective functional and the state with respect to the geometry and compactness of the family of domains over which the optimization takes 409

410

Chapter 8. Shape Continuity and Optimization

place. Since it is not possible to give a complete general theory of shape optimization problems within this book, we choose to concentrate on a few simple generic examples that are illustrative of the underlying technicalities involved. Several examples of optimization problems involving the Lp -topology on measurable sets via the characteristic function were given in Chapter 5. In this chapter, we first characterize the continuity of the transmission problem and the upper semicontinuity of the first eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of definition since they require different constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial differential equations. In the above problems the strong continuity of solutions of the elliptic equation or the eigenvector equation with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains. Those issues have received a lot of attention for the Laplace equation with homogeneous Dirichlet boundary conditions. With a sequence of domains in a fixed holdall D associate a sequence of extensions by zero of the solutions in the fixed space H01 (D). The Poincar´e inequality is uniform for that sequence, as the first eigenvalue of the Laplace equation in each domain is dominated by the one associated with the larger holdall D. By a classical compactness argument, the sequence of extensions converges to some limiting element y in H01 (D). To complete the proof of the continuity, two more fundamental questions remain. Is y the solution of the Laplace equation for the limit domain? Does y satisfy the Dirichlet boundary condition on the boundary of the limit domain? The first issue can be resolved by assuming that the Sobolev spaces associated with the moving domains converge in the Kuratowski sense, i.e., with the property that any element in the Sobolev space associated with the limit domain can be approached by a sequence of elements in the moving Sobolev spaces. That property is obtained in examples where the compactivorous property 1 follows from the choice of the definition of convergence of the domains. If the domains are open subsets of D, then the complementary Hausdorff topology has that property. The family of subsets of D satisfying a uniform fat segment property has that property. For the second issue it is necessary to impose some constraints at least on the limit domain. The stability condition introduced by J. Rauch and M. Taylor [1] and used by E. N. Dancer [1] and D. Daners [1] precisely assumes that the limit domain is sufficiently smooth, just enough to have the solution in H01 (Ω). Of course, assuming such a regularity on the limit domain makes things easier but is a little bit artificial. The more fundamental issue is to identify the families of domains for which this stability property is preserved in the limit. The uniform cone property was used in the context of shape optimization by D. Chenais [1, 2] in ´sio [3, 5, 6, 8, 9] showed that this is true 1973. In 1994 D. Bucur and J.-P. Zole under general capacitary conditions and have constructed new compact subfamilies 1 A sequence of open sets converging to a limit open set in some topology has the compactivorous property if any compact subset of the limit set is contained in all sets of the sequence after a certain rank: the sequence eats up the compact set after a certain rank (cf. Theorem 2.4 (iii) in Chapter 6).

1. Introduction and Generic Examples

411

of domains with respect to the Hausdorff complementary topology. Furthermore ˇ ´ k [1, 2] in D. Bucur [1] proved that the condition given in 1993 by V. Sver a dimension 2 involving a bound on the number of connected components of the complement of the domain can be recovered from the more general capacity conditions that are sufficient in the case of the Laplacian with homogeneous Dirichlet boundary conditions. In a more recent paper D. Bucur [5] proved that they are almost necessary. Intuitively, those capacity conditions are such that, locally, the complement of the domains in the family under consideration has enough capacity to preserve and retain the homogeneous Dirichlet boundary condition in the limit.

1.1

First Generic Example

The first example is the optimization of the first eigenvalue of an associated elliptic differential operator:  sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω  , (1.1) inf 1 λ (Ω) = A |ϕ|2 dx 0 = ϕ∈H (Ω) inf λ (Ω) 0 Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D.

1.2

Second Generic Example

As a second generic example, consider the transmission problem over the fixed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: find y = y(χΩ ) ∈ H01 (D) such that  ∀ϕ ∈ H01 (D), (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (1.2) D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials.

1.3

Third Generic Example

As a third example, consider the minimization of the objective function  def 1 |uΩ − g|2 dx J(Ω) = 2 Ω

(1.3)

over the family of open subsets Ω of a bounded open holdall D of RN , where g in L2 (D), uΩ ∈ H01 (Ω) is the solution of the variational problem  ∀ϕ ∈ H01 (Ω), A∇uΩ · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , (1.4) Ω

A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗A = A and αI ≤ A ≤ βI for some coercivity and continuity constants 0 < α ≤ β, and f |Ω denotes the restriction of f in H −1 (D) to H −1 (Ω).

412

1.4

Chapter 8. Shape Continuity and Optimization

Fourth Generic Example

As a fourth generic example, consider the previous example but with the homogeneous Dirichlet boundary value problem replaced by the homogeneous Neumann boundary value problem of the mathematician for the Laplacian ∂y = 0 on ∂Ω. ∂n

−∆y + y = f in Ω,

(1.5)

The classical (physical) homogeneous Neumann boundary value problem ∂y = 0 on ∂Ω ∂n

−∆y = f in Ω, has a unique solution up to a constant if

2

 D

(1.6)

f dx = 0.

Upper Semicontinuity and Maximization of the First Eigenvalue

Given a bounded open nonempty domain D ⊂ RN , consider the minimization of the Raleigh quotient for an open nonempty subset Ω of D:   A∇ϕ · ∇ϕ dx def D inf1 A∇ϕ · ∇ϕ dx, (2.1) λA (Ω) = = inf |ϕ|2 dx 0=ϕ∈H0 (Ω;D) ϕ∈H01 (Ω;D) D D ϕL2 =1

where A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗

and αI ≤ A(x) ≤ βI

A(x) = A(x)

(2.2)

for some coercivity and continuity constants 0 < α ≤ β. We set λA (∅) = +∞. The minimizers are characterized as follows. Theorem 2.1. (i) Given a bounded open nonempty domain Ω in RN , there exist uΩ ∈ H01 (Ω; D), uΩ = 0, and λA (Ω) > 0 such that  A∇uΩ · ∇ϕ − λA (Ω) uΩ ϕ dx = 0 (2.3) ∀ϕ ∈ H01 (Ω; D), Ω

and  ∀ϕ ∈

H01 (Ω; D),

A

λ (Ω) =

D

A∇uΩ · ∇uΩ dx  ≤ |uΩ |2 dx D

 D

A∇ϕ · ∇ϕ dx  . |ϕ|2 dx D

Conversely, if there exist u ∈ H01 (Ω; D), u = 0, and λ such that  A∇u · ∇ϕ − λ u ϕ dx = 0 ∀ϕ ∈ H01 (Ω; D), Ω

(2.4)

(2.5)

2. Upper Semicontinuity and Maximization of the First Eigenvalue and

 ∀ϕ ∈ H01 (Ω; D),

then

 λ=

D

λ≤

D

A∇ϕ · ∇ϕ dx  , |ϕ|2 dx D

A∇u · ∇u dx  = inf |u|2 dx ϕ∈H01 (Ω;D) D

 D

413

(2.6)

A∇ϕ · ∇ϕ dx  , |ϕ|2 dx D

u is a nonzero solution of the minimization problem (2.1), and λ is equal to the first eigenvalue λA (Ω) associated with Ω. (ii) Given a bounded open nonempty domain D, there exists λA D > 0 (that depends only on the diameter of D and A) such that for all open subsets Ω of D, λA (Ω) ≥ λA (D) ≥ λA D > 0, where λD (Ω) is the first eigenvalue of the symmetrical linear operator ϕ → −div (A ∇ϕ) on D. Proof. (i) For any bounded open Ω, the infimum is bounded below by 0 and hence finite. Let {ϕn } be a minimizing sequence such that ϕn L2 (Ω) = 1. By coercivity, α ∇ϕn 2L2 (Ω) is bounded. Hence {ϕn } is a bounded sequence in H01 (Ω) and there exist ϕ ∈ H01 (Ω) and a subsequence, still indexed by n, such that ϕn  ϕ in H 1 (Ω)-weak. By the Rellich–Kondrachov compactness theorem, the subsequence strongly converges in L2 (Ω). Then     ϕ2n dx → ϕ2 dx and A∇ϕ · ∇ϕ dx ≤ lim inf A∇ϕn · ∇ϕn dx 1= n→∞ Ω Ω Ω  Ω  A∇ϕ · ∇ϕ dx A∇ϕn · ∇ϕn dx ⇒ Ω  2 ≤ lim inf Ω  2 = λA (Ω). n→∞ ϕ dx ϕ dx Ω Ω n By definition of λA (Ω), 0 = ϕ ∈ H01 (Ω) is a minimizing element and inequality (2.4) is verified. Since ϕ = 0, the Rayleigh quotient is differentiable in ϕ and its directional derivative in the direction ψ is given by    A∇ϕ · ∇ψ dx ϕ ψ dx Ω 2 −2 A∇ϕ · ∇ϕ dx Ω 4 . 2 ϕL2 (Ω) ϕL2 (Ω) Ω A nonzero solution ϕ of the minimization problem on Ω is necessarily a stationary point, and for all ψ ∈ H01 (Ω),     A∇ϕ · ∇ϕ dx A Ω A∇ϕ · ∇ψ dx = ϕ ψ dx = λ (Ω) ϕ ψ dx ϕ2L2 (Ω) Ω Ω Ω and we get (2.3). Conversely, any λ and nonzero u ∈ H01 (Ω; D) that verify (2.5) yield λ ≥ λA (Ω). But, from inequality (2.6), λ ≤ λA (Ω) and λ = λA (Ω). Hence, u is a minimizer of the Rayleigh quotient over H01 (Ω; D).

414

Chapter 8. Shape Continuity and Optimization

(ii) Let diam (D) be the diameter of D and x ∈ RN be a point such that D ⊂ Bx = B(x, diam (D)). Therefore, from Theorem 2.3 in Chapter 2, H01 (Ω; Bx ) ⊂ H01 (D; Bx ) ⊂ H01 (Bx ),

H01 (Ω; D) ⊂ H01 (D)

with the associated isometries. Hence, by definition of the minimum of a functional over an increasing sequence of sets, λA (Ω) ≥ λA (D) ≥ λA (Bx ). If λA (Bx ) = 0, we repeat the above construction with H01 (Bx ) in place of H01 (Ω; Bx ) and end up with an element ϕ ∈ H01 (Bx ) such that    ϕ2 dx = 1 and α |∇ϕ|2 dx ≤ A∇ϕ · ϕ dx = 0, Bx

Bx

Bx

which is impossible in H01 (Bx ). Finally λA (Bx ) is independent of the choice of x since the eigenvalue is invariant under a translation of the domain: it depends only on the diameter of D. The case λA (∅) = +∞ is compatible with the results. Remark 2.1. For A equal to the identity operator I, denote by λ(Ω), λ(D), and λD the quantities 1 λA (Ω), λA (D), and λA D of Theorem 2.1. For any open Ω ⊂ D and any ϕ ∈ H0 (Ω; D) 1 ϕL2 (D) ≤ √ ∇ϕL2 (D) λD 4 1 ⇒ ∀Ω open ⊂ D, ϕH 1 (Ω) ≤ 1 + ∇ϕL2 (Ω) λD

(2.7) (2.8)

and the spaces H01 (Ω) and H01 (D) will be endowed with the respective equivalent norms ∇ϕL2 (Ω) and ∇ϕL2 (D) . Before considering the existence of minimizers/maximizers, we investigate the question of the continuity of the first eigenvalue λA (Ω) with respect to Ω for the uniform complementary Hausdorff topology on  def

Cdc (D) = d
Ω : ∀Ω open subset in D and Ω = RN (cf. (2.11) of Definition 2.2 in Chapter 6). Theorem 2.2. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). The mapping d
Ω → λA (Ω) : Cdc (D) ⊂ C0 (D) → R ∪{+∞}

(2.9)

is upper semicontinuous. Proof. Since λA (Ω) is finite in Ω = ∅ and +∞ in Ω = ∅, it is upper semicontinuous in ∅. For Ω = ∅, consider a sequence {Ωn } of open subsets of D such that d
Ωn → d
Ω in C(D) and the associated sequence {λA (Ωn )}. We want to show that lim sup λA (Ωn ) ≥ λA (Ω).

2. Upper Semicontinuity and Maximization of the First Eigenvalue

415

Consider the new sequence {Ωn ∩Ω}. Since Ω ⊂ D, for all n, (Ωn ∩Ω) = Ωn ∪Ω ⊃ D = ∅ and d
(Ωn ∩Ω) = d
Ωn ∪
Ω = min{d
Ωn , d
Ω } → d
Ω in C(D). Let un ∈ H01 (Ωn ∩ Ω), un L2 (D) = 1, be an eigenvector for the eigenvalue λn = λA (Ωn ∩ Ω). From Theorem 2.1 un is solution of the variational equation (2.3):   A∇un · ∇ϕn dx = λn un ϕn dx. (2.10) ∀ϕn ∈ H01 (Ωn ; D), D

D

Since Ω = ∅, there exists a compact ball Kρ = B(x, ρ) ⊂ Ω for some x ∈ Ω and ρ > 0. By the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, there exists an integer N (Kρ ) such that for all n ≥ N (Kρ ), Kρ ⊂ Ωn ∩ Ω and by Theorem 2.1 (ii), 0 < λn ≤ λA (B(x, ρ)). Therefore, the sequence of eigenvalues {λn } is bounded. By coercivity α∇un }2L2 (D) ≤ λn ∇un }2L2 (D) = λn and {un } is bounded in H01 (Ω; D) since Ωn ∩ Ω ⊂ Ω. Letting λ∗ = lim sup λn , there exist subsequences, still indexed by n, and u ∈ H01 (Ω; D) such that λn → λ ∗ ,

un L2 (D) = 1,

un  u in H01 (Ω; D)-weak

⇒ uL2 (D) = 1.

Since for all n, λA (Ωn ) ≤ λA (Ωn ∩ Ω), it is readily seen that lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ and that it is sufficient to show that λ∗ = λA (Ω) to get the upper semicontinuity. We cannot directly go to the limit in (2.10), since the test functions ϕn depend on Ωn . To get around this difficulty, recall the density of D(Ω) in H01 (Ω; D) and use the compactivorous property a second time. For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn ∩ Ω. Therefore, for all n ≥ N (K)   1 1 un ∈ H0 (Ωn ∩ Ω; D) ⊂ H0 (Ω; D), A∇un · ∇ϕ dx = λn un ϕ dx. D

By letting n → ∞



A∇u · ∇ϕ dx = λ∗

∀ϕ ∈ D(Ω), D

D

 u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is verified in H01 (Ω; D). Hence, u ∈ H01 (Ω; D), uL2 (Ω) = 1, satisfies the variational equation   A∇u · ∇ϕ dx = λ∗ u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

416

Chapter 8. Shape Continuity and Optimization

Therefore, since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ∗ . To show that λ∗ = λA (Ω), it remains to prove that λ∗ ≤ λA (Ω). For each n  A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ∩ Ω; D), λ (Ωn ∩ Ω) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ω ∩ Ωn for all n ≥ N . Therefore for all n ≥ N  A∇ϕ · ∇ϕ dx A λ (Ωn ∩ Ω) ≤ D ϕ2L2 (D) and by taking the lim sup



∀ϕ ∈ D(Ω),

∗

λ ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ∗ ≤ λA (Ω). As a result λ∗ = λA (Ω). Finally, λA (Ωn ) ≤ λA (Ωn ∩ Ω)

⇒ lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ = λA (Ω),

and we have the upper semicontinuity of the map d
Ω → λA (Ω). In view of the previous theorem, we now deal with the maximization of the first eigenvalue λA (Ω) associated with the operator A with respect to a family of open subsets Ω of D. Theorem 2.3. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) Then λA (Ω) = λA (∅) = +∞.

sup

(2.11)

Ω∈Cdc (D)

The conclusion remains true when Cdc (D) is replaced by Cdc (D) = {d
Ω : Ω convex and open ⊂ D}

(2.12)

(cf. Theorem 8.6 in Chapter 6). (ii) Given an open set E, ∅ = E ⊂ RN , consider the compact family c Cd,loc (E; D)

=

d
Ω

∃x ∈ RN , ∃A ∈ O(N) such that : x + AE ⊂ Ω open ⊂ D

&

of Theorem 2.6 in Chapter 6. Then, there exists an open Ω∗ such that d
Ω∗ ∈ c Cd,loc (E; D) and λA (Ω∗ ) =

sup c Ω∈Cd,loc (E;D)

λA (Ω) ≤ β λI (E).

(2.13)

3. Continuity of the Transmission Problem

417

c (E; D) is replaced by Cdc (E; D) or The conclusion remains true when Cd,loc c c the subfamilies of convex subsets Cd (E; D) and Cd,loc (E; D) of Theorem 8.6 in Chapter 6.

(iii) There exists Ω∗ ∈ L(D, r, O, λ) such that λA (Ω∗ ) =

sup

λ(Ω) ≤ βλI (O).

(2.14)

Ω∈L(D,r,O,λ)

Proof. (i) From Theorem 2.2 and the compactness of Cdc (D) in C(D). (ii) All the families are compact by Theorems 2.6 and 8.6 in Chapter 6. So there is a maximizer d
Ω∗ of λA (Ω). The upper bound is obvious. (iii) By compactness of L(D, r, O, λ) in W 1,p (D) and hence in Cdc (D) for the C(D)-topology.

3

Continuity of the Transmission Problem

As a second generic example, consider the transmission problem over the fixed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: find y = y(χΩ ) ∈ H01 (D) such that  (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (3.1) ∀ϕ ∈ H01 (D), D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials. Theorem 3.1. Let D be a bounded open domain in RN and k1 and k2 be some strictly positive constants such that α = max{k1 , k2 } > 0 and f ∈ H −1 (D). Denote by y(χ) the solution of the transmission problem (3.1) associated with χ ∈ X(D). The mapping χ → y(χ) : X(D) → H01 (D) is continuous when X(D) is endowed with the strong Lp (D)-topology, 1 ≤ p < ∞. The same result is true with co X(D) in place of X(D). Remark 3.1. This theorem applies to the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2) since the convergence of bΩn → bΩ in W 1,p (D) implies the convergence of χΩn → χΩ in Lp (D). It also applies to the compact family of convex sets C(D) = {χΩ : Ω convex subset of D} introduced in (3.24) in section 3.4 of Chapter 5 (cf. Theorem 3.5 (iii) and Corollary 1 to Theorem 6.3 in section 6.1 of Chapter 5).

418

Chapter 8. Shape Continuity and Optimization

´sio [2], [24, Proof of Theorem 3.1. The proof follows the arguments of J.-P. Zole sect. 4.2.1, pp. 446–448]. It is sufficient to prove that for any convergent sequence χn → χ in L2 (D)-strong, the corresponding solutions yn = y(χn ) ∈ H01 (D) converge to y = y(χ) in H01 (D)-strong. First, the sequence {yn } is bounded in H01 (D):  α∇yn 2L2 (D) ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D

⇒ α∇yn 2L2 (D)

= f, yn  ≤ f H −1 (D) yn H01 (D) 4 1 ≤ f H −1 (D) yn H01 (D) ≤ f H −1 (D) 1 + ∇yn L2 (Ω) λD

since we have the equivalence of norms from (2.7) in Remark 2.1. So there is a subsequence, still denoted by {yn }, that weakly converges to some y ∈ H01 (D). To show that y is solution of (3.1) corresponding to χ, we go to the limit in  ∀ϕ ∈ H01 (D), (k1 χn + k2 (1 − χn )) ∇ϕ · ∇yn dx = f, ϕ. D

Since (k1 χn +k2 1−χn ))∇ϕ → (k1 χ+k2 (1−χ))∇ϕ in L2 (D)-strong and ∇yn → ∇y in L2 (D)-weak, we get that y ∈ H01 (D) is the unique solution of  ∀ϕ ∈ H01 (D), (k1 χ + k2 (1 − χ)) ∇y · ∇ϕ dx = f, ϕ. D

Finally, we prove the strong convergence. Indeed  (k1 χn + k2 (1 − χn )) ∇(yn − y) · ∇(yn − y) dx α ∇(yn − y)2L2 (D) ≤ D  ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D  (k1 χn + k2 (1 − χn )) ∇yn · ∇y dx −2  D (k1 χn + k2 (1 − χn )) ∇y · ∇y dx + D  (k1 χn + k2 (1 − χn )) ∇y · ∇y dx =f, yn  − 2 f, y + D  (k1 χ + k2 (1 − χ)) ∇y · ∇y dx = 0. → −f, y + D

Since every strongly convergent subsequence converges to the same limit, the whole sequence strongly converges to y in H01 (D).

4 4.1

Continuity of the Homogeneous Dirichlet Boundary Value Problem Classical, Relaxed, and Overrelaxed Problems

We have seen in section 3.5 of Chapter 5 two relaxations of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets Ω of a bounded

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

419

open holdall D by introducing the following closed subspaces of H01 (D):  def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D ,

 def H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D

(4.1) (4.2)

(cf. (3.27) and (3.28) in Theorem 3.6 of Chapter 5). It is readily seen that H1 (Ω; D) ⊂ H•1 (Ω; D). For Ω open, H01 (Ω; D) ⊂ H1 (Ω; D) ⊂ H•1 (Ω; D) and the three closed subspaces of H01 (D) are generally not equal as can be seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The choice of a relaxation is very much problem dependent and should be guided by the proper modeling of the underlying physical or technological phenomenon at hand. In general, there is a price to pay for the above relaxations. Both H1 (Ω; D) and H•1 (Ω; D) depend on the equivalence class of measurable sets [Ω] and not specifically on Ω. In general, there is not even an open representative of Ω in the class. From section 3.3 in Chapter 5, [Ω] = [I], where I is the measure theoretic interior of Ω,   m(B(x, r) ∩ Ω) N I = x ∈ R : lim =1 , r0 m(B(x, r)) int I = Ω1 , where

    Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) is open, and int Ω ⊂ int I and ∂I ⊂ ∂Ω (cf. Definition 3.3 and Theorem 3.3 in Chapter 5). In view of the cleaning operation in the definition of I, all cracks of zero measure will be deleted or not seen by the functions of the spaces H1 (Ω; D) and H•1 (Ω; D). When mN (∂Ω) = 0, [int I] = [Ω] and int I = Ω1 is an open representative in the class [Ω]. Let D be a bounded open subset of RN and let f ∈ H −1 (D), where H −1 (D) is the topological dual of H01 (D). For all open subsets Ω of D ϕ → f, ϕH −1 (D)×H01 (D) : D(Ω) → R

(4.3)

is well-defined and continuous with respect to the H 1 (Ω)-topology and extends to an element f |Ω ∈ H −1 (Ω). The map ϕ → f, ϕH −1 (D)×H01 (D) : H1 (Ω; D) → R is also well-defined and continuous with respect to the closed subspace H1 (Ω; D) of H01 (D) and defines an element f |Ω of the dual H1 (Ω; D) of H1 (Ω; D). Similarly, the map ϕ → f, ϕH −1 (D)×H01 (D) : H•1 (Ω; D) → R is also well-defined and continuous with respect to the closed subspace H•1 (Ω; D) of H01 (D) and defines an element f |•Ω of the dual H•1 (Ω; D) of H•1 (Ω; D). Let A ∈ L∞ (D; L(RN , RN )) be a symmetrical matrix function on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.4)

420

Chapter 8. Shape Continuity and Optimization

for some coercivity and continuity constants 0 < α ≤ β. Consider the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian ∃y(Ω) ∈ H01 (Ω; D) such that,  A∇y(Ω) · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , ∀ϕ ∈ H01 (Ω; D),

(4.5)

D

the relaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H1 (Ω; D) such that,  1 A∇y(χΩ ) · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H (Ω; D),

(4.6)

D

and the overrelaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H•1 (Ω; D) such that,  A∇y(χΩ ) · ∇ϕ dx = f |•Ω , ϕH•1 (Ω;D) ×H•1 (Ω;D) . ∀ϕ ∈ H•1 (Ω; D),

(4.7)

D

By convention, set y(∅) = 0 and y(χ∅ ) = 0 as elements of H01 (D). In addition we shall keep the terminology homogeneous Dirichlet boundary value problem for what we just called the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian. Since, in general, H01 (Ω; D) is a distinct closed subspace of H1 (Ω; D), a solution of (4.6) is not necessarily a solution of (4.5). In the other direction, a solution y = y(Ω) ∈ H01 (Ω; D) of (4.5) belongs to H1 (Ω; D) and is a solution of ∃y ∈ H1 (Ω; D) such that,  A∇y · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H01 (Ω; D),

(4.8)

D

but y is not necessarily unique in H1 (Ω; D) since this variational equation is verified only on the closed subspace H01 (Ω; D) of H1 (Ω; D). However, the space H1 (Ω; D) is very close to the space H01 (Ω; D) and the two spaces will coincide on smooth sets and more generally on crack-free sets that will be introduced in section 7 (cf. Definition 7.1 (ii) and Theorem 7.3 (ii)). The following terminology has been introduced by J. Rauch and M. Taylor [1]. Definition 4.1. An open subset Ω of D is said to be stable with respect to D if H01 (Ω; D) = H1 (Ω; D). The same considerations apply to the overrelaxed problem (4.7) in H•1 (Ω; D) with the additional observation that even for sets with a very smooth boundary, the solution does not coincide with the one of the (classical) Dirichlet problem when the Ω has more than one connected component as seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The problem in H•1 (Ω; D) is associated with different physical phenomena.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.2

421

Classical Dirichlet Boundary Value Problem

We have the following general results. Theorem 4.1. Let D be a bounded open nonempty subset in RN . Associate with each open subset Ω of D the solution y(Ω) ∈ H01 (Ω; D) of the classical Dirichlet problem (4.5) on Ω. (i) Let Ω and the sequence {Ωn } be open subsets of D such that d
Ωn → d
Ω in C(D). Then y(Ωn ∩ Ω) → y(Ω) in H01 (D)-strong

(4.9)

and there exist a subsequence {Ωnk } and y0 ∈ H1 (Ω; D) such that y(Ωnk )  y0 in H01 (D)-weak

(4.10)

and y0 is a (not necessarily unique) solution of (4.8). (ii) The function d
Ω → y(Ω) : Cdc (D) → H01 (D)-strong

(4.11)

is continuous for each Ω such that H01 (Ω; D) = H1 (Ω; D). Remark 4.1. We have the additional information that y0 ∈ H1 (Ω; D) ⊂ H01 (int Ω; D) in part (i) of the theorem (cf. Theorem 7.3 (i)). Remark 4.2. The projection operator PΩ : H01 (D) → H01 (Ω; D)

(4.12)

is defined for u ∈ H01 (D) as the solution of the variational equation   ∃PΩ u ∈ H01 (Ω; D), ∀ϕ ∈ H01 (Ω; D), ∇(PΩ u) · ∇ϕ dx = ∇u · ∇ϕ dx. D

D

Applying the theorem to a sequence of open subsets {Ωn } of D converging to Ω in the complementary Hausdorff topology with  def f, ϕ = ∇u · ∇ϕ dx, D

there exist u0 ∈ H1 (Ω; D) and a subsequence {Ωnk } such that d
Ωn → d
Ω in C(D),

PΩnk u  u0 in H01 (D)-weak.

The projection operator PΩ is continuous at Ω if H01 (Ω; D) = H1 (Ω; D), but this is only a sufficient condition.

422

Chapter 8. Shape Continuity and Optimization

Proof. (i) The proof of the continuity of (4.9) follows the same steps as the proof of the upper semicontinuity of the first eigenvalue of the Laplacian in Theorem 2.2. As for the convergence (4.10), it is readily seen that the sequence {yn = y(Ωn )} of elements of H01 (Ωn ; D) is bounded in the bigger space H01 (D): by coercivity 4 1 2 α ∇yn L2 (D) ≤ f H −1 (D) yn H01 (D) ≤ 1 + f H −1 (D) ∇yn L2 (D) λD from Remark 2.1, where λD > 0 is the first eigenvalue of the Laplacian on D. There exist a subsequence, still indexed by n, and some y0 in H01 (D) such that yn  y0 in H01 (D)-weak and hence yn → y0 in L2 (D)-strong. From Corollary 1 to Theorem 4.4 in Chapter 6 ∀x ∈ RN , ⇒ ∀ a.a. x ∈ RN ,

lim inf χΩn (x) ≥ χΩ (x) n→∞

0 = lim inf (1 − χΩn (x))|yn (x)|2 ≥ (1 − χΩ (x))|y0 (x)|2 n→∞

since (1 − χΩn (x)) ≥ 0, |yn | ≥ 0, and |yn |2 → 0 a.e. in D. By definition, y0 ∈ H1 (Ω; D). Finally, by the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, for each ϕ ∈ D(Ω) there exists N such that for all n > N , supp ϕ ⊂ Ωn . Therefore, by substituting ϕ in (4.5) on Ωn , for all n > N  A∇yn · ∇ϕ dx = f |Ωn , ϕH −1 (Ωn )×H01 (Ωn ) = f |Ω , ϕH −1 (Ω)×H01 (Ω) (4.13) 2

D

and by going to the limit



∀ϕ ∈ D(Ω), D

A∇y0 · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω)

(4.14)

and, by density of D(Ω) in H01 (Ω; D), y0 ∈ H1 (Ω; D) is a solution of (4.8). (ii) If, in addition, H1 (Ω; D) = H01 (Ω; D), then y0 ∈ H01 (Ω; D) is the unique solution of (4.5). Hence y0 = y(Ω), solution of the classic problem (4.5), is independent of the choice of the converging subsequence. Therefore the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-weak. The strong continuity in H01 (D) now follows from the convergence of un → u0 2 in L (D)-strong: α∇(un − u0 )2L2 (D)  ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D    = A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx D D  D A∇un · ∇u0 dx = f, un  + f, u0  − 2 D  A∇u0 · ∇u0 dx = 0 → 2 f, u0  − 2 D

and the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-strong. Therefore, the function (4.11) is continuous in Ω.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.3

423

Overrelaxed Dirichlet Boundary Value Problem

In this section we concentrate on the overrelaxation (4.7) of the homogeneous Dirichlet boundary value problem. To show the continuity with respect to the characteristic function of domains, we first approximate its solution by transmission problems. 4.3.1

Approximation by Transmission Problems

The overrelaxed Dirichlet problem (4.7) corresponding to χ = χΩ ∈ X(D) is first approximated by a transmission problem over D indexed by ε > 0 going to zero: find yε = yε (χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D)     1 χ + (1 − χ) ∇yε · ∇ϕ dx = χf ϕ dx. ε D D

(4.15)

They are special cases of the general transmission problem (3.1) over the fixed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D for k1 = 1 and k2 = 1/ε and χ = χΩ . Theorem 4.2. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). For ε, 0 < ε < 1, let yε be the solution of problem (4.15) in H01 (D). Then y(χΩ ) = lim yε in H01 (D)-strong. ε→0

Proof. For ε, 0 < ε < 1, substitute ϕ = yε in (4.15) to get the bounds √ ∇yε L2 (D) ≤ c(D) f L2 (D) , (1 − χΩ ) ∇yε L2 (D) ≤ ε c(D) f L2 (D) . There exist a sequence εn  0 and y0 ∈ H01 (D) such that yn = yεn  y0 in H01 (D)weak and (1 − χΩ ) ∇yn → 0 in L2 (D)-strong. Therefore, (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D), (4.15) reduces to   ∇yε · ∇ϕ dx = χΩ f ϕ dx ∀ϕ ∈ H•1 (Ω; D), D

and as n → ∞

D

 ∀ϕ ∈

H•1 (Ω; D),

 ∇y0 · ∇ϕ dx =

D

χΩ f ϕ dx. D

Hence y0 ∈ H•1 (Ω; D) is the solution of (4.7). Since the limit is the same for all convergent sequences, yε → y0 in H01 (D)-weak. By subtracting the last two equations   ∀ϕ ∈ H•1 (Ω; D), [∇yε − ∇y0 ] · ∇ϕ dx = 0 ⇒ [∇yε − ∇y0 ] · ∇y0 dx = 0. D

D

424

Chapter 8. Shape Continuity and Optimization

Finally, we estimate the following term:    2 |∇(yε − y0 )| dx = [∇yε − ∇y0 ] · ∇yε dx − [∇yε − ∇y0 ] · ∇y0 dx D D D [∇yε − ∇y0 ] · ∇yε dx = D   ∇yε · ∇yε dx − ∇y0 · ∇yε dx = D  D = χΩ f yε dx − ∇y0 · ∇yε dx D D χΩ f y0 dx − ∇y0 · ∇y0 dx = 0. → D

As a result yε → y0 in 4.3.2

D

H01 (D)-strong.

Continuity with Respect to X(D) in the Lp (D)-Topology

Theorem 4.3. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong

(4.16)

is continuous for all Lp (D)-topologies, 1 ≤ p < ∞, on X(D). Proof. Given χ = χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χ = χΩ . In view of Theorem 4.2, let εn > 0 be such that the approximating solution yn ∈ H01 (D) of (4.15),     1 1 χΩ n + ∀ϕ ∈ H0 (D), (1 − χΩn ) ∇yn · ∇ϕ dx = χΩn f ϕ dx, εn D D is such that yn − y(χΩn )H 1 (D) < 1/n and recall the bounds ∇yn L2 (D) ≤ c(D) f L2 (D) , √ (1 − χΩn ) ∇yn L2 (D) ≤ εn c(D) f L2 (D) ≤ c(D) f L2 (D) .

(4.17) (4.18)

Therefore the sequence {y(χΩn } is also bounded in H01 (D) and there exist subsequences, still indexed by n, and y0 ∈ H01 (D) such that yn  y0 and y(χΩn )  y0 in H01 (D)-weak. Moreover, (1 − χΩn ) ∇y(χΩn ) = 0. Hence (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D)     1 χΩ n + ∀ϕ ∈ H01 (D), (1 − χΩn ) ∇yn · (χΩ ∇ϕ) dx = χΩn f ϕ dx. εn D D

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

425



χΩ f ϕ dx. The left-hand side can be rewritten as    1 χΩ χΩn ∇yn · ∇ϕ + χΩ (1 − χΩn ) (1 − χΩn ) ∇yn · ∇ϕ dx εn D   χΩ ∇y0 · ∇ϕ dx = ∇y0 · ∇ϕ dx →

The right-hand side converges to

D

D

D

since χΩ χΩn → χΩ and χΩ (1 − χΩn ) → 0 in L2 (D)-strong and the bound (4.18). Finally, y0 ∈ H•1 (Ω; D) is solution of   1 ∇y0 · ∇ϕ dx = χΩ f ϕ dx. ∀ϕ ∈ H• (Ω; D), D

D

Since the limit point y0 = y(χΩ ) is independent of the choice of the sequence, y(χΩn )  y(χΩ ) in H01 (D)-weak. Finally,  2 |∇ (y(χΩn ) − y(χΩ ))| dx D    2 2 = |∇y(χΩn )| dx + |∇y(χΩ )| dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx D D   D χΩn f y(χΩn ) dx + χΩ f y(χΩ ) dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx = D D D   χΩ f y(χΩ ) dx − 2 ∇y(χΩ ) · ∇y(χΩ ) dx = 0 →2 D

D

and y(χΩn ) → y(χΩ ) in

H01 (D)-strong.

Remark 4.3. The penalization technique used in this section can be applied to higher-order elliptic, and even parabolic, problems and the Navier–Stokes equation (cf. R. Dziri ´sio [6]) provided that the relaxation is acceptable in the modeling and J.-P. Zole of the physical phenomenon. The continuity of the classical homogeneous Dirichlet problem for the Laplacian will be given in section 6 under capacity conditions.

4.4

Relaxed Dirichlet Boundary Value Problem

In this section we consider the relaxation (4.6) of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets of a bounded open holdall D for the space H1 (Ω; D). Theorem 4.4. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H1 (Ω; D) be the solution of the relaxed Dirichlet problem (4.1). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong is continuous for all Lp (D)-topology, 1 ≤ p < ∞, on X(D).

(4.19)

426

Chapter 8. Shape Continuity and Optimization

Proof. Given χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χΩ and let y(χΩ ) ∈ H1 (Ω; D) and y(χΩn ) ∈ H1 (Ωn ; D) be the corresponding solutions of (4.1). Since H1 (Ω; D) ⊂ H•1 (Ω; D), we have the continuity χΩ → y(χΩ ) from Theorem 4.3. To complete the proof we need to show that the limiting element y(χΩ ) in the proof of Theorem 4.3 is solution of the variational equation (4.1) in the smaller space H1 (Ω; D). Indeed, for each n, (1 − χΩn )y(χΩn ) = 0 implies (1 − χΩ )y(χΩ ) = 0 and y(χΩ ) ∈ H1 (Ω; D). Finally, since y(χΩ ) is solution of the variational equation (4.7) for all ϕ ∈ H•1 (Ω; D), it still is solution for ϕ in the smaller subspace H1 (Ω; D). Hence y(χΩ ) ∈ H1 (Ω; D) is the solution of (4.1).

5

Continuity of the Homogeneous Neumann Boundary Value Problem

As a fourth generic example, consider the homogeneous Neumann boundary value problem of mathematicians for the Laplacian ∂y = 0 on ∂Ω, −∆y + y = f in Ω, ∂n   ∇y · ∇ϕ + y ϕ dx = f ϕ dx. ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.1) (5.2)

Ω

The classical homogeneous Neumann boundary value problem −∆y = f in Ω, ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω),

∂y = 0 on ∂Ω, ∂n  ∇y · ∇ϕ dx = f ϕ dx Ω

(5.3) (5.4)

Ω

 has a unique solution up to a constant if D f dx = 0. One way to get around the nonuniqueness is to introduce the closed subspace    def Ha1 (Ω) = ϕ ∈ H 1 (Ω) : ϕ dx = 0 Ω

of functions with zero average. Then there exists a unique y ∈ Ha1 (Ω) such that   ∀ϕ ∈ Ha1 (Ω), ∇y · ∇ϕ dx = f ϕ dx. (5.5) Ω

Ω

Moreover, the solution of (5.5) can be approximated by the ε-problems   ∇yε · ∇ϕ + ε yε ϕ dx = f ϕ dx ∃yε ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.6)

Ω

and yε → y in Ha1 (Ω)-strong. Note that, by assumption on f , yε ∈ Ha1 (Ω). One of the key conditions to get the continuity with respect to the domain Ω is the following density property.

5. Continuity of the Homogeneous Neumann Boundary Value Problem

427

Definition 5.1. An open set Ω is said to have the H 1 -density property if the set {ϕ|Ω : ϕ ∈ C01 (RN )}

(5.7)

is dense in H 1 (Ω). This property is not verified for a domain Ω with a crack in R2 , since elements of H 1 (Ω) have different traces on each side of the crack. Recall from section 6.1 in Chapter 2 that this condition is verified for a large class of sets Ω that can be characterized by the geometric segment property that is equivalent to the property that the set be locally a C 0 -epigraph (cf. section 6.2 of Chapter 2). We recall Theorem 6.3 of Chapter 5. Theorem 5.1. If the open set Ω has the segment property, then the set

 f |Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to Ω is dense in W m,p (Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular C k (Ω) is dense in W m,p (Ω) for any m ≥ 1 and k ≥ m. Lemma 5.1. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let Ω and the sequence {Ωn } be open subsets of D. Denote by yn ∈ H 1 (Ωn ) the solution of (5.1) in Ωn and by y ∈ H 1 (Ω) the solution of (5.1) in Ω. Assume the following: (i) χΩn → χΩ in L2 (D); (ii) d
Ωn → d
Ω in C(D); (iii) Ω has the H 1 -density property. Then 9n ) → (˜ : in (L2 (D) × L2 (D))-weak, y , ∇y) (˜ yn , ∇y where z˜ denotes the extension by zero of a function z from Ω or Ωn to D. Proof. The proof of the lemma follows the approach of W. Liu and J. E. Rubio [1] in 1992. Since f ∈ L2 (D), we readily have the following estimates: 2 2 2 9n 2 2 ˜ yn 2L2 (D) + ∇y L (D) = yn L2 (Ωn ) + ∇yn L2 (Ωn ) ≤ f L2 (D) .

There exist subsequences, still indexed by n, z ∈ L2 (D), and Z ∈ L2 (D)N such that y˜n  z weakly in L2 (RN )

9n  Z weakly in L2 (RN )N . and ∇y

By the H 1 -density property of Ω, the variational equation (5.2) on Ωn can be rewritten in the form   / . 9n · ∇ϕ + y˜n ϕ dx = χΩn ∇y χΩn f ϕ dx (5.8) ∀ϕ ∈ C01 (RN ), RN

RN

428

Chapter 8. Shape Continuity and Optimization

and we can go to the limit as n → ∞:   ∀ϕ ∈ C01 (RN ), χΩ [Z · ∇ϕ + z ϕ] dx = RN

χΩ f ϕ dx.

(5.9)

RN

By definition of the distributional derivative, for all n   ∂i yn ϕn dx = − yn ∂i ϕn dx, ∀ϕn ∈ D(Ωn ). Ωn

Ωn

By the compactivorous property (cf. Theorem 2.4 (iii) in Chapter 6) and the convergence d
Ωn → d
Ω in C(D), for each ϕ ∈ D(Ω), there exists N such that for all n > N , supp ϕ ⊂ Ωn . Hence for all n > N   y ϕ dx = − y˜n ∂i ϕ dx, ∀ϕ ∈ D(Ω) ∂9 i n Ω

Ω

and by going to the limit   Zi ϕ dx = − z ∂i ϕ dx, Ω

∀ϕ ∈ D(Ω),

⇒ ∇z = Z ∈ L2 (Ω)

Ω

and from (5.9), z ∈ H 1 (Ω) is solution of the variational equation   1 N ∀ϕ ∈ C0 (R ), ∇z · ∇ϕ + z ϕ dx = f ϕ dx. Ω

Ω

By the H 1 -density property of Ω, this extends to all ϕ ∈ H 1 (Ω), which implies that z = y. Since all weakly convergent subsequences converge to the same limit, the whole sequence weakly converges and this completes the proof. Remark 5.1. We know that the H 1 -density condition holds for the family of domains satisfying the segment property of Definition 6.1 in Chapter 2 and not only for H 1 -spaces but also W m,p -spaces (cf. Theorem 6.3 in Chapter 2), thus extending the property and the results to a broader class of partial differential equations. We have shown in Theorem 6.5 of Chapter 2 that a domain satisfying the segment property is locally a C 0 -epigraph in the sense of Definition 5.1 of Chapter 2. Theorem 5.2. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let L(D, r, O, λ) be the compact family of sets in W 1,p (D), 1 ≤ p < ∞, verifying the uniform fat segment property. Then the function  : L(D, r, O, λ) → (L2 (D) × L2 (D))-weak, Ω → (˜ y (Ω), ∇y(Ω)) where z˜ denotes the extension by zero of a function z from Ω to D, is continuous. It is also true when L(D, r, O, λ) is replaced by the family of convex sets  def

Cb (D) = bΩ : Ω ⊂ D, ∂Ω = ∅, Ω convex (cf. Theorem 10.2 in Chapter 7).

(5.10)

6. Elements of Capacity Theory

429

Proof. In L(D, r, O, λ) endowed with the W 1,p (D)-topology, 1 ≤ p < ∞, the three assumptions of the lemma are verified. Remark 5.2. Therefore the previous arguments are verified for the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2). However, the subsets of a bounded holdall verifying a uniform segment property do not include the very important domains with cracks ´sio for which some specific results have been obtained by D. Bucur and J.-P. Zole [2, 4] (and D. Bucur and N. Varchon [1] in dimension 2). An issue that was not tackled is the nonhomogeneous boundary condition that involves a boundary integral in the variational formulation: ∂y = g on ∂Ω, (5.11) −∆y + y = f in Ω, ∂n    ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), ∇y · ∇ϕ + y ϕ dx = f ϕ dx + g ϕ dΓ. (5.12) Ω

6

Ω

Γ

Elements of Capacity Theory

Recall the relaxation of the homogeneous Dirichlet boundary value problem for the Laplacian (4.6) in section 4.4 by using the space  def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D .

(6.1)

The classical solution lies in the closed subspace H01 (Ω; D) of H1 (Ω; D), and we know that they are not equal when Ω is not stable with respect to D. The difference between the two spaces is that a function in H01 (Ω; D) is zero in D\Ω as an H01 (D)function and not just almost everywhere as an L2 (D)-function. Some information is lost in the relaxation, and this requires the finer notion of quasi-everywhere that will characterize the set over which an H01 (D)-function is zero. This section introduces the basic elements and results from capacity theory that will play an essential role in establishing the continuity of the classical homogeneous Dirichlet problem (4.5) with respect to its underlying domain. The characterization of the space H01 (Ω; D) in terms of capacity will be given in Theorem 6.2. By introducing capacity constraints on sets, it will be possible to construct larger compact families that include domains with cracks.

6.1

Definition and Basic Properties

For p > N the elements of W 1,p (RN ) can be redefined on sets of measure zero to be continuous functions (cf. R. A. Adams [1, Thm. 5.4, p. 97]): W 1,p (RN ) → C 1,λ (RN ),

0 < λ ≤ 1 − N/p.

(6.2)

430

Chapter 8. Shape Continuity and Optimization

But for p ≤ N this is no longer the case. We first recall the definition of (1, p)capacity with respect to RN and with respect to an open subset G of RN along with a number of basic technical results. Definition 6.1. Let p, 1 < p < ∞, be a real number and N ≥ 1 be an integer. (i) (L. I. Hedberg [2, p. 239]) The capacity is defined as follows: for a compact subset K ⊂ RN   def Cap1,p (K) = inf |∇ϕ|p dx : ϕ ∈ C0∞ (RN ), ϕ > 1 on K ; RN

for an open subset G ⊂ RN

 def Cap1,p (G) = sup Cap1,p (K) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

 def Cap1,p (E) = inf Cap1,p (G) : ∀G ⊃ E, G open . In what follows, we shall use the simpler notation Capp as in L. C. Evans and R. F. Gariepy [1, pp. 147] instead of Cap1,p . (ii) (L. C. Evans and R. F. Gariepy [1, pp. 160] and J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 87]) A function f is said to be quasi-continuous on D ⊂ RN if, for each ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in RN is said to be quasi-open if, for all ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and E ∪ Gε is open. In the above definitions the capacity is defined with respect to RN . In J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 2, p. 27], the capacity is defined relative to an open subset D of RN instead of the whole space RN . Definition 6.2. Let N ≥ 1 be an integer, let p, 1 < p < ∞, and let D be a fixed open subset of RN . (i) The (1, p)-capacity is defined as follows: for a compact subset K ⊂ D   def p ∞ |∇ϕ| dx : ϕ ∈ C0 (D), ϕ > 1 on K ; Cap1,p (K, D) = inf D

for an open subset G ⊂ D

 def Cap1,p (G, D) = sup Cap1,p (K, D) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

 def Cap1,p (E, D) = inf Cap1,p (G, D) : ∀G open such that E ⊂ G ⊂ D .

6. Elements of Capacity Theory

431

(ii) A function f is said to be quasi-continuous in D if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in D is said to be quasi-open if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and E ∪ Gε is open. Clearly, by definition, ∀E ⊂ D,

Cap1,p (E) ≤ Cap1,p (E, D).

(6.3)

The number Cap1,p (E, D) ∈ [0, ∞] is called the (variational) (p, mN )-capacity of the condenser (E, G), where mN is the N -dimensional Lebesgue measure. It can easily be shown that for a quasi-open set there exists a decreasing sequence {Ωn } of open sets such that Ωn ⊃ E and capD (Ωn \E) goes to zero as n goes to infinity. We say that a property holds quasi-everywhere (q.e.) in D if it holds in the complement D\E of a set E of zero capacity. A set of zero capacity has zero measure, but the converse is not true. The capacity is a countably subadditive set function, but it is not additive even for disjoint sets. Hence the union of a countable number of sets of zero capacity has zero capacity.

6.2

Quasi-continuous Representative and H 1 -Functions

The following theorem is from L. I. Hedberg [2] (sect. 2, p. 241 and Thm. 1.1, p. 237, and footnote, p. 238 referring to T. Wolff and the paper by L. I. Hedberg and Th. H. Wolff [1]). See also J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.4, p. 89] Theorem 6.1. Let p, 1 < p < ∞, be a real number, N ≥ 1 be an integer, and D ⊂ RN be open. (i) Every f in W 1,p (D) has a quasi-continuous representative: there exists a quasi-continuous function f1 defined on D such that f1 = f almost everywhere in D (hence f1 is a representative of f in W 1,p (D) ). Any two quasicontinuous representatives f1 and f2 of f ∈ W 1,p (D) such that f1 = f2 almost everywhere are equal quasi-everywhere in D. (ii) Let f ∈ W 1,p (RN ). Let K ⊂ RN be closed, and suppose that f |K = 0 (trace defined quasi-everywhere). Then f |
K ∈ W01,p (K). The following key lemma completes the picture (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.5, p. 90]). See also L. I. Hedberg [2, p. 241]. Lemma 6.1. (i) Let Ω be an open subset of RN , let p, 1 ≤ p < ∞, and consider an element u of W 1.p (Ω). Then u ∈ W01.p (Ω) if and only if there exists a quasicontinuous representative u1 of u in RN such that u1 = 0 quasi-everywhere in Ω and u1 = u almost everywhere in Ω.

432

Chapter 8. Shape Continuity and Optimization

(ii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W01.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. (iii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W 1.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. A function ϕ in W 1,p (D) is said to be zero quasi-everywhere in a subset E of D if there exists a quasi-continuous representative of ϕ which is zero quasi-everywhere in E. This makes sense since any two quasi-continuous representatives of an element ϕ of W 1,p (D) are equal quasi-everywhere. For any ϕ ∈ W01,p (D) and t ∈ R, the set {x ∈ D : ϕ(x) > t} is quasi-open. Moreover, the subspace W01,p (Ω; D) introduced in section 2.5.3 in Chapter 2 can now be characterized by the capacity. Theorem 6.2. Given two bounded open subsets Ω and D of RN ,   W01,p (Ω; D) = ϕ ∈ W01,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D,

 W01,p (Ω; D) = ϕ ∈ W 1,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D.

(6.4) (6.5)

The characterization of W01,p (Ω; D) requires the notion of capacity in a very essential way. It cannot be obtained by saying that the function and its derivatives are zero almost everywhere in D\Ω. Recall the definition of the other extension H1 (Ω; D) of H 1 (Ω) to a measurable set D containing Ω (cf. Chapter 5, section 3.5, Theorem 3.6, identity (3.27)). Its definition readily extends from H 1 to W 1,p as follows:   def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. in D . By definition W01,p (Ω; D) ⊂ W1,p (Ω; D), but the two spaces are generally not equal, as can be seen from the following example. Denote by Br , r > 0, the open ball of radius r > 0 in R2 . Define Ω = B2 \∂B1 and D = B3 . The circular crack ∂B1 in Ω has zero measure but nonzero capacity. Since ∂B1 has zero measure, W1,p (Ω; D) contains functions ψ ∈ W01,p (B2 ) whose restrictions to B2 are not zero on the circle ∂B1 and hence do not belong to W01,p (Ω). Yet for Lipschitzian domains and, more generally, for domains Ω that are stable with respect to D in the sense of Definition 4.1, the two spaces are indeed equal.

6.3

Transport of Sets of Zero Capacity

Any Lipschitz continuous transformation of RN which has a Lipschitz continuous inverse transports sets of zero capacity onto sets of zero capacity. Lemma 6.2. Let D be an open subset of RN and let T be an invertible transformation of D such that both T and T −1 are Lipschitz continuous. For any E ⊂ D Cap1,2 (E, D) = 0 ⇐⇒ Cap1,2 (T (E), D) = 0.

6. Elements of Capacity Theory

433

Proof. (a) E = K, K compact in D. Observe that, in the definition of the capacity, it is not necessary to choose the functions ϕ in C0∞ (D). They can be chosen in a larger space as long as  |∇ϕ|2 dx < ∞ and ϕ > 1 on K. D

So the capacity is also given by   def Cap1,2 (K, D) = inf |∇ϕ|2 dx : ϕ ∈ H01 (D) ∩ C(D), ϕ > 1 on K . D

But the space

H01 (D)

∩ C(D) is stable under the action of T :

ϕ ∈ H01 (D) ∩ C(D) ⇐⇒ T ◦ ϕ ∈ H01 (D) ∩ C(D). Consider the matrix A(x) = | det(DT (x))| ∗DT (x)( ∗DT (x))−1 . def

By assumption on T , the elements of the matrix A belong to L∞ (D) and αI ≤ A(x) ≤ α−1 I a.e. in D.

∃α > 0,

Define E(K) = {ϕ ∈ H01 (D) ∩ C(D) : ϕ > 1 on K}. For any ϕ ∈ E(T (K)), ϕ ◦ T ∈ E(K) and    |∇ϕ|2 dx = A ∇(ϕ ◦ T ) · ∇(ϕ ◦ T ) dx ≥ α |∇(ϕ ◦ T )|2 dx. D

D

D

Let K be a compact subset such that Cap1,2 (K, D) = 0. For each ε > 0 there exists ϕ ∈ E(K) such that   2 |∇ϕ| dx ≤ ε ⇒ ∀ε, |∇(ϕ ◦ T )|2 dx ≤ ε/α or ϕ ◦ T ∈ E(T −1 (K)) D

D

⇒ ∀ε,

Cap1,2 (T −1 (K), D) ≤ ε/α

⇒ Cap1,2 (T −1 (K), D) = 0

and we can repeat the proof with T −1 in place of T . (b) E = G, G open. Let

 Cap1,2 (G, D) = sup Cap1,2 (K, D) : ∀K ⊂ G, K compact = 0. Therefore capD (K) = 0, which implies that Cap1,2 (T (K), D) = 0 and hence

 Cap1,2 (T (G), D) = sup Cap1,2 (K  , D) : ∀K  ⊂ T (G), K  compact 

= sup Cap1,2 (T (K), D) : ∀K ⊂ G, K compact = 0. (c) General case. Let def

Cap1,2 (E, D) = inf {capD (G) : ∀G ⊃ E, G open} .

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Chapter 8. Shape Continuity and Optimization

Therefore, for all ε > 0, there exists G ⊃ E open such that Cap1,2 (E, D) ≤ Cap1,2 (G, D) ≤ ε

⇒ Cap1,2 (T (E), D) ≤ Cap1,2 (T (G), D)

since T (E) ⊂ T (G) is open. By definition of Cap1,2 (G, D) we have ∀K ⊂ G,

Cap1,2 (K, D) ≤ ε

⇒ Cap1,2 (T (K), D) ≤ αε

⇒ Cap1,2 (T (G), D) = sup{Cap1,2 (T (K), D) : K ⊂ G} ≤ αε. Finally, for all ε > 0, Cap1,2 (T (E), D) ≤ αε and hence Cap1,2 (T (E), D) = 0.

7

Crack-Free Sets and Some Applications

In this section we show that the result for the second question raised in the introduction to this chapter is true for measurable sets in the following large family.

7.1

Definitions and Properties

Definition 7.1. (i) The interior boundary of a set Ω is defined as def

∂i Ω = ∂Ω\∂Ω.

(7.1)

The exterior boundary of a set Ω is defined as def

∂e Ω = ∂Ω\∂Ω.

(7.2)

(ii) A subset Ω of RN is said to be crack-free if ∂i Ω = ∅; its complement Ω is said to be crack-free if ∂e Ω = ∅. The intuitive terminology crack-free arises from the following identities: ∂i Ω = ∂Ω\∂Ω = int Ω\int Ω,

∂e Ω = ∂Ω\∂Ω = int Ω\int Ω.

(7.3)

A crack-free set Ω does not have internal cracks that would disappear after the cleaning operation with respect to the closure of the set2 . It is verified for sets that 2 Since ∂Ω = Ω ∩ Ω ⊂ Ω ∩ Ω = ∂Ω, the definition of a crack-free set Ω is equivalent to the property ∂Ω = ∂Ω. In 1994 A. Henrot [2] introduced the terminology Carath´ eodory set for such ¨ki, J. Sprekels, and D. Tiba [1], a set that was later adopted by D. Tiba [3], P. Neittaanma and A. Henrot and M. Pierre [4]. However, this terminology does not seem to be standard. For instance, in the literature on polynomial approximations in the complex plane C, a Carath´ eodory set is defined as follows.

Definition 7.2 (O. Dovgoshey [1] or D. Gaier [1]). A bounded subset Ω of C is said to be a Carath´ eodory set if the boundary of Ω coincides with the boundary of the unbounded component of the complement of Ω. A Carath´ eodory domain is a Carath´ eodory set if, in addition, Ω is simply connected. In V. A. Martirosian and S. E. Mkrtchyan [1], Ω is further assumed to be measurable. This definition excludes not only interior cracks but also bounded holes inside the set Ω as can be seen from the example of the annulus Ω = {x ∈ R2 : 1 < |x| < 2} in R2 . So, it is more restrictive than ∂Ω = ∂Ω. In order to avoid any ambiguity, we choose to keep the intuitive terminology crack-free.

7. Crack-Free Sets and Some Applications

435

are locally the epigraph of a C 0 -function. Yet, it is also verified for sets that are not locally the epigraph of a C 0 -function. The following equivalent characterization of a crack-free set in terms of the oriented distance function was used as its definition in M. C. Delfour and J.´sio [43]. P. Zole Theorem 7.1. A subset Ω of RN is crack-free if and only if bΩ = bΩ . Its complement Ω is crack-free if and only if b
Ω = b
Ω (or, equivalently, bΩ = bint Ω ). Proof. If ∂Ω = ∂Ω, then |bΩ | = d∂Ω = d∂Ω = |bΩ |. Hence, dΩ + d
Ω = |bΩ | = |bΩ | = dΩ + d
Ω implies d
Ω = d
Ω and bΩ = bΩ . Conversely, if bΩ = bΩ , then d∂Ω = |bΩ | = |bΩ | = d∂Ω and ∂Ω = ∂Ω since both sets are closed. We also have the following equivalences: b
Ω = bint
Ω ⇐⇒ bΩ = bΩ ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω

(7.4)

and bΩ = bint Ω ⇐⇒ b
Ω = b
Ω ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω.

(7.5)

Remark 7.1. An open set Ω is crack-free if and only if Ω = int Ω. Indeed, by definition, Ω is crackfree if and only if Ω = Ω. But since Ω is open Ω = Ω and Ω = Ω = int Ω. We give the following general technical theorem and its corollary. ´sio [43]). Let K be a compact Theorem 7.2 (M. C. Delfour and J.-P. Zole subset of RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on K implies that v has a quasi-continuous representative v ∗ such that v ∗ |int K ∈ W01,p (int K). Corollary 1. Let Ω be a bounded set in RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on Ω implies that v has a quasi-continuous representative v ∗ such that v ∗ |int Ω ∈ W01,p (int Ω). Proof. Choose the quasi-continuous representative  1 def v dy v ∗ (x) = lim δ0 |B(x, δ)| B(x,δ)

(7.6)

of v and let {Gε } be the associated family of open subsets of (1, p)-capacity less than ε of Definition 6.1 (ii). (a) If there exists ε > 0 such that Gε ⊂ int K, then K ⊂ Gε and v ∗ is continuous in K. But, by assumption, v = 0 almost everywhere in the open set K. Hence, by definition of v ∗ , v ∗ = 0 everywhere in the open subset K ⊂ K, and, by continuity, v ∗ = 0 everywhere in K.

436

Chapter 8. Shape Continuity and Optimization

(b) If for all ε > 0, Gε ∩ K = ∅, then either x ∈ ∪ε>0 Gε ∩ K or x ∈ ∩ε>0 Gε ∩ ∂K, since K = int K ∪ ∂K = K ∪ ∂K. In the first case, v ∗ (x) = 0 since x ∈ K. In the second case, Capp (∩ε>0 Gε ∩ ∂K) ≤ Capp (Gε ) < ε and ∩ε>0 Gε ∩ ∂K has zero (1, p)-capacity. From (a) and (b), v ∗ = 0 quasi-everywhere in K. Therefore by Theorem 5 in L. I. Hedberg and Th. H. Wolff [1], which generalizes Theorem 1.1 in L. I. Hedberg [2], v ∗ |int K ∈ W01,p (int K). Remark 7.2. In L. I. Hedberg [2, 3] a compact set K is said to be (1, p)-stable if v = 0 quasieverywhere on K implies that v|int K ∈ W01,p (int K). A sufficient condition (cf. Theorem 6.4 in L. I. Hedberg [2]) for the (1, p)-stability of K is that K be (1, p)-thick (1, p) quasi-everywhere on ∂K. In the proof of Theorem 7.2 we have shown that, under the weaker sufficient condition v = 0 almost everywhere on K, v = 0 quasi-everywhere on K. However, this condition would not be sufficient for the (2, p)-stability as shown in the example of L. I. Hedberg [3] (cf. Thm. 3.14, p. 100). Remark 7.3. There is also a result in dimension N = 2 in Theorem 7 of L. I. Hedberg and Th. H. Wolff [1]: for p, 2 ≤ p < ∞, and q, q −1 + p−1 = 1, K is (1, q)stable if and only if Capq (∂K ∩ e1,q (K)) = 0, where e1,q (K) = {x ∈ RN : K is (1, q)-thin at x}. It says that the part of ∂K where K is thin has zero (1, q)capacity. However, in the equivalence of parts (b) and (d) of Theorem 7, there seems to be a typo: the (1, q)-stability is defined as follows: “K is (1, q)-stable if v = 0 q.e. on K implies that v ∈ W01,q (K).” It should probably read “v ∈ W01,q (int K).” We now turn to our specific problem. Given a measurable subset Ω of a bounded open holdall D ⊂ RN , define the closed subspace   def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. (7.7) of W01,p (D). This space provides a relaxation of the Dirichlet boundary value problem to domains that are only measurable. The next theorem will show that in fact W1,p (Ω; D) can be embedded in W01,p (int Ω). Furthermore, for crack-free measurable subsets Ω of D, W1,p (Ω; D) coincides with W01,p (int Ω). This will yield the continuity of the homogeneous Dirichlet problem with respect to the open domains Ω in the family L(D, r, O, λ) of subsets of D verifying a uniform fat segment property. Theorem 7.3. Let D be bounded open in RN . (i) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ 1,p v ∈ W (Ω; D) =⇒ 1,p such that v ∗ | int Ω ∈ W0 (int Ω) and W01,p (int Ω; D) ⊂ W1,p (Ω; D) ⊂ W01,p (int Ω; D).

(7.8)

7. Crack-Free Sets and Some Applications (ii) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ v ∈ W 1,p (Ω; D)  =⇒ such that v ∗ |int Ω ∈ W01,p (int Ω) and Ω crack-free

437

(7.9)

and W01,p (int Ω; D) = W1,p (Ω; D) = W01,p (int Ω; D). Proof. (i) From Theorem 7.2 with K = Ω and v˜, the extension by zero of v from D to RN . Given ϕ ∈ W01,p (int Ω), the zero extension e0 ϕ ∈ W01,p (int Ω; D) and (1 − χint Ω )ϕ = 0 almost everywhere in D. This implies that (1 − χΩ )ϕ = 0 almost everywhere in D and ϕ ∈ W1,p (Ω; D). (ii) From (i) and the equivalences (7.4): int Ω = int Ω. Remark 7.4. The condition that v ∈ W1,p (Ω; D) is weaker than the capacitary condition of ´sio [12] and hence its conclusion that v ∗ ∈ W01,p (int Ω) D. Bucur and J.-P. Zole is also weaker. It gives information only about the function on ∂Ω rather than on the whole ∂Ω. The “crack-free” property of Definition 7.1 is purely geometric and even set theoretic. It is different from and probably stronger than the capacitary ´sio [12]. Instead of forcing Ω to be condition of D. Bucur and J.-P. Zole “uniformly thick” at quasi all points of the boundary ∂Ω in order to preserve the zero Dirichlet condition on the whole boundary, it erases the “interior boundary” ∂Ω\∂Ω. Its advantage is that it does not require an a priori knowledge of the differential operator and does not necessitate the Maximum Principle or harmonic functions; its disadvantage is that it cannot handle problems where the trace of the function on ∂Ω\∂Ω is an important feature of the problem at hand. Yet, it should readily be extendable to Sobolev spaces on C 1,1 -submanifolds of codimension 1 to deal with the Laplace–Beltrami or shell equations.

7.2 7.2.1

Continuity and Optimization over L(D, r, O, λ) Continuity of the Classical Homogeneous Dirichlet Boundary Condition

In light of Theorem 4.1, it remains to find families of sets with properties that would enable us to sharpen part (i) or that would verify the stability condition H01 (Ω; D) = H1 (Ω; D). In the second case, the family of subsets of a bounded holdall D that satisfy the uniform fat segment property meets all the requirements. Theorem 7.4. Let D be a bounded open nonempty subset in RN and Ω be open subsets of D. Denote by y(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → y(Ω) : L(D, r, O, λ) → H01 (D)-strong

(7.10)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), d
Ω in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χ
Ω in Lp (D).

438

Chapter 8. Shape Continuity and Optimization

Proof. Since all topologies are equivalent on L(D, r, O, λ), consider a domain Ω and a sequence {Ωn } such that d
Ωn → d
Ω in W 1,p (D) for some p, 1 ≤ p < ∞. By Theorem 6.1 (ii) of Chapter 2, all sets in L(D, r, O, λ) are crack-free since int Ω = ∅,

Ω = int Ω = Ω

⇒ b Ω = bΩ .

By Theorem 7.3 (ii), H1 (Ω; D) = H01 (Ω; D). Finally, from part (ii) of Theorem 4.1, we get the continuity of the solutions of the classical Dirichlet problem. 7.2.2

Minimization/Maximization of the First Eigenvalue

Consider the first eigenvalue λA (Ω) introduced in (2.1)  A∇ϕ · ∇ϕ dx def A D λ (Ω) = inf1 |ϕ|2 dx 0=ϕ∈H0 (Ω;D) D

(7.11)

for the Laplace equation on the open Ω of D. It was shown in Theorem 2.2 that λA (Ω) is an upper semicontinuous function of Ω and that there is a solution to the maximization problem with respect to Ω. We now turn to the minimization of λA (Ω) with respect to Ω that requires the lower semicontinuity and hence the continuity of λA (Ω) with respect to Ω. To do that we choose the smaller family L(D, r, O, λ) of sets verifying a uniform fat segment property. Theorem 7.5. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) The mapping d
Ω → λA (Ω) : L(D, r, O, λ) → R ∪{+∞}

(7.12)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), d
Ω in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χ
Ω in Lp (D). (ii) The minimization and maximization problems inf

λ(Ω) and

Ω∈L(D,r,O,λ)

sup

λ(Ω)

(7.13)

Ω∈L(D,r,O,λ)

have solutions in L(D, r, O, λ). Moreover, 0 < λA (D) ≤

inf Ω∈L(D,r,O,λ)

λ(Ω) ≤

sup

λ(Ω) ≤ β λI (O).

(7.14)

Ω∈L(D,r,O,λ)

Proof. (i) Continuity. Go back to the proof of Theorem 2.2. Given Ω ∈ L(D, r, O, λ), consider a sequence {Ωn } in L(D, r, O, λ) such that d
Ωn → d
Ω in C(D). Let λA (Ωn ) and un ∈ H01 (Ωn ; D), un L2 (D) = 1, be the eigenvalue and an eigenvector associated with Ωn . By definition of L(D, r, O, λ), for all x ∈ ∂Ω, there exists Ax ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, y + Ax O ⊂ Ω. Fix x0 ∈ ∂Ω. There exist z ∈ B(x0 , r) ∩ Ω

7. Crack-Free Sets and Some Applications

439

and ε > 0 such that B(z, ε) ⊂ B(x0 , r) ∩ Ω. As a result B(z, ε) + Ax O ⊂ Ω and z + Ax O ⊂ Ω. By the compactivorous property, there exists N such that, for all n > N , z + Ax O ⊂ Ωn and ∀n > N,

λn = λA (Ωn ) ≤ λA (z + Ax O) ≤ βλI (z + Ax O) = βλI (O)

since the eigenvalue is independent of a translation and a rotation of the set. This means that the sequence {λn } of eigenvalues is bounded. By coercivity α∇un }2L2 (D) ≤ λn un 2L2 (D) = λn and {un } is bounded in H01 (D). There exist λ0 and u0 ∈ H1 (Ω; D) and subsequences, still indexed by n, such that λn → λ 0 ,

un L2 (D) = 1,

⇒ u0 L2 (D) = 1.

un  u0 in H01 (D)-weak

But for all Ω ∈ L(D, r, O, λ) we have H1 (Ω; D) = H01 (Ω; D) and u0 ∈ H01 (Ω; D). For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn . Therefore for all n ≥ N (K)   A∇un · ∇ϕ dx = λn un ϕ dx. un ∈ H01 (Ωn ; D), D

By letting n → ∞

D





∀ϕ ∈ D(Ω),

A∇u · ∇ϕ dx = λ0 D

u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is verified in H01 (Ω; D). Hence, u0 ∈ H01 (Ω; D), uL2 (Ω) = 1, satisfies the variational equation   A∇u0 · ∇ϕ dx = λ0 u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

Since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ0 . To show that λ0 = λA (Ω), it remains to prove that λ0 ≤ λA (Ω). For each n  A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ; D), λ (Ωn ) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ωn and ϕ ∈ D(Ωn ). Therefore for all n > N  A∇ϕ · ∇ϕ dx A λ (Ωn ) ≤ D ϕ2L2 (D) and by going to the limit ∀ϕ ∈ D(Ω),

 λ0 ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

440

Chapter 8. Shape Continuity and Optimization

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ0 ≤ λA (Ω). As a result λ0 = λA (Ω). Therefore the whole sequence {λA (Ωn )} converges to λA (Ω) and we have the continuity of the map d
Ω → λA (Ω). The strong continuity in H01 (D) now follows from the convergence of un → u0 in L2 (D)-strong: α∇(un − u0 )2L2 (D)  ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D    A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx = D D   D un · un dx + λ0 u0 · u0 dx − 2 A∇un · ∇u0 dx = λn D D D   u0 · u0 dx − 2 A∇u0 · ∇u0 dx = 0. → 2λ0 D

D

(ii) This follows from the continuity and the compactness of L(D, r, O, λ) in all the mentioned topologies. The inequalities have been proven in part (i). Corollary 1. Under the assumptions of Theorem 7.5, the maximization and minimization problems (7.13) also have solutions in L(D, r, O, λ) under an equality or an inequality constraint of the form m(Ω) = α, m(Ω) ≤ α, or m(Ω) ≥ α provided that there exists Ω ∈ L(D, r, O, λ) such that m(Ω ) = α, m(Ω ) ≤ α, or m(Ω ) ≥ α. Proof. This follows from Theorem 13.1 and its Corollaries 1 and 2 of section 13 in Chapter 7, where it is shown that the convergence is strong not only in the Hausdorff topologies but also for the associated characteristic functions in L1 (D), thus making the volume function continuous with respect to open domains in L(D, r, ω, λ). For the spaces of convex subsets recall from Theorem 8.6 of Chapter 6 and Theorem 10.2 of Chapter 7 that the compactness remains true in the W 1,1 (D)-topology.

8

Continuity under Capacity Constraints

We have established the continuity of the homogeneous Dirichlet problem, that is, the continuity of the function d
Ω → u(Ω) : Cdc (D) → H01 (D)-strong for the family of domains verifying a uniform fat segment property. Domains with cracks lie between families of L(D, r, O, λ) type and Cdc (D). Topologies using the characteristic function χΩ are excluded since they do not see interior boundaries of zero measure. Families that contains sets with cracks must be chosen in such a way that the homogeneous Dirichlet boundary condition on the sequence of solutions u(Ωn ) in the converging sets Ωn is preserved by the solution u(Ω) on the boundary of the limit domain Ω in section 4. We have to be careful since the relative capacity of the

8. Continuity under Capacity Constraints

441

complement of the domains near the boundary must not vanish. In order to handle this point, we introduce the following concepts and terminology related to the local capacity of the complement near the boundary points. The following definition is due to J. Heinonen, T. Kilpelainen, and O. Martio [1, pp. 114–115]. Definition 8.1. (i) Given r > 0, a set E ⊂ RN , and a point x, the function def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

(8.1)

is called the capacity density function. (ii) We say that a set E is thick at a point x if 

1



0

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.2)

This is called the Wiener criterion. (iii) We say that an open set Ω satisfies the Wiener density condition if  ∀x ∈ ∂Ω, 0

1



Capp (Ω ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.3)

This means that the set Ω is thick at every boundary point x ∈ ∂Ω (that is, Ω has “sufficient” capacity around each boundary point). Remark 8.1. If Ω satisfies the (r, c)-capacity density condition, the complement is thick in any point of the boundary. The Wiener criterion can be strengthened by introducing the following condition (J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 127]). Definition 8.2. (i) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r , 0 < r < r,

Capp (Ω ∩ B(x, r  ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

≥ c.

(8.4)

We also say that Ω is uniformly (r, c)-thick. (ii) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the strong (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r  , 0 < r < r,

Capp (Ω ∩ B(x, r  ), B(x, 2r )) ≥ c. Capp (B(x, r ), B(x, 2r ))

We also say that Ω is strongly uniformly (r, c)-thick.

(8.5)

442

Chapter 8. Shape Continuity and Optimization

(iii) For r, 0 < r < 1, and c > 0 define the following family of open subsets of D: & Ω satisfies the strong def . (8.6) Oc,r (D) = Ω ⊂ D : (r, c)-capacity density condition

Example 8.1. Only a few capacities can be computed explicitly. The computation of the spherical condensers Capp (B(x, r), B(x, R)) and Capp (B(x, r), B(x, R)) leads to some simplifications in the denominator of the above expression (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 35]). For 0 < r < R < ∞ Capp (B(x, r), B(x, R)) = Capp (B(x, r), B(x, R)

(8.7)

and for p > 1   p−1 p−N 1−p |N − p|  p−N N −1   , R p−1 − r p−1 ω p−1 Capp (B(x, r), B(x, R)) =  1−N  R  N −1  ω , log r

p = N, p = N,

where ω N −1 is the volume (Lebesgue measure) of the ball in RN −1 . In particular Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p , N

Capp (B(x, r)) = Capp (B(x, r), R ) = c2 (N, p) r Capp ({x0 }, R ) = c3 (N, p) r N

N −p

1 < p, N −p

,

1 < p < N,

,

(8.8)

p < N,

for constants ci (N, p) that depend only on p and N . Remark 8.2. Definition 8.2 (i) uses a slightly different definition of the previous capacity density function of Definition 8.1: def

capx,r (E) = instead of def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r)) Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

.

Note that Capp (B(x, r), B(x, 2r)) = Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p ,

1 < p,

and Capp (E ∩ B(x, r), B(x, 2r)) ≤ Capp (E ∩ B(x, r), B(x, 2r)) imply that ∀r > 0, ∀x,

capx,r (E) ≥ capx,r (E)

8. Continuity under Capacity Constraints

443

and the strong (r, c)-capacity density condition implies the (r, c)-capacity density condition. But for an open subset Ω of the bounded open subset D of RN , the following two conditions are equivalent: ∀x ∈ ∂Ω, ∀ 0 < r  < r, and ∀x ∈ ∂Ω, ∀0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥c Capp (B(x, r ), B(x, 2r ))

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r  ), B(x, 2r ))

≥ c,

(8.9)

(8.10)

and also from the Oc,r (D) family viewpoint the two definitions are equivalent. It is clear that (8.9) implies (8.10), but the converse is not obvious. Recall Theorem 4.1 and the notation un for the solution of (4.5) in H01 (Ωn ; D) and u ∈ H1 (Ω; D) for the weak limit in H01 (D) which satisfies (4.8). The following theorem gives the main continuity result. Theorem 8.1. Let D be a bounded open nonempty subset of RN of class C 2 for N ≥ 3 and Ω be an open subset of D. Assume that A is a matrix function that satisfies the conditions (4.4) and that the elements of A belong to C 1 (D). Denote by u(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → u(Ω) : Oc,r (D) → H01 (D)-strong

(8.11)

is continuous with respect to the uniform topology C(D) on Cdc (D). Proof. When the dimension N of the space is equal to 1, H 1 (D) ⊂ C(D) and no approximation of f is necessary. When N ≥ 2, we first prove the result for f ∈ H s (D), s > N/2 − 2 (since for D of class C 2 the corresponding solution will belong to H s+2 (D) ⊂ C(D)), and then, by approximation of f , we prove it for f ∈ H −1 (D). (i) First consider f ∈ H s (D), s > N/2−2, for N ≥ 2 (s = −1 for N = 1). The proof will make use of the following results from J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 3, Thm. 3.70, p. 80]. Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let v be an A-harmonic function; that is, v ∈ 1,p Wloc (Ω) is a weak solution of equation  A∇v(Ω) · ∇ϕ dx = 0 (8.12) ∀ϕ ∈ D(Ω), D

in the open set Ω. Then there exists a continuous function u on Ω such that v(Ω) = u almost everywhere. Note that the continuous representative from Lemma 8.1 is in fact a quasicontinuous H01 (Ω) representative. Indeed, let v1 = v almost everywhere and let v1 be continuous on Ω. We want to show that v1 is a quasi-continuous representative of v. There exists a quasi-continuous representative v2 of v, which is equal to v almost

444

Chapter 8. Shape Continuity and Optimization

everywhere. So v1 is continuous, v2 is quasi-continuous, and v1 = v2 almost everywhere. Using J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.12], we get v1 = v2 quasi-everywhere. Always from J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 6.27, p. 122] we get the following lemma. Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let Ω belong to Oc,r (D). If θ ∈ W 1,p (Ω) ∩ C(Ω) and if h is an A-harmonic function in Ω such that h − θ ∈ W01,p (Ω), then ∀x0 ∈ ∂Ω,

lim h(x) = θ(x0 ).

x→x0

Note that the fact that Ω belongs to Oc,r (D) involves the notion of thickness in any point of its boundary, which necessarily occurs in the proof of the lemma (cf. Definition 8.1). Returning to Theorem 8.1, it will be sufficient to prove the continuity for a subsequence of {Ωn }. By Theorem 4.1 there exists a subsequence of {Ωn }, still indexed by n, such that un weakly converges to u in H01 (D), and u ∈ H1 (Ω; D) satisfies (4.8) in Ω. We now prove that under our assumptions uΩ = u|Ω ∈ H01 (Ω), which implies that u = eΩ (uΩ ) and u ∈ H01 (Ω; D) since (1 − χΩ )u = (1 − χΩ )eΩ (uΩ ) = 0 almost everywhere in D. For that purpose we use Lemma 6.1, which says that it is sufficient to prove that u = 0 quasi-everywhere in D\Ω for some quasi-continuous representative u. From Theorem 7.3 (i) H01 (int Ω; D) ⊂ H1 (Ω; D) ⊂ H01 (int Ω; D) and hence u = 0 quasi-everywhere in D\Ω. So it remains to show that u = 0 quasi-everywhere in ∂Ω ∩ D. From the Banach–Saks theorem (cf. I. Ekeland and R. Temam [1]) there exists a sequence of averages def

ψn =

Nn

αkn un ,

0 ≤ αkn ≤ 1 ,

k=n

Nn

αkn = 1

k=n

such that ψn → u in H01 (D). Because of the strong convergence of {ψn } to u in H01 (D), we have ψn (x) → u(x) q.e. in D for a subsequence of {ψn }, still indexed by n. Let G0 be the set of zero capacity on which {ψn (x)} does not converge to u(x). Given x ∈ D\(Ω ∪ G0 ), we prove that, for all ε > 0, |u(x)| < ε. We have |u(x)| ≤ |u(x) − ψn (x)| + |ψn (x)|. There exists Nε,x > 0 such that for all n > Nε,x , |u(x) − ψn (x)| < ε/2. It remains to show that there exists N  such that for all n ≥ N  , |un (x)| < ε/2 implies ψn (x)| < ε/2. Denote by uD the solution of (4.5) in D. By assumption on D and f , the solution uD is continuous in D. Subtracting the corresponding equations, we obtain  A∇(uD − un ) · ∇ϕ dx = 0, ∀ϕ ∈ H01 (Ωn ; D),  D A∇(uD − uΩn ) · ∇ϕ dx = 0. ∀ϕ ∈ H01 (Ωn ), Ωn

8. Continuity under Capacity Constraints

445

Define ˜ n def = uD − un , h

def

hn = uD |Ωn − uΩn ,

def

θn = uD |Ωn .

˜ n to Ωn is A-harmonic in Ωn and, from Lemma 8.1, Therefore the restriction hn of h continuous in Ωn . Moreover we have the continuity of uΩn in the closure Ωn , and uΩn is zero on the boundary. To show that, we use Lemma 8.2 with hn and θn . By definition, hn −θn = −uΩn belongs to H01 (Ωn ). From the continuity of uD we obtain ˜ n of hn is equal to uD on ∂Ωn . Hence the extension that the continuous extension h un of uΩn to the boundary is zero. Using J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 6.44], we obtain that if hn is δ-H¨olderian on ∂Ωn , then there exists δ1 , δ ≤ δ1 < 1, such that it is δ1 -H¨olderian on all Ωn . We have that uD is (s + 2 − N/2)-H¨ olderian on D (with s + 2 − N/2 > 0), with a constant M , because of the assumption on f and D. Finally, we get ∀x, y ∈ ∂Ωn ,

|hn (x) − hn (y)| = |uD (x) − uD (y)| ≤ M |x − y|s−N/2+2 .

So there exist δ1 = δ1 (N, β/α, c) and M1,n = 80M r−2 max{1, (diam (Ωn ))2} ≤ 80M r−2 max{1, (diam (B))2} = M1 such that ∀x, y ∈ Ωn ,

|hn (x) − hn (y)| ≤ M1,n |x − y|δ1 ≤ M1 |x − y|δ1 .

By a simple argument we obtain that this inequality holds in D, and hence there exists δ2 , δ1 ≤ δ2 < 1, such that for all x, y ∈ D we have ˜ n (x) − h ˜ n (y)| + |uD (x) − uD (y)| |un (x) − un (y)| ≤ |h ≤ M1 |x − y|δ1 + M |x − y| ≤ M2 |x − y|δ2 . Choose R > 0 such that M2 Rδ2 < ε/2. Because of the H c -convergence of Ωn to Ω there exists an integer nR > 0 such that for all n ≥ nR we have (D\Ωn ) ∩ B(x, R) = ∅. For xn ∈ (D\Ωn ) ∩ B(x, R), |un (x)| = |un (x) − un (xn )| ≤ M2 |x − xn |δ2 ≤ M2 Rδ2 ≤ ε/2 because un (xn ) = 0. So ∀n > nR ,

Nn Nn n ε ε |ψn (x)| = αk un (x) ≤ αkn = . 2 2 k=n

k=n

Finally, we obtain |u(x)| ≤ ε. Because ε was arbitrary, we have u(x) = 0 quasieverywhere on D\Ω, which implies that u ∈ H01 (Ω; D) and uΩ = u|Ω ∈ H01 (Ω). The strong convergence of {un } to u now follows from Theorem 4.1 and u = eΩ (uΩ ). (ii) In the next step we prove that the continuity is preserved for f ∈ H −1 (D). The main idea is to use the continuous dependence of the solution uΩ,f ∈ H01 (Ω; D)

446

Chapter 8. Shape Continuity and Optimization

with respect to f , which is uniform in Ω. Indeed, let Ω ⊂ D, and let f, g ∈ H −1 (D). Then, by a simple subtraction of the equations, we get  |∇(uΩ,f − uΩ,g )|2 dx = f − g, uΩ,f − uΩ,g  D

⇒ uΩ,f − uΩ,g H01 (D) ≤ f − gH −1 (D) . So, let f ∈ H −1 (D) and fε = f ∗ ρε for some mollifier ρε . Letting ε → 0 we have fε − f H −1 (D) → 0. Let {Ωn } ⊂ Oc,r (D) be a sequence that converges in the H c -topology to an open set Ω (d
Ωn → d
Ω ). Then we have H 1 (D)

0 uΩ,fε uΩn ,fε −→

because fε ∈ H s (D) and, from the previous considerations, H 1 (D)

0 uΩn ,f uΩn ,fε −→

uniformly in Ωn . Given δ > 0, we get uΩn ,f − uΩ,f H01 (D) ≤ uΩn ,f − uΩn ,fε H01 (D) + uΩn ,fε − uΩ,fε H01 (D) + uΩ,fε − uΩ,f H01 (D) . Choose ε sufficiently small such that fε − f H −1 (D) < δ/4, and for each ε choose nε,δ > 0 such that ∀n > nε,δ ,

uΩn ,fε − uΩ,fε H01 (D) < δ/2

⇒ ∀n > nε,δ ,

uΩn ,f − uΩ,f H01 (D) ≤ δ.

As δ was arbitrary, the proof is complete. ˇ ´ k [2] in dimension N = 2 using We can now recover the result of V. Sver a the fact that for a fixed number c > 0 the family {d
Ω : Ω open ⊂ D and #(Ω) ≤ c} is compact in C(D), where #(Ω) denotes the number of connected components of Ω (cf. Theorem 2.5 (iv) in Chapter 6 and Definition 2.2 in Chapter 2). Theorem 8.2. Let the assumptions of Theorem 8.1 on the matrix function A be satisfied. Let N = 2 and let c > 0 be a positive integer. Define the set def

Oc = {Ω : Ω open ⊂ D and #(Ω) ≤ c}. Then the set Oc is compact and the map Ω → u(Ω) : Oc → H01 (D)-strong is continuous for the C(D)-topology on Cdc (D).

9. Compact Families Oc,r (D) and Lc,r (O, D)

447

ˇ ´ k [2] is given in D. Bucur [1] as a conseA proof of the result of V. Sver a quence of Theorem 8.1. The main idea of the proof is to consider f ≥ 0 (because of the decomposition of f = f + − f − ) and a sequence {Ωn } ⊂ Oc such that Ωn converges in the H c -topology to an open set Ω, and uΩn  u in H01 (D). He constructs extensions Ω+ n of the domains Ωn with the following properties: −→ d
Ω+ , d
Ω+ n

cap(Ω+ \Ω) = 0,

{Ω+ n } ⊂ Oc,r (D),

where c and r are suitable constants. In fact it can be proved that Ω+ n satisfy this capacity density condition, because in an open connected set any two points are linked by a continuous curve that lies in the set, and in the bidimensional case a curve has a positive capacity. From the previous theorem we get uΩ+  uΩ+ . We n easily obtain that uΩ+ = uΩ ≥ u ≥ 0, which will imply that u = 0 quasi-everywhere on Ω, and this concludes the proof. For details, see D. Bucur [1].

Compact Families Oc,r (D) and Lc,r (O, D)

9

We first prove the compactness of the family Oc,r (D) for the topology of uniform convergence of the distance function of the complement in D. Then, by analogy with the “fat segment property,” we specialize to a “thick set property” that generalizes ´sio [10]. the flat cone property of D. Bucur and J.-P. Zole

9.1

Compact Family Oc,r (D)

Theorem 9.1. Let D ⊂ RN be bounded open, let c > 0, and let r > 0. (i) The subset {d
Ω : Ω ∈ Oc,r (D)}

(9.1)

of Cdc (D) is compact for the uniform topology of C(D). (ii) Given a sequence of open subsets {Ωn } and Ω in Oc,r (D), and the solutions u(Ωn ) ∈ H01 (Ωn ; D) and u(Ω) ∈ H01 (Ω; D) of the Dirichlet problem (4.5) on Ωn and Ω, then d
Ωn → d
Ω in C(D)

⇒ u(Ωn ) → u(Ω) in H 1 (D)-strong.

(9.2)

We shall need the following technical lemma. Lemma 9.1. 3 Let D, D = ∅, be a bounded open subset of RN . Consider a sequence {Ωn }, Ωn = ∅, of open subsets of D. Assume that d
Ωn → d
Ω0 in C 0 (D) for some open Ω0 ⊂ D, ∂Ω0 = ∅. 3 Part (ii) was proved by D. Bucur and J.-P. Zole ´sio [10, Lem. 3.1, p. 686] by a contradiction argument. We give a direct proof using the function bΩ .

448

Chapter 8. Shape Continuity and Optimization

(i) There exists N such that ∂Ωn = ∅.

∀n > N,

(ii) There exists a subsequence {Ωnk }, ∂Ωnk = ∅, of {Ωn : n > N } such that for all ε > 0, there exists K such that ∀k > K,

def

∂Ω0 ⊂ Uε (∂Ωnk ) = {x ∈ RN : d∂Ωnk (x) < ε}

and ∀x0 ∈ ∂Ω0 , ∃ {xnk }, xnk ∈ ∂Ωnk such that xnk → x0 . Proof. (i) By the boundedness assumption on D, Ωn = ∅ and Ω0 = ∅. Since ∂Ω0 = ∅, Ω0 = ∅, and Ω0 = ∅, def

∃x0 ∈ Ω0 such that d = d
Ω0 (x0 ) > 0. There exists N such that ∀n > N,

d
Ωn − d
Ω0 C 0 (D) < d/2

⇒ d
Ωn (x0 ) ≥ d
Ω0 (x0 ) − |d
Ωn (x0 ) − d
Ω0 (x0 )| > d − d/2 > 0 ∀n > N, Ωn = ∅ ⇒ ∂Ωn = ∅. (ii) Since, for n > N , ∂Ωn = ∅, the subsequence {bΩn : n > N } belongs to the compact family Cb (D). Therefore there exist a subsequence, denoted by {bΩn }, and Ω ⊂ D, ∂Ω = ∅, such that bΩn → bΩ . Hence d
Ωn → d
Ω and d∂Ωn → d∂Ω ⇒ Ω0 = Ω

⇒ Ω0 = Ω = int Ω

⇒ ∂Ω0 ⊂ ∂Ω.

Finally, for all ε > 0, there exists N  > N such that ∀n > N  ,

d∂Ωn − d∂Ω C 0 (D) < ε

⇒ ∀n > N  ,

⇒ ∀x ∈ ∂Ω0 ,

d∂Ωn (x) < ε

def

∂Ω0 ⊂ Uε (∂Ωn ) = {x ∈ RN : d∂Ωn (x) < ε}.

As a result, for all x ∈ ∂Ω0 , there exists {xn ∈ ∂Ωxn } such that |xn − x| = d∂Ωn (x) → d∂Ω (x) = 0. Proof of Theorem 9.1. (i) It is sufficient to prove that for all c > 0 and all r, 0 < r < 1, Oc,r (D) is closed in C(D) for the uniform convergence topology. Consider a Cauchy sequence {d
Ωn } in Oc,r (D). There exists an open set Ω ⊂ D such that d
Ωn → d
Ω in C(D). If ∂Ω = ∅, then Ω ∩ Ω = ∅, Ω ⊂ Ω ⊂ Ω, and Ω = ∅ ∈ Oc,r (D). If ∂Ω = ∅, by Lemma 9.1, there exist K  ≥ K and a subsequence {Ωnk }k>K  such that ∀k > K  ,

d
Ωn − d
Ω C(D) ≤ ε (⇒ Ωnk ⊂ (Ω)ε = {x ∈ RN : d
Ω (x) ≤ ε}) k

9. Compact Families Oc,r (D) and Lc,r (O, D)

449

and, for each x ∈ ∂Ω, there exists xnk ∈ ∂Ωnk such that ∀k > K  ,

|xnk − x| ≤ ε.

Define the translations k > K .

def

τk (y) = y + xnk − x, Then, for all ρ > 0, τk (B(xnk , ρ)) = B(x, ρ) and dτk (
Ωn

k

)

− d
Ωn C(D) ≤ |xnk − x| ≤ ε k

⇒ τk (Ωnk ) ⊂ (Ωnk )ε ⊂ ((Ω)ε )ε ⊂ (Ω)2ε . The implication ((Ω)ε )ε ⊂ (Ω)2ε follows from ∀y ∈ (Ω)ε , d
Ω (z) ≤ d
Ω (y) + |z − y| ≤ ε + |z − y| ⇒ ∀z ∈ ((Ω)ε )ε ,

d
Ω (z) ≤ 2ε

⇒ d
Ω (z) ≤ ε + d(
Ω)ε (z)

⇒ ((Ω)ε )ε ⊂ (Ω)2ε .

From the identities τk (Ωnk ) ⊂ (Ω)2ε and τk (B(xnk , ρ)) = B(x, ρ), for all r , 0 < r  < r, c≤

Capp (Ωnk ∩ B(xnk , r ), B(xnk , 2r  )) Capp (B(xnk , r ), B(xnk , 2r  )) =

Capp (τk (Ωnk ∩ B(xnk , r )), τk (B(xnk , 2r ))) Capp (τk (B(xnk , r )), τk (B(xnk , 2r  )))

=

Capp ((Ω)2ε ∩ B(x, r ), B(x, 2r )) Capp (τk (Ωnk ) ∩ B(x, r ), B(x, 2r  )) ≤ , Capp (B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

since Capp is invariant with respect to the translation that is a special case of an affine isometry. Now (Ω)2ε is monotone decreasing as ε → 0 and (Ω)2ε  Ω = ∩ε>0 (Ω)2ε . From L. C. Evans and R. F. Gariepy [1, Thm. 2 (ix), p. 151], lim Capp ((Ω)2ε ∩ B(x, r ), B(x, 2r )) = Capp (Ω ∩ B(x, r  ), B(x, 2r ))

ε→0

⇒ ∀x ∈ ∂Ω, ∀r , 0 < r < r,

c≤

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

and Ω ∈ Oc,r (D). We conclude that Oc,r (D) is compact as a closed subset of the compact set Cdc (D) in the topology of uniform convergence on D. (ii) From part (i) and Theorem 8.1.

450

9.2

Chapter 8. Shape Continuity and Optimization

Compact Family Lc,r (O, D) and Thick Set Property

By analogy with the “fat segment property,” we introduce a “thickness property.” Definition 9.1. Given r > 0 and c > 0, let O be a bounded subset of RN such that 0 ∈ O\O and O satisfies the strong (r, c)-capacity density condition in 0: ∀r  , 0 < r < r,

Capp (O ∩ B(0, r  ), B(0, 2r )) ≥ c. Capp (B(0, r ), B(0, 2r ))

(9.3)

(i) An open set Ω is said to satisfy the (r, c, O)-capacity density condition if ∀x ∈ ∂Ω, ∃Ax ∈ O(N) such that x + Ax O ⊂ Ω.

(9.4)

(ii) For r, 0 < r < 1, and c > 0, define the following family of open subsets of D: def

Lc,r (O, D) = {Ω ⊂ D : Ω satisfies the (r, c, O)-capacity density condition.} (9.5)

As a special case of O we have the flat cone property introduced by D. Bucur ´sio [1, 5] in 1993: given an aperture ω, 0 < ω < π/2, a height λ > 0, and J.-P. Zole and a unit normal ν,   1 def Oω,λ,ν = y ∈ RN : PH (y) < y · eN < λ ∩ {ν}⊥ , (9.6) tan ω where PH is the orthogonal projection onto the hyperplane H = {eN }⊥ orthogonal to the direction eN (cf. Figure 2.6 in Chapter 2). The intersection with the hyperplane {ν}⊥ makes the cone flat. This cone satisfies the strong (r, c)-capacity density condition in 0 of Definition 9.1 from the scaling property of this cone. Theorem 9.2. Given r > 0, Oω,λ,ν associated with an aperture ω, 0 < ω < π/2, a height λ > 0, and a unit normal ν through identity (9.6), and def

c=

Capp (Oω,λ,ν ∩ B(0, r), B(0, 2r)) , Capp (B(0, r), B(0, 2r))

(9.7)

then 0 < c < ∞, for all r , 0 < r ≤ r, ∀r , 0 < r  < r,

Capp (Oω,λ,ν ∩ B(0, r  ), B(0, 2r )) ≥ c, Capp (B(0, r ), B(0, 2r ))

(9.8)

and Oω,λ,ν satisfies the strong (r, c)-capacity density condition in 0. Proof. Since Oω,λ,ν is a flat cone in the hyperplane {ν}⊥ , the capacity of its intersection with B(0, r), Capp (Oω,λ,ν ∩ B(0, r), B(0, 2r)), is strictly positive and finite.

9. Compact Families Oc,r (D) and Lc,r (O, D)

451

From L. C. Evans and R. F. Gariepy [1, Thm. 2, p. 151],   N −p r   Capp (B(0, r), B(0, 2r)), Capp (B(0, r ), B(0, 2r )) = r   N −p r   Capp (Oω,λ,ν ∩ B(0, r ), B(0, 2r )) = Capp ((r/r  Oω,λ,ν ) ∩ B(0, r), B(0, 2r)), r Capp (Oω,λ,ν ∩ B(0, r  ), B(0, 2r )) Capp ((r/r  Oω,λ,ν ) ∩ B(0, r), B(0, 2r)) = . (9.9) Capp (B(0, r ), B(0, 2r )) Capp (B(0, r), B(0, 2r)) Now,

  r 1 r N PH (y) < y · eN < λ ∩ {ν}⊥ Oω,λ,ν =  y ∈ R : r r tan ω    r  r 1 r r N = y∈R : PH  y <  y · eN < λ  ∩ {ν}⊥ r tan ω r r r   1 r PH (z) < z · eN < λ  ∩ {ν}⊥ = z ∈ RN : tan ω r   1 N PH (y) < y · eN < λ ∩ {ν}⊥ = Oω,λ,ν ⊃ y∈R : tan ω

since r/r ≥ 1. Finally from (9.9) Capp (r/r Oω,λ,ν ∩ B(0, r), B(0, 2r)) Capp (Oω,λ,ν ∩ B(0, r  ), B(0, 2r )) = Capp (B(0, r ), B(0, 2r )) Capp (B(0, r), B(0, 2r)) Capp (Oω,λ,ν ∩ B(0, r), B(0, 2r)) = c. Capp (B(0, r), B(0, 2r)) satisfies the strong (r, c)-capacity density condition in 0. ≥

Therefore, Oω,λ,ν Lemma 9.2.

(i) For all c > 0 and r > 0, Lc,r (O, D) ⊂ Oc,r (D),

where Oc,r (D) is the family of open subsets Ω of D satisfying the strong (r, c)capacity density condition of Definition 8.2 (iii). (ii) Given a sequence of open subsets {Ωn } and Ω in Lc,r (O, D), and the solutions u(Ωn ) ∈ H01 (Ωn ; D) and u(Ω) ∈ H01 (Ω; D) of the Dirichlet problem (4.5) on Ωn and Ω, then d
Ωn → d
Ω in C(D)

⇒ u(Ωn ) → u(Ω) in H 1 (D)-strong.

(9.10)

Proof. (i) By Definition 9.1 (i) for an open set Ω satisfying the following (r, c, O)capacity density condition: for all x ∈ ∂Ω and all r , 0 < r  < r, x + Ax O ⊂ Ω and Capp ([x + Ax O] ∩ B(x, r ), B(x, 2r  )) Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥ . Capp (B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

(9.11)

452

Chapter 8. Shape Continuity and Optimization

From L. C. Evans and R. F. Gariepy [1, Thm. 2, p. 151] since y → x + Ax y is an affine isometry and for all ρ > 0, Ax B(x, ρ) = B(x, ρ), Capp ([x + Ax O] ∩ B(x, r ), B(x, 2r )) Capp (O ∩ B(0, r  ), B(0, 2r )) = ≥ c (9.12) Capp (B(x, r ), B(x, 2r )) Capp (B(0, r ), B(0, 2r )) ⇒ ∀x ∈ ∂Ω, ∀r  , 0 < r < r,

Capp (Ω ∩ B(x, r  ), B(x, 2r )) ≥ c, Capp (B(x, r ), B(x, 2r ))

(9.13)

and Ω satisfies the strong (r, c)-capacity density condition. (ii) From part (i) and Theorem 8.1. Theorem 9.3. Let D be a bounded open subset of RN and c > 0 and r > 0 be constants. The subset {d
Ω : Ω ∈ Lc,r (O, D)}

(9.14)

of Cdc (D) is compact for the uniform topology of C(D). Proof. Consider an arbitrary sequence {d
Ωn }, Ωn ∈ Lc,r (O, D). By compactness of Cdc (D), there exist an open subset Ω0 of D and a subsequence of {d
Ωn }, still denoted by {d
Ωn }, that converges to d
Ω0 in Cdc (D). If ∂Ω0 = ∅, then Ω ∈ Lc,r (O, D). If ∂Ω0 = ∅, then, by Lemma 9.1 (ii), for each x ∈ ∂Ω0 , there exist a subsequence {Ωnk } of {Ωn } and a sequence {xk }, xk ∈ ∂Ωnk , that converges to x. Since {Ωnk } ⊂ Lc,r (O, D), by Definition 9.1 of Lc,r (O, D), for each k ∃Ak ∈ O(N)

such that xk + Ak O ⊂ Ωnk

⇒ dxk +Ak O ≥ d
Ωn . k

But since Ak ∈ O(N), ∗ dxk +Ak O (y) = dO (A−1 k (y − xk )) = dO ( Ak (y − xk )).

There exist a subsequence of {Ak }, still denoted by {Ak }, and A ∈ O(N) such that −1 (y − x)) = dx+AO (y). Ak → A and since xk → x, dO (A−1 k (y − xk )) → dO (A Finally, since d
Ωn → d
Ω , k

dxk +Ak O ≥ d
Ωn

⇒ dx+AO ≥ d
Ω

k

⇒ ∀x ∈ ∂Ω, x + AO ⊂ Ω

and Ω ∈ Lc,r (O, D).

9.3

Maximizing the Eigenvalue λA (Ω)

A first example of extremal domain follows directly from Theorem 9.1. Theorem 9.4. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). Given an open E, ∅ = E ⊂ D, consider the compact c (E; D) ∩ Oc,r (D), where families Cdc (E; D) ∩ Oc,r (D) and Cd,loc def

Cdc (E; D) = {d
Ω : E ⊂ Ω open ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open ⊂ D N

d
Ω :

&

(9.15) (9.16)

9. Compact Families Oc,r (D) and Lc,r (O, D)

453

are the compact families defined in Theorem 2.6 (i) of Chapter 6. Then the maximization problems

 sup λA (Ω) : ∀Ω ∈ Oc,r (D) such that E ⊂ Ω ⊂ D , (9.17) & ∃x ∈ RN , ∃A ∈ O(N) such that sup λA (Ω) : ∀Ω ∈ Oc,r (D) such that (9.18) x + AE ⊂ Ω open ⊂ D have solutions.4 c Proof. From Theorem 2.6 (i) in Chapter 6, Cdc (E; D) and Cd,loc (E; D) are comA pact in C(D). Since sets of that family are nonempty, λ (Ω) < +∞ and, from Theorem 2.2, the function d
Ω → λA (Ω) : Cdc (D) ⊂ C0 (D) → R ∪{+∞} is upper semicontinuous and the function d
Ω → λA (Ω) : Cdc (E; D) ∩ Oc,r (D) → R is upper semicontinuous. So there exist maximizers in the compact set Cdc (E; D) ∩ Oc,r (D). c (E; D) ∩ Oc,r (D). The same applies for Cd,loc

9.4

State Constrained Minimization Problems

We now give the following general existence theorem. Theorem 9.5. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let uΩ be the solution of (4.5) for Ω in Oc,r (D). If h is continuously defined from H01 (D) into R, then J(Ω) = h(eΩ (uΩ )) is continuously defined from Oc,r (D) into R and reaches its extremal values on that set. Example 9.1. Consider the following shape function for which we can get the existence of minimizing domains even if the assumptions of Theorem 9.5 on h are not satisfied. Given  α > 0, 1 def 1 , (9.19) J(Ω) = |∇uΩ − z¯|2 dx + α  2 Ω |uΩ |2 dx Ω where uΩ is the solution of (4.5) and z¯ ∈ L2 (D; RN ). From Theorem 2.3 in Chapdef ter 2, u = eΩ (uΩ ) and ∇uΩ are zero almost everywhere in D\Ω, and the function can be rewritten as   1 1 1 J(Ω) = − |∇u − z¯|2 dx + α  |¯ z |2 dx. (9.20) 2 dx 2 D 2 |u| D\Ω D The last term is not of the form h(uΩ ) for some h, but is simply of the form h(Ω). It turns out that J is not continuous but only lower semicontinuous for the H c topology from Theorem 4.1 of Chapter 6 applied to the last term. Hence it can be minimized. The minimization problem of Example 9.1 can now be formulated as follows on D. Let D be a bounded open nonempty domain in RN , f be an element of 4 In

some sense the maximizing solution is an Oc,r (D)-approximation of the set E.

454

Chapter 8. Shape Continuity and Optimization

H −1 (D), and B be a ball containing D. Let µ be a positive measure on D and a be a positive constant such that 0 < a < µ(D) < +∞. Let uΩ be the solution of (4.5), and let u = eΩ (uΩ ), its extension by zero in H01 (D). For J(Ω) defined in (9.19) consider the following problem: min{J(Ω) : Ω ∈ Oc,r (B), Ω ⊂ D, µ(Ω) ≤ a}.

(9.21)

Theorem 9.6. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. For any constants α > 0, c > 0 and r, 0 < r < 1, problem (9.21) has at least one solution. Proof. It is sufficient to notice that the first term in (9.20) is continuous under the assumptions of Theorem 9.5. The second term is lower semicontinuous from Theorem 4.1 of Chapter 6 (with the measure of density |¯ z |2 , that is, dµ = z¯|2 dz). Hc

To complete the proof, recall that if Ωn ⊂ D and Ωn −→ Ω, then Ω ⊂ D. As α > 0, a minimizing sequence cannot converge to ∅. In fact, for any admissible domain Ω0 and any optimal solution uΩ0 ,  α . |uΩ0 |2 dx ≥ J(Ω 0) Ω

9.5

Examples with a Constraint on the Gradient

Let D be a fixed bounded smooth open domain in RN , let f in H −1 (D), and let g in L2 (D). For any open subset Ω of D, let uΩ be the solution of the Dirichlet problem (4.5) in H01 (Ω). Given α > 0, M > 0, and an open subset E of D, consider the following minimization problem: inf {J(Ω) : Ω open , E ⊂ Ω ⊂ D, ess sup |∇uΩ | ≤ M } ,  def |uΩ − g|2 dx. J(Ω) = m(Ω) + α

(9.22)

Ω ∞

It is understood that if |∇u| is not in L (D), then the ess sup |∇u| is +∞. In a first step it is easy to check that the problem   inf m(Ω) + α |uΩ − g|2 dx Ω  (9.23) : ess sup |∇uΩ | ≤ M, E ⊂ Ω, Ω ∈ Oc,r (D) has minimizing solutions. Let {Ωn } be a minimizing sequence. By assumption there exist Ω and a subsequence, still indexed by n, such that Ωn → Ω in the H c -topology (d
Ωn → d
Ω in C(D)) and ess sup |∇uΩn | ≤ M. Then |∇un | converges in L2 (D) to |∇u| from the previous sections. For any ϕ ∈ L2+ (D) we have  (|∇un | − M ) ϕ dx ≤ 0, D

9. Compact Families Oc,r (D) and Lc,r (O, D)

455

and then, in the limit, we get the same inequality with u. Then, the function being lower semicontinuous with respect to that topology, we get the existence. Lemma 9.3. For any f ∈ H −1 (D) and any r > 0 and c > 0, problem (9.23) has solutions in Oc,r (D) In fact, the presence of the constraint on the gradient of u is helpful for getting the continuity of the application Ω → u = eΩ (uΩ ), and we directly get the existence of solutions to the original problem (9.22). In the proof of Theorem 8.1 we only needed the equicontinuity of the family of solutions {un = eΩn (uΩn )} and the fact that if x ∈ Ω, then u(x) = 0 for a quasi-continuous representative. If Ω ∈ Oc,r (D), those two assumptions are readily satisfied. Notice that the boundedness of the gradients in (9.22) implies the equicontinuity of any minimizing sequence {un }. So, in order to obtain a continuity result with constraints on the boundedness of the gradient, we only have to notice that if u ∈ H01 (Ω), ess sup |∇u| ≤ M . Then u ∈ W 1,∞ (Ω) and u is Lipschitzian with the constant M , or more exactly, there exists a Lipschitzian function almost everywhere equal to u, which is also a quasi-continuous representative of u. To obtain that u(x) = 0 in any point of the complement of Ω, we have to introduce the following capacity constraints on Ω: we require that Ω be capacity extended (see D. Bucur [1]), i.e., that Ω = Ω∗ , where  def

Ω∗ = x ∈ RN : ∃εx > 0, such that capD (B(x, εx ) ∩ Ω) = 0 .

(9.24)

Indeed, let u be continuous on D, u = 0 quasi-everywhere on Ω and Ω = Ω∗ . Then for all x ∈ Ω, for all ε > 0 we have capD (B(x, εx ) ∩ Ω) > 0 and there exists a point xε in B(x, εx ) ∩ Ω, where u(xε ) = 0. By continuity we get u(x) = 0. We recall from D. Bucur [1] the main properties of the capacity extension. Proposition 9.1. For any open Ω ⊂ D, the set Ω∗ is open, Ω ⊂ Ω∗ ,

capD (Ω∗ \Ω) = 0,

and

(Ω∗ )∗ = Ω∗ .

As capD (Ω∗ \Ω) = 0 we get eΩ (uΩ ) = e∗Ω (uΩ∗ ) and any minimizing sequence can be made up only of capacity-extended domains. Theorem 9.7. Given f, g ∈ L2 (D), α > 0, and M > 0, problem (9.22) has minimizing solutions. Proof. From the boundedness of the gradient we obtain that {uΩn } are uniformly Lipschitz continuous, and we can apply the same arguments as in Theorem 8.1, avoiding the capacity conditions. Consider the penalized version of problem (9.22). Given M > 0 and β > 0, define def (9.25) Jβ (Ω) = J(Ω) + β sup ess (|∇uΩ | − M )+ D

and consider the existence of solutions to the following problem: inf {Jβ (Ω) : E ⊂ Ω open ⊂ D} .

(9.26)

456

Chapter 8. Shape Continuity and Optimization

Theorem 9.8. Given f and g in L2 (D), α > 0, β > 0, and M ≥ 0, problem (9.26) has minimizing solutions. Proof. Let {Ωn } be a minimizing sequence for the function Jβ , Ωn = Ω∗n . We have

 ess sup |∇uΩn | ≤ max M, β −1 J(Ω1 ) = M  for all n ≥ 1. Then, from the previous considerations we have the strong convergence in H01 (D) of un = eΩn (uΩn ) to u = eΩ (uΩ ) for some subsequence {Ωn } that converges to Ω in the H c -topology. Hence Ω is a minimizing domain. To complete the proof note that the map Ω → sup ess (|∇u| − M )+ D

is not lower semicontinuous from the H c -topology to R. Nevertheless, if {Ωn } is a minimizing sequence for problem (9.26), the required property is satisfied. We can assume that the sequence is chosen such that   sup ess |∇un | → c = lim inf sup ess |∇un | . n→∞

D

D

For any ε > 0 there exists nε > 0 such that for n ≥ nε , supD ess |∇uΩn | ≤ c + ε. Because of the H01 (D)-strong convergence of un to u, we get sup ess |∇u| ≤ c + ε D   ⇒ sup ess |∇u| ≤ lim inf sup ess |∇u| . ∀ε > 0,

D

D

Remark 9.1. We can change the L∞ -norm for a differentiable one; that is, we need ∇u1,p ≤ M,

∀p > N.

By the continuous inclusion W 1,p (D) ⊂ W ε,∞ (D), ε small, the result still holds.

Chapter 9

Shape and Tangential Differential Calculuses 1

Introduction

In Chapters 3, 5, 6, and 7, we have constructed nonlinear and nonconvex complete metric spaces of geometries. The spaces F(Θ) and X(D) have group structures. For the groups F(Θ), Θ ⊂ C 0,1 (RN , RN ), it is possible to construct C 1 -paths in F(Θ) and we have shown that the tangent space is Θ in all points of F(Θ) in Chapter 4. It is more difficult to fully characterize tangent spaces for the spaces of characteristic functions or distance functions. Yet, paths constructed from velocity fields can also be used to obtain C 1 -paths in those spaces. In the absence of sharper results, we shall concentrate on the notion of semiderivatives or derivatives of a shape functional (cf. section 3 of Chapter 4) with respect to a velocity field. This point of view was used in Chapter 4 to give an equivalent characterization of the continuity of a shape functional with respect to Courant metrics in terms of the continuity along continuous paths generated by velocities. Moreover the velocity approach readily extends to the constrained case. As we pointed out in section 3.2 of Chapter 4, two types of semiderivatives are of interest: the weaker Gateaux style semiderivative (3.6) and the stronger Hadamard style semiderivative of Definition 3.2 for which the chain rule is available. This chapter has been structured along those general directions. In section 2 we give a self-contained review of semiderivatives and derivatives in topological vector spaces in order to prepare the ground for shape derivatives. Section 3 gives the definitions and the main properties of first-order shape semiderivatives and derivatives of shape functionals (Eulerian, Hadamard, Gateaux, and Fr´echet). For the Hadamard semidifferentiability we get the Courant metric continuity of the shape functional as for Banach spaces. A complete structure theorem is given for the shape gradient in section 3.4. Before going to second-order derivatives, the main elements of the shape calculus are introduced in section 4 and the basic formulae for domain and boundary integrals are given in section 4.1 and section 4.2. Their application is illustrated in a series of examples in section 4.3. 457

458

Chapter 9. Shape and Tangential Differential Calculuses

The final expressions of the gradients in the examples of section 4 always lead to a domain and a boundary expression. The boundary expression contains fundamental properties of the gradients, and the natural way to untangle some of the resulting terms is to use the tangential calculus. Section 5 gives the main elements of that calculus for a C 2 -submanifold of RN of codimension 1, including Stokes’s and Green’s formulae in section 5.5 and the relationship between tangential and covariant derivatives in section 5.6. This is applied to the derivative of the integral of the square of the normal derivative of section 4.3.3. Section 6 extends definitions and structure theorems to second-order derivatives. In order to develop a better feeling for the abstract definitions the secondorder derivative of the domain integral is computed in section 6.1 using the combined strengths of the shape and tangential calculuses. A basic formula for the secondorder semiderivative of the domain integral is given in section 6.2. This is completed with structure theorems in the nonautonomous case in section 6.3 and the autonomous case in section 6.4. The shape Hessian is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket in section 6.5. This symmetrical part is itself decomposed into a symmetrical part that depends only on the normal component of the velocity field and a symmetrical term made up of the gradient acting on a generic group of terms that occurs in all examples considered in section 6.

2

Review of Differentiation in Topological Vector Spaces

In this section we review some elements of semiderivatives and derivatives in topological vector spaces that will be useful for defining shape semiderivatives and derivatives. In that context we begin with the weaker notion of Gateaux semiderivative. Its simplicity makes it useful for computing a semiderivative. Yet, the chain rule for the derivative of the composition of such functions does not hold and the function itself need not be continuous. The more interesting notion is the stronger one of Hadamard semiderivative that makes the function continuous and builds the chain rule in its definition. It also readily extends to the tangential semiderivative on smooth submanifolds. In the Euclidean space the Hadamard derivative coincides with the total differential or Fr´echet derivative, but it is both weaker and more general in infinite-dimensional vector spaces where the existence of a metric on the space is not required. The chain rule is the central ingredient of a good differential or semidifferentiable calculus.

2.1

Definitions of Semiderivatives and Derivatives

The notion of semiderivative corresponds to the intuitive notion of differential that was formalized by Newton and Leibniz, who also introduced the notation in 1684. Definition 2.1. Let f be a real-valued function defined in a neighborhood of a point x of a topological vector space E.

2. Review of Differentiation in Topological Vector Spaces

459

(i) We say that f has a Gateaux semiderivative 1 at a point x ∈ U in the direction v ∈ E if the following limit exists: f (x + εv) − f (x) ; ε0 ε lim

(2.1)

when it exists it will be denoted by df (x; v). (ii) We say that f has a Hadamard semiderivative 2 at x ∈ U in the direction v ∈ E if the following limit exists: lim

ε0 w→v

f (x + εw) − f (x) ; ε

(2.2)

when it exists it will be denoted by dH f (x; v). The above definitions extend from a real-valued function to a function f into another topological vector space F . Remark 2.1. Our definition of Hadamard semiderivative is simple and general, but it somewhat hides the original definition of J. Hadamard [2] that builds into the definition the chain rule for the derivative of the composition of functions and the continuity of the function, as we shall see in Theorem 2.5. It is readily seen that Definition 2.1 (i) is equivalent to lim

ε0 ∀ϕ:[0,τ ]→E such that + ϕ(0 )=x, ϕ (0+ )=v

f (ϕ(ε)) − f (x) , ε

(2.3)

where the limit is independent of the choice of path ϕ : [0, τ ] → E such that limε0 ϕ(ε) = x and limε0 (ϕ(ε) − ϕ(0))/ε = v. Since we are using paths, this definition extends to paths on manifolds M with v in the tangent space to M in x. It is clear that if dH f (x; v) exists, the semiderivative df (x; v) exists and is equal to dH f (x; v), but the converse is not true without additional assumptions, as 1 In his original work (published after his death in 1914) R. Gateaux [2] introduces as differential the directional derivative (ε goes to 0 not only by positive values but also by negative values yielding the property df (x − v) = −df (x; v)), but he does not a priori assume that this differential is linear with respect to the direction. So he cannot get the differential of the norm |x|, but his definition of differential is very close to our version. 2 In 1923 J. Hadamard [2] introduces a notion of differential that builds in the chain rule for ´chet [1] shows in 1937 that “the definition of the total the composition of functions. M. Fre differential of Stolz-Young is equivalent to the definition of J. Hadamard [2] (in finite dimension). However, when the latter is extended to functionals (functions of functions - infinite dimensional spaces), it is more general than the one of the author (M. Fr´echet) and necessarily verifies the chain ´chet [1, rule for the differentiation of the composition of functions.” In the same paper M. Fre p. 239] weakens the notion of differential by dropping the linearity with respect to the direction. Again this amounts to letting ε go to zero by positive and negative values in our definition. As in the case of Gateaux his definition is very close to our semiderivative version of the derivative of Hadamard that can be found in J.-P. Penot [1, p. 250] in 1978 and earlier in A. Bastiani [1, Def. 3.1, p. 17] in 1964 as a seemingly well-known notion.

460

Chapter 9. Shape and Tangential Differential Calculuses

can be seen from the following example. The Gateaux semiderivative is generally neither linear nor continuous with respect to the direction. Example 2.1. Consider the following function f : R2 → R:  x5  , f (x, y) = (y − x2 )2 + x8  0,

if (x, y) = (0, 0),

(2.4)

if (x, y) = (0, 0).

By definition, df ((0, 0); (0, 0)) is always 0, but here dH f ((0, 0); (0, 0)) does not exist. Choose the sequences   1 1 1 → (0, 0) as n → +∞ tn =  0 and wn = , n n n3 and consider the quotient def

qn =

f ((0, 0) + tn wn ) − f (0, 0) . tn

As n goes to infinity, qn =

( n12 )5 1 1 8 n ( n2 )

= n4 → +∞

and dH f ((0, 0); (0, 0)) does not exist. The function f is Gateaux semidifferentiable at (0, 0) in all directions v = (v1 , v2 ) in R2 and df (0, 0); (v1 , v2 )) =

v1 , if v2 = 0, 0, if v2 = 0,

(2.5)

but it is neither linear nor continuous with respect to v (vn = (1, 1/n) → (1, 0)). The Hadamard semiderivative is also generally not linear in v. Example 2.2. Let E be a normed vector space. The Hadamard semiderivative of the norm f (x) = |x|E at x = 0 is given by ∀v ∈ E,

dH f (0; v) = |v|E ,

(2.6)

which is continuous but not linear in v. We shall also need the following notions of (full) derivatives. Definition 2.2. Let f be a real-valued function defined in a neighborhood of a point x of a topological vector space E. (i) The function f has a Gateaux derivative at x if ∀v ∈ E, df (x; v) exists and v → df (x; v) : E → R is linear and continuous.

(2.7)

Whenever it exists the linear map (2.7) will be denoted by ∇f : E → E  .

2. Review of Differentiation in Topological Vector Spaces

461

(ii) The function f has a Hadamard derivative at x if ∀v ∈ E,

def

dH f (x; v) = lim

t0 w→v

f (x + tw) − f (x) exists and t

v → dH f (x; v) : E → R is linear and continuous. The above definitions extend from a real-valued function to a function f into a normed vector space F . The next example is a Gateaux differentiable function in 0 that is not continuous in 0 and not Hadamard differentiable in 0. Example 2.3. Consider the function f : R2 → R:  x6  , def f (x, y) = (y − x2 )2 + x8  0,

if (x, y) = (0, 0),

(2.8)

if (x, y) = (0, 0).

It is readily seen that f has a Gateaux semiderivative at (0, 0) in all directions,   (2.9) ∀v ∈ R2 , df (0, 0); v = 0,   and that v → df (0, 0); v = 0 is trivially linear and continuous with respect to v. But it is not Hadamard semidifferentiable in (0, 0) in the direction (0, 0) by the same argument as in Example 2.1. Similarly, for ε > 0 by choosing the directions w(ε) = (1, ε) → v = (1, 0) as ε → 0,     f εw(ε) − f (0, 0) = ε and dH f ((0, 0); (1, 0)) does not exist. Finally, f this follow the path (ε, ε2 ) as ε goes to 0: 6 f (ε, ε2 ) − f (0, 0) = ε = 1 ε8 ε2 then

2.2

1 → +∞ ε3 is not continuous in (0, 0). To see

→ +∞ as ε → 0.

Derivatives in Normed Vector Spaces

We now specialize to normed vector spaces. In a normed vector space E, we have the strong (norm) topology and the weak topology. As a result we have two notions of Hadamard semiderivatives, def

dsH f (x; v) =

def

dw H f (x; v) =

lim

f (x + tw) − f (x) , t

(2.10)

lim

f (x + tw) − f (x) , t

(2.11)

t0 w→v in E-strong

t0 wv in E-weak

462

Chapter 9. Shape and Tangential Differential Calculuses

and two notions of derivatives, where w → v denotes the strong convergence and w  v denotes the weak convergence. By definition, the weak Hadamard semiderivative is a stronger condition than the strong Hadamard semiderivative: dw H f (x; v) exists

⇒

dsH f (x; v) exists.

(2.12)

The two notions coincide when E is finite-dimensional. Definition 2.3. Let f be a real-valued function defined in a neighborhood of a point x of a normed vector space E. (i) The function f has a strong Hadamard derivative in x if def

∀v ∈ E,

dsH f (x; v) =

lim

t0 w→v in E-strong

f (x + tw) − f (x) exists and t

v → dsH f (x; v) : E → R is linear and continuous. (ii) The function f has a weak Hadamard derivative in x if ∀v ∈ E,

def

dw H f (x; v) =

lim

t0 wv in E-weak

f (x + tw) − f (x) exists and t

v → dw H f (x; v) : E → R is linear and continuous. (iii) The function f has a Fr´echet derivative in x if there exists a continuous linear mapping L(x) : E → R such that lim

|h|E →0

f (x + h) − f (x) − L(x)h → 0. |h|E

The above definitions extend from a real-valued function to a function f into another normed vector space F . If f has a Fr´echet derivative in x, then it has a Gateaux derivative at x. Indeed for all v = 0 and t  0, h = tv → 0 in norm, and f (x + tv) − f (x) − L(x)tv f (x + tv) − f (x) →0 − L(x)v = |v| t |tv|E ⇒ df (x; v) = L(x)v and v → df (x; v) : E → R is linear and continuous. By definition ∇f (x), vE = L(x)v, where ·, ·E denotes the duality pairing between E  and E. The following theorem makes explicit the connections and equivalences between the three notions of derivatives. Theorem 2.1. Let E be a normed vector space and f : V (x) → R be a function defined in a neighborhood of a point x ∈ E.

2. Review of Differentiation in Topological Vector Spaces

463

(i) If f has a Fr´echet derivative in x, then f has a weak Hadamard derivative in x and dw H f (x; v) = ∇f (x), vE = L(x)v. If f has a weak Hadamard derivative in x, then f has a strong Hadamard derivative in x. If f has a strong Hadamard derivative in x, then f has a Gateaux derivative in x. (ii) If E is a reflexive Banach space, f has a weak Hadamard derivative in x if and only if f has a Fr´echet derivative in x. For a function f , the four notions are well ordered: Fr´echet derivative ⇒ weak Hadamard derivative ⇒ strong Hadamard derivative and the last one implies that f has a Gateaux derivative in x. In view of Theorem 2.1, the first three notions are equivalent in finite dimension. Theorem 2.2. When E is finite-dimensional, the strong Hadamard derivative, the weak Hadamard derivative, and the Fr´echet derivative of Definition 2.3 are equivalent and equal. Proof of Theorem 2.1. (i) Given t > 0 and w ∈ E, consider the quotient def

q(t, w) =

f (x + tw) − f (x) t

and for h ∈ E the quotient   f (x + h) − f (x) − L(x)h , def |h|E Q(h) =  0,

h = 0, h = 0.

Then q(t, w) = Q(h(t, w)) |w|E + L(x)w. def

Given v ∈ E, as w  v and t  0, h(t, w) = tw → 0 in the E-norm. Hence Q(h(t, w)) → 0 since f has a Fr´echet derivative in x and L(x)w → L(x)v as w  v by continuity. Therefore lim q(t, w) = L(x)v

wv t0

w and dw H f (x; v) exists and v → dH f (x; v) = L(x)v : E → R is linear and continuous with respect to v. The second statement follows from (2.12) and the last one by definition of the Gateaux and Hadamard derivatives. (ii) Define the element L(x) of E  as

∀v ∈ E,

def

L(x)v = ∇f (x), vE = dw H f (x; v).

464

Chapter 9. Shape and Tangential Differential Calculuses

Denote by Q the limsup of |Q(h)| as |h| → 0, which can possibly be infinite. Let {hn } be a sequence in E such that |hn | → 0 and |Q(hn )| → Q. For h = 0   h . Q(h) = Q |h| |h| Since E is a reflexive Banach space there exist a subsequence of {hn }, still denoted by {hn }, and v ∈ E such that def

wn =

hn v∈E |hn |

and |Q(hn )| → Q.

By letting tn = |hn |, f (x + tn wn ) − f (x) − ∇f (x), wn E tn = q(tn , wn ) − ∇f (x), wn E → dw H f (x; v) − ∇f (x), vE = 0.

Q(hn ) =

Therefore Q = 0 and f is Fr´echet differentiable in x. We complete this section with a classical sufficient condition for the Fr´echet differentiability. Theorem 2.3. Given a normed vector space E and a map f : V (x) → R, defined in a neighborhood V (x) of x ∈ E, assume that (i) for all y ∈ V (x), f has a Gateaux derivative ∇f (y), and (ii) the map ∇f : V (x) → E  -strong is continuous in x. Then f has a Fr´echet derivative in x. Proof. There exists ρ > 0 such that the open ball B(x, ρ) is contained in V (x). For h ∈ E, 0 < |h| < ρ, consider the quotient def

Q(h) =

f (x + h) − f (x) − ∇f (x), hE . |h|E

There exists α(h), 0 < α(h) < 1, such that f (x + h) − f (x) = ∇f (x + α(h)h), hE < ; h ⇒ Q(h) = ∇f (x + α(h)h) − ∇f (x), |h|E E ⇒ |Q(h)| ≤ |∇f (x + α(h)h) − ∇f (x)|E  → 0 as h → 0. This shows that f is Fr´echet differentiable in x.

2. Review of Differentiation in Topological Vector Spaces

2.3

465

Locally Lipschitz Functions

The following theorem gives a sufficient condition for the equivalence of Gateaux and Hadamard semiderivatives (resp., Gateaux and Hadamard derivatives). Theorem 2.4. Let E be a normed vector space and f : E → R be a function that is uniformly Lipschitzian in a neighborhood U of x in E, that is, ∃c(x) > 0, ∀y, z ∈ U,

|f (y) − f (z)| ≤ c(x) |y − z|E .

(2.13)

(i) If df (x; v) exists, then dsH f (x; v) exists and dsH f (x; v) = df (x; v). Moreover, if df (x; v) exists for all v ∈ E, then ∀v2 , v1 ∈ E,

|dsH f (x; v2 ) − dsH f (x; v1 )| ≤ c |v2 − v1 |E .

(ii) If f has a Gateaux derivative at x, then ∀v, dsH f (x; v) exists and v→ dsH f (x; v) : E → R is linear and continuous and f has a strong Hadamard derivative in x. Proof. (i) There exist ε > 0 and a neighborhood W of v in E such that ∀w ∈ W, ∀ε, 0 < ε ≤ ε,

x + εw ∈ U and x + εv ∈ U.

Then  1  1  1 f (x + εw) − f (x) = f (x + εw) − f (x + εv) + f (x + εv) − f (x) ε ε ε and

1  f (x + εw) − f (x) − df (x; v) ε 1  ≤ f (x + εv) − f (x) − df (x; v) + c(x)|w − v|E . ε

So as ε → 0 and w → v, dsH f (x; v) = df (x; v). (ii) By definition, from part (i).

2.4

Chain Rule for Semiderivatives

The family of Hadamard semidifferentiable functions is extremely interesting since this property is stable under composition. They form a sufficiently large family of nondifferentiable functions that include continuous convex functions as we shall see in section 2.5. Theorem 2.5. Let E and F be two topological vector spaces and let h be the composition of two mappings f and g   h(x) = f g(x) (2.14)

466

Chapter 9. Shape and Tangential Differential Calculuses

in a neighborhood U of a point x in E, where g:U ⊂E→F

and

f : g(U ) → R .

(2.15)

Assume that (i) g has a Gateaux (resp., Hadamard) semiderivative at x in the direction v, and (ii) dH f (g(x); dg(x; v)) exists, where the Hadamard semiderivative is taken in the general sense of Definition 2.1 (ii) with respect to the topologies of E and F .3 Then dh(x; v) = dH f (g(x); dg(x; v))

 resp., dH h(x; v) = dH f (g(x); dH g(x; v)) .



(2.16)

Proof. (a) For ε > 0 small enough let m(ε) =

g(x + εv) − g(x) − dg(x; v) in F. ε

By assumption m(ε) → 0 as ε → 0. By the definition of dh(x; v) we want to find the limit of the differential quotient     f g(x + εv) − f g(x) , d(ε) = ε which can be rewritten as      f g(x) + ε dg(x; v) + m(ε) − f g(x) d(ε) = , ε where dg(x; v) + m(ε) → dg(x; v) as ε → 0. So by the definition of dH f ,   lim d(ε) = dH f g(x); dg(x; v) .

ε0

(b) When g is Hadamard semidifferentiable we replace m(ε) and d(ε) by m(ε, w) =

g(x + εw) − g(x) − dg(x; v) ε

and

   / 1.  f g(x) + ε dg(x; v) + m(ε, w) − f g(x) ε and proceed as in part (a). d(ε, w) =

3 For normed vector spaces this theorem gives two sets of results: one for the strong topology and one for the weak topology.

2. Review of Differentiation in Topological Vector Spaces

467

In general we cannot improve the semiderivative of h by improving the semiderivative of g when f is not Hadamard semidifferentiable. Example 2.4. Consider the composition of the function f : R2 → R in Example 2.3 and the map g : R → R2 ,

g(x) = (x, x2 ).

(2.17)

The map f is Gateaux, but not Hadamard semidifferentiable, and the map g is infinitely differentiable. However, the composition   (2.18) h(x) = f g(x) = 1/x2 is not even Gateaux semidifferentiable in 0. We reiterate that the Hadamard semidifferentiability is a key property for the “chain rule.” In general the composition h = f ◦ g of two maps will fail to have a semiderivative unless f has a Hadamard semiderivative, even if the map g : E → F is Fr´echet differentiable at the point x.

2.5

Semiderivatives of Convex Functions

Finally the class of functions that are Hadamard semidifferentiable is quite large since it contains the classical continuously differentiable functions and the convex continuous functions. Theorem 2.6. Let f : U ⊂ E → R be a function defined in an open convex subset of a topological vector space E. (i) f is convex in U if and only if ∀x ∈ U, ∀v ∈ E, df (x; v) exists, ∀x ∈ U, ∀v ∈ E, df (x; v) + df (x; −v) ≥ 0, ∀x, y ∈ U, f (y) ≥ f (x) + df (x; y − x).

(2.19) (2.20) (2.21)

(ii) If E is a normed vector space and if f is convex in U and continuous in x ∈ U , there exists a neighborhood V (x) of x such that ∀y ∈ V (x), ∀v ∈ E,

dsH f (y; v) exists.

(2.22)

Remark 2.2. def In a normed vector space E, the norm nE (x) = |x|E is a convex and Lipschitzian function in E. Hence for all x, v ∈ E, dH nE (x; v) exists and dH nE (0; v) = |v|E . Proof. Part (ii) is a consequence of part (i) and the fact that a continuous convex function at x is locally Lipschitzian in a neighborhood of x (cf. I. Ekeland and R. Temam, [1, pp. 11–12]) by direct application of Theorem 2.4. We give the proof

468

Chapter 9. Shape and Tangential Differential Calculuses

of part (i) for completeness. Assume that f is convex in U . Let θ ∈ ]0, 1]. Given x ∈ U and v ∈ E ∃α, 0 < α < 1, such that x − αv ∈ U, ∃θ0 , 0 < θ0 < 1, such that ∀θ ∈]0, θ0 ], x + θv ∈ U. For this fixed α, we show that f (x) − f (x − αv) f (x + θv) − f (x) ≤ . (2.23) α θ This follows from the identity θ α (x + θv) + (x − αv) x= α+θ α+θ and the convexity of f , α θ f (x) ≤ f (x + θv) + f (x − αv). α+θ α+θ This can be rewritten as   α θ f (x) − f (x − αv) ≤ f (x + θv) − f (x) θ+α θ+α and yields (2.23). Define f (x + θv) − f (x) ϕ(θ) = , 0 < θ < θ0 . θ Now we show that ϕ is a monotone increasing function of θ > 0. For all θ1 and θ2 , 0 0 < θ 1 < θ2 < θ0 :  1  θ1 θ1 x − f (x) f (x + θ1 v) − f (x) = f (x + θ2 v) + 1 − θ2 θ2   θ1 θ1 ≤ f (x + θ2 v) + 1 − f (x) − f (x), θ2 θ2 ∀θ ∈ ]0, θ0 [ ,

which implies that ϕ(θ1 ) ≤ ϕ(θ2 ). As ϕ is a monotone increasing function which is bounded below, its limit exists as θ goes to 0. By definition it is equal to df (x; v). Now that we know that df (x; v) exists in all directions v, go back to inequality (2.23) and let α go to zero to get (2.20). Conversely, apply inequality (2.19) twice to x and x + θ(y − x) and to y and x + θ(y − x) with θ ∈ [0, 1]: f (x) ≥ f (x + θ(y − x)) + df (x + θ(y − x); −θ(y − x)), f (y) ≥ f (x + θ(y − x)) + df (x + θ(y − x); (1 − θ)(y − x)). Multiply the first inequality by 1 − θ and the second by θ and add up each side: (1 − θ)f (x) + θf (y) ≥ f (x + θ(y − x)) + (1 − θ)θdf (x + θ(y − x); −(y − x)) + θ(1 − θ)df (x + θ(y − x); y − x). By using property (2.20), we get f (θy + (1 − θ)x) ≤ θf (y) + (1 − θ)f (x) and the convexity of f in U .

2. Review of Differentiation in Topological Vector Spaces

2.6

469

Hadamard Semiderivative and Velocity Method

In section 3 of Chapter 4 we have drawn an analogy between Gateaux and Hadamard semiderivatives on one hand and shape semiderivatives obtained by a perturbation of the identity and the velocity method on the other hand. The next theorem relates the Hadamard semiderivative and the semiderivative obtained by the velocity method for real functions defined on RN . Theorem 2.7. Let f : NX → R be a real function defined in a neighborhood NX of a point X in RN . Then f is Hadamard semidifferentiable at (X, v) if and only if there exists τ > 0 such that for all velocity fields V : [0, τ ] → R satisfying the following assumptions:   (a) (V1 ) ∀x ∈ RN , V (·, x) ∈ C [0, τ ]; RN ; (b) (V2 ) ∃c > 0, ∀x, y ∈ RN ,

V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|;

(c) the limit def

df (X; v) =

lim

∀V, V (0)=v t0

  f Tt (V )(X) − f (X) t

(2.24)

exists and is independent of the choice of V satisfying (a) and (b), where def

Tt (V )(X) = x(t; X),   dx (t; X) = V t, x(t; X) , 0 < t < τ, dt

x(0; X) = X.

Proof. (⇒) Let V be a vector field satisfying conditions (a), (b), and (c). Define def

w(t) =

1 t



t

  V s, x(s) ds,

0 < t ≤ τ.

0

It is continuous on ]0, τ ] and 1 w(t) − v = t

 t. /   V s, x(s) − V (0, X) ds. 0

Therefore |w(t) − v| ≤ c max |x(s) − X| + max |V (s, X) − V (0, X)| [0,t]

[0,t]

⇒ lim w(t) = v. t0

So

    f x(t; X) − f (X) f X + tw(t) − f (X) = lim = dH f (X; v), lim t0 t t w(t)→v t0

470

Chapter 9. Shape and Tangential Differential Calculuses

since f is Hadamard semidifferentiable at (X, v) and the limit depends only on V (0) = v. (⇐) Conversely saying that the dH f (x; v) exists is equivalent to saying that for the sequence tn = 2−n and any sequence wn → v the limit f (x + tn wn ) − f (x) tn exists and depends only on v. We now associate with the sequence {wn } a velocity field V satisfying (a) to (c) such that V (tn , X) = wn and Ttn (V )(X) = X + tn wn . Then from properties (2.24) and V (o) = v we conclude that dH f (x; v) exists. Set t0 = 1, tn = 2−n , and xn = X + tn wn and observe that tn − tn+1 = −2−(n+1) . Define for m ≥ 1 the following C m -interpolation in (0, 1]: for t in [tn+1 , tn ]  t − tn (xn+1 − xn ) Tt (X) = xn + p tn+1 − tn   t − tn (tn+1 − tn )wn + q0 tn+1 − tn   t − tn (tn+1 − tn )wn+1 , + q1 tn+1 − tn 

def

def

T0 (X) = X,

where p, q0 , q1 ∈ P 2m+1 [0, 1] are polynomials of order 2m + 1 on [0, 1] such that p(0) = 0, p(1) = 1, and p(
) (0) = 0 = p(
) (1), 1 ≤  ≤ m, q0 (0) = 0 = q0 (1), (
) (
) q0 (0) = 1, q0 (1) = 0, and q0 (0) = 0 = q0 (1), 2 ≤  ≤ m, and q1 (0) = 0 = q1 (1), (
) (
) q1 (0) = 0, q1 (1) = 1, and q1 (0) = 0 = q1 (1), 2 ≤  ≤ m. By definition m N T (·, X) ∈ C ((0, 1]; R ). Moreover, for 1 ≤  ≤ m, ∂T ∂T (tn , X) = wn , (tn+1 , X) = wn+1 , Ttn (X) = xn , Ttn+1 (X) = xn+1 ,   ∂t  ∂t  t − tn xn+1 − xn t − tn ∂T + q0 wn (t, X) = p ∂t tn+1 − tn tn+1 − tn tn+1 − tn   t − tn + q1 wn+1 . tn+1 − tn We now show that T satisfies conditions (a) and (b), which are equivalent to condition (4.2) in Chapter 4 on V . First observe that Tt (X) − X and ∂T /∂t are both independent of X and (Tt − I)(X) ∈ C m ((0, 1]; RN ). Define def

f (t) = Tt (X) − X

⇒

∂f ∂T (t, X) = . ∂t ∂t

Hence T and ∂T /∂t clearly satisfy conditions (T1) in (4.8) in Chapter 4. It satisfies condition (T2) since Tt−1 (Y ) = Y − f (t), and, a fortiori, Tt−1 satisfies conditions (T3). Therefore, the velocity field ∂f def ∂T V (t, x) = (t, Tt−1 (x)) = (t) ∂t ∂t

3. First-Order Shape Semiderivatives and Derivatives

471

satisfies conditions (V) given by (4.2) of Chapter 4. Thus properties (a) and (b) are satisfied. It remains to show that ∂f /∂t(t) → v as t → 0. It is easy to check that by construction4 −p (τ ) + q0 (τ ) + q1 (τ ) = 1 in [0, 1] and that on each interval [tn+1 , tn ] xn+1 − xn = wn+1 − 2wn , tn+1 − tn     t − tn t − tn ∂f (wn+1 − 2wn ) + q0 wn (t) = p ∂t tn+1 − tn tn+1 − tn   t − tn + q1 wn+1 , tn+1 − tn   ∂f t − tn [(wn+1 − v) − 2(wn − v)] (t) − v = p ∂t tn+1 − tn     t − tn t − tn   + q0 (wn − v) + q1 (wn+1 − v). tn+1 − tn tn+1 − tn The left-hand side now clearly converges since the three polynomials are bounded by a constant independent of n. This is sufficient to prove the theorem.

3 3.1

First-Order Shape Semiderivatives and Derivatives Eulerian and Hadamard Semiderivatives

Go back to Definition 3.1 of a shape functional in section 3.1 of Chapter 4 and the discussion of Gateaux and Hadamard semiderivatives in section 3.2 of the same chapter, where we have shown that results based on perturbations of the identity or asymptotic developments can be readily recovered by direct application of the velocity method with nonautonomous velocity fields V (t, x). In view of this, we give all the basic definitions, constructions, and theorems within the context of the velocity method and emphasize the Hadamard semiderivative, which is important in situations where the shape is parametrized and the chain rule is necessary. We give the definitions in the constrained case under assumptions (5.5) in Chapter 4 that can be specialized to the unconstrained case (D = RN ).5 Consider a velocity field V : [0, τ ] × D → RN

(3.2)

4 By interpolation of the function g(τ ) = τ − 1, g(τ ) = g(0)p(τ ) + g
(0)q (τ ) + g
(1)q (τ ) = 0 1 −p(τ ) + q0 (τ ) + q1 (τ ), and necessarily 1 = g
(τ ) = −p
(τ ) + q0
(τ ) + q1
(τ ). 5 In the unconstrained case, everything reduces to the conditions (V) on the velocity field given by (4.2) in Chapter 4: there exists τ = τ (V ) > 0 such that

(V)

∀x ∈ RN , ∃c > 0, ∀x, y ∈ RN ,





V (·, x) ∈ C [0, τ ]; RN ,

V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|.

(3.1)

472

Chapter 9. Shape and Tangential Differential Calculuses

verifying condition (V1D ) and the equivalent form (V2C ) of condition (V2D )6 for some τ = τ (V ) > 0:   ∀x ∈ D, V (·, x) ∈ C [0, τ ]; RN , ∃c > 0, (V1D ) (3.3) ∀x, y ∈ D, V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|, (V2C )

∀t ∈ [0, τ ], ∀x ∈ D,

V (t, x) ∈ LD (x) = {−CD (x)} ∩ CD (x),

where LD (x) = CD (x) ∩ {−CD (x)} is the linear tangent space to D at the point x ∈ ∂D and CD (x) is Clarke’s tangent cone to D at x (Theorem 5.2 in Chapter 4). We shall refer to the set of conditions (3.2)–(3.3) as condition (VD ). Introduce the following linear subspace of Lip (D, RN ):  def

Lip L (D, RN ) = θ ∈ Lip (D, RN ) : ∀X ∈ ∂D, θ(X) ∈ LD (X) .

(3.4)

Under the action of a velocity field V satisfying conditions (3.2)–(3.3), a set Ω in D is transformed into a new domain, def

Ωt (V ) = Tt (V )(Ω) = {Tt (V )(X) : ∀X ∈ Ω} ,

(3.5)

also contained in D. The transformation Tt : D → D associated with the velocity def field x → V (t)(x) = V (t, x) : D → RN is given by def

Tt (X) = x(t, X),

t ≥ 0, X ∈ D,

(3.6)

where x(·, X) is solution of the differential equation dx (t, X) = V (t, x(t, X)), t ≥ 0, dt

x(t, X) = X.

(3.7)

Recall Definition 3.1 of section 3 in Chapter 4 of a shape functional. Definition 3.1. Given a nonempty subset D of RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D will be referred to as the underlying holdall or universe. A shape functional is a map J :A→E (3.8) from some admissible family A of sets in P(D) into a topological space E. For instance, A could be the set X (Ω) = {F (Ω) : ∀F ∈ F(Θ)}. Definition 3.2. Let J be a real-valued shape functional. 6 We

replace condition (V2D ) in (5.5) of Chapter 4 (V2D )

∀x ∈ D, ∀t ∈ [0, τ ],

±V (t, x) ∈ TD (x)

by its equivalent form (V2C ) by using Theorem 5.2 in Chapter 4.

3. First-Order Shape Semiderivatives and Derivatives

473

(i) Let V be a velocity field satisfying conditions (3.2)–(3.3). J has an Eulerian semiderivative at Ω in the direction V if the limit   J Ωt (V ) − J(Ω) (3.9) lim t0 t exists. It will be denoted by dJ(Ω; V ). For simplicity, when V (t, x) = θ(x), θ ∈ Lip L (D, RN ) an autonomous velocity field, we shall write dJ(Ω; θ). (ii) Let Θ be a topological vector subspace of Lip L (D, RN ) satisfying conditions (3.2)–(3.3). J has a Hadamard semiderivative at Ω in the direction θ ∈ Θ if J(Ωt (V )) − J(Ω) t ([0,τ ];Θ)

lim

0

V ∈C V (0)=θ t0

(3.10)

exists, depends only on θ, and is independent of the choice of V satisfying conditions (3.2)–(3.3). It will be denoted by dH J(Ω; θ). If dH J(Ω; θ) exists, J has an Eulerian semiderivative at Ω in the direction V (t, x) = θ(x) and dH J(Ω; θ) = dJ(Ω; V (0)) = dJ(Ω; V ). (iii) Let Θ be a topological vector subspace of Lip L (D, RN ) satisfying conditions (3.2)–(3.3). J has a Hadamard derivative at Ω with respect to Θ if it has a Hadamard semiderivative at Ω in all directions θ ∈ Θ and the map θ → dJ(Ω; θ) : Θ → R

(3.11)

is linear and continuous. The map (3.11) will be denoted G(Ω) and referred to as the gradient of J in the topological dual Θ of Θ. The definition of an Eulerian semiderivative is quite general. For instance, it readily applies to shape functionals defined on closed submanifolds D of RN . It includes cases where dJ(Ω; V ) is dependent not only on V (0) but also on V (t) in a neighborhood of t = 0. We shall see that this will not occur under some continuity assumption on the map V → dJ(Ω; V ). When dJ(Ω; V ) depends only on V (0), the analysis can be specialized to autonomous vector fields V and the semiderivative can be related to the gradient of J. Example 3.1. For any measurable subset Ω of RN , consider the volume function  J(Ω) = dx.

(3.12)

Ω

For Ω with finite volume and V in C([0, τ ]; C01 (RN , RN )), consider the transformations Tt (Ω) of Ω:    J(Tt (Ω)) = dx = | det(DTt )| dx = det(DTt ) dx Tt (Ω)

Ω

Ω

474

Chapter 9. Shape and Tangential Differential Calculuses

for t small since det(DT0 ) = det I = 1. This yields the Eulerian semiderivative  dJ(Ω; V ) = div V (0) dx. (3.13) Ω

By definition it depends only on V (0) and hence J has a Hadamard semiderivative  div V (0) dx (3.14) dH J(Ω; V (0)) = Ω

and even a gradient G(Ω) for Θ = C01 (RN , RN ). The following simple continuity condition can also be used to obtain the Hadamard semidifferentiability. Theorem 3.1. Let Θ be a Banach subspace of Lip L (D, RN ) satisfying conditions (3.2)–(3.3), J be a real-valued shape functional, and Ω be a subset of RN . (i) Given θ ∈ Θ, if ∀V ∈ C([0, τ ]; Θ) such that V (0) = θ,

dJ(Ω; V ) exists,

(3.15)

and if the map V → dJ(Ω; V ) : C([0, τ ]; Θ) → R

(3.16)

is continuous for the subspace of V ’s such that V (0) = θ, then J is Hadamard semidifferentiable at Ω in the direction θ with respect to Θ and   (3.17) ∀V ∈ C([0, τ ]; Θ), V (0) = θ, dJ(Ω; V ) = dH J Ω; θ . (ii) If for all V in C([0, τ ]; Θ), dJ(Ω; V ) exists and the map V → dJ(Ω; V ) : C([0, τ ]; Θ) → R

(3.18)

is continuous, then J is Hadamard semidifferentiable at Ω in the direction V (0) with respect to Θ and     ∀V ∈ C([0, τ ]; Θ), dJ(Ω; V ) = dJ Ω; V (0) = dH J Ω; V (0) . (3.19) Proof. Given V in C([0, τ ]; Θ) such that V (0) = θ, construct the sequence def

Vn (t) =

V (t),   V nτ ,

0 ≤ t ≤ τ /n, τ /n < t ≤ τ,

for all integers n ≥ 1.

Note that for each n ≥ 1, dJ(Ω; Vn ) = dJ(Ω; V ) since for t < τ /n, Tt (Vn ) = Tt (V ). By continuity of V , {Vn } converges in C([0, τ ]; Θ) to the autonomous field V
(t) = V (0): ∀n, ∀V ∈ C([0, τ ]; Θ), ∀ε > 0, ∃δ > 0, ∀t , |t | < δ, V (t ) − V (0)Θ < ε.

3. First-Order Shape Semiderivatives and Derivatives

475

Hence τ <δ n sup V (t) − V (0)Θ < ε ⇒ Vn → V
. ∀ε > 0, ∃N > 0, ∀n ≥ N,

⇒

τ t∈[0, n ]

Hence by continuity of the map (3.18)     ⇒ dJ(Ω; V ) = dJ Ω; V
. dJ(Ω; Vn ) → dJ Ω; V
  But since dJ Ω; V
is dependent only on V (0), for all W ∈ Θ such that W (0) =   V (0) we have the identity dJ(Ω; W ) = dJ Ω; V
. Therefore, J has a Hadamard semiderivative at Ω in the direction V (0) and   dH J(Ω; V (0)) = dJ Ω; V
= dJ(Ω; V ). This concludes the proof of the theorem. For simplicity, the last theorem was given only for a Banach space Θ. This includes the spaces C0k (RN ), C k (RN ), C 0,1 (RN ), and Bk (RN , RN ) considered in Chapters 3 and 4. However, its conclusions are not limited to Banach spaces. Other constructions can be used as illustrated below in the unconstrained case. Consider velocity fields in   →m,k def − = lim VKm,k : ∀K compact in RN , (3.20) V −→ K

where for m ≥ 0,   def VKm,k = C m [0, τ ]; Dk (K, RN ) ∩ Lip(RN , RN )

(3.21)

and lim denotes the inductive limit set with respect to K endowed with its natural −→ inductive limit topology. For autonomous fields, this construction reduces to

 →k def − V = lim VKk : ∀K compact in RN , (3.22) −→ K

def

VKk =

D (K, RN ) ∩ Lip (K, RN ), k = 0, 1 ≤ k ≤ ∞. Dk (K, RN ), 0

(3.23)

→ − For k ≥ 1, V k = Dk (RN , RN ). In all cases the conditions (3.1) of the unconstrained case D = RN are verified. Theorem 3.2. Let J be a real-valued shape functional, Ω be a subset of RN , and k ≥ 0 be an integer. → − (i) Given θ ∈ V k , assume that → − ∀V ∈ V 0,k , V (0) = θ, dJ(Ω; V ) exists (3.24)

476

Chapter 9. Shape and Tangential Differential Calculuses and that the map → − V → dJ(Ω; V ) : V 0,k → R

(3.25)

is continuous for all V ’s such that V (0) = θ. Then J is Hadamard semidifferentiable in Ω in the direction θ with respect to V k and     → − (3.26) ∀V ∈ V 0,k , V (0) = θ, dH J Ω; θ = dJ(Ω; V ) = dJ Ω; V (0) . → − (ii) Assume that for all V ∈ V 0,k , dJ(Ω; V ) exists and that the map → − V → dJ(Ω; V ) : V 0,k → R

(3.27)

is continuous. Then J is Hadamard semidifferentiable in Ω in the direction V (0) with respect to V k and     → − ∀V ∈ V 0,k , dH J Ω; V (0) = dJ(Ω; V ) = dJ Ω; V (0) . (3.28) Proof. It is sufficient to prove the theorem for any compact subset K of RN and 0,k k k = C([0, τ ]; VK ), where VK is a Banach space conhence only for velocities in VK k N N tained in C0 (R , R ). So the theorem follows from Theorem 3.1.

3.2

Hadamard Semidifferentiability and Courant Metric Continuity

We now relate the Hadamard semidifferentiability of a shape functional with the Courant metric continuity studied in section 6 of Chapter 4. Recall the generic metric spaces F(Θ) of Micheletti with metric d and the quotient group F(Θ)/G with the Courant metric dG in (2.31) of Chapter 3. Theorem 3.3. Let Θ be equal to C0k+1 (RN , RN ), C k+1 (RN , RN ), or C k,1 (RN , RN ), k ≥ 0. Assume that G = G(Ω) for Ω closed or open and crack-free. If a shape functional J is Hadamard semidifferentiable for all θ ∈ Θ, then it is continuous with respect to the Courant metric dG(Ω) . Proof. By definition of the Hadamard semiderivative and Theorem 6.1 for Θ = C0k+1 (RN , RN ), Theorem 6.2 for Θ = C k+1 (RN , RN ), and Theorem 6.3 for Θ = C k,1 (RN , RN ) of Chapter 4.

3.3

Perturbations of the Identity and Gateaux and Fr´ echet Derivatives

In the unconstrained case (D = RN ), we have introduced in sections 3 and 4.2 of Chapter 4 a notion of directional semiderivative associated with first- and secondorder perturbations of the identity. Even if this approach does not naturally extend to the constrained case, it is interesting to compare the definitions and results with those associated with the velocity method of Definition 3.2.

3. First-Order Shape Semiderivatives and Derivatives

477

Recall the generic metric spaces F(Θ) of Micheletti with metric d and the quotient group F(Θ)/G with the Courant metric dG in (2.31) in Chapter 3. Assume that G = G(Ω) for Ω closed or open and crack-free and that there exist a ball Bε of radius ε > 0 in Θ and a constant c > 0 such that ∀f ∈ Bε ,

[I + f ]−1 − IΘ ≤ c f Θ .

(3.29)

Except for C 0 (RN , RN ) and B0 (RN , RN ), this is true in all Banach spaces Θ ⊂ C 0,1 (RN , RN ) considered in Chapters 3 and 4 (cf. Theorem 2.14 in section 2.5.2 and Theorem 2.17 in section 2.6.2 of Chapter 3). Hence the maps f → [I + f ] → [I + f ] : Bε ⊂ Θ → F(Θ) → F(Θ)/G(Ω) are well-defined and continuous in f = 0 since dG (I, I + f ) ≤ d(I, I + f ) ≤ f Θ + [I + f ]−1 − IΘ ≤ (1 + c) f Θ . For a shape functional J, the map def

[I + f ] → JΩ (f ) = J([I + f ](Ω)) : F(Θ)/G(Ω) → R is well-defined since J is invariant on G(Ω) and the map def

f → JΩ (f ) = J([I + f ](Ω)) : Θ → R is continuous in f = 0 if J is continuous in Ω for the Courant metric on F(Θ)/G(Ω). The following definitions are now the extension of the standard definitions of section 2 for a function JΩ (f ) defined on the ball Bε in the normed vector space Θ to topological vector spaces. They parallel the ones of Definition 3.2. Definition 3.3. Let J be a real-valued shape functional and Θ be a topological vector subspace of Lip (RN , RN ). For f ∈ Θ, denote [I + f ](Ω) by Ωf . (i) JΩ is said to have a Gateaux semiderivative at f in the direction θ ∈ Θ if the following limit exists and is finite:   J [I + f + tθ](Ω) − J([I + f ](Ω)) def . (3.30) dJΩ (f ; θ) = lim t0 t (ii) JΩ is said to be Gateaux differentiable at f if it has a Gateaux semiderivative at f in all directions θ ∈ Θ and the map θ → dJΩ (f ; θ) : Θ → R

(3.31)

is linear and continuous. The map (3.31) is denoted ∇JΩ (f ) and referred to as the gradient of JΩ in the topological dual Θ of Θ.

478

Chapter 9. Shape and Tangential Differential Calculuses

(iii) If, in addition, Θ is a normed vector space, we say that J is Fr´echet differentiable at f if J is Gateaux differentiable at f and |J([I + f + θ](Ω)) − J([I + f ](Ω)) − ∇JΩ (f ), θΘ | = 0. θΘ θΘ →0 lim

(3.32)

The semiderivatives of J and JΩ are related. Theorem 3.4. Let J be a real-valued shape functional, let f ∈ Θ, and let I + f ∈ F(Θ). (i) Assume that JΩ has a Gateaux semiderivative at f in the direction θ ∈ Θ; then J has an Eulerian semiderivative at Ωf in the direction Vθf and dJΩ (f ; θ) = dJ(Ωf ; Vθf ),

Vθf (t) = θ ◦ [I + f + tθ]−1 def

(3.33)

for t sufficiently small. (ii) If J has a Hadamard semiderivative at Ωf in the direction θ ◦ [I + f ]−1 , then JΩ has a Gateaux semiderivative at f in the direction θ and dJΩ (f ; θ) = dH J(Ωf ; θ ◦ [I + f ]−1 ).

(3.34)

Conversely, if JΩ has a Gateaux semiderivative at f in the direction θ ◦[I +f ], then J has a Hadamard semiderivative at Ωf in the direction θ. If either dJΩ (f ; θ) or dH J(Ωf ; θ) is linear and continuous with respect to all θ in Θ, so is the other and ∇JΩ (f ), θΘ = G(Ωf ), θ ◦ [I + f ]−1 Θ , G(Ωf ), θΘ = ∇JΩ (f ), θ ◦ [I + f ]Θ .

∀θ ∈ Θ,

(3.35)

  J [I + f + tθ](Ω) − J([I + f ](Ω)) dJΩ (f ; θ) = lim t0 t   J Tt ([I + f ](Ω)) − J([I + f ](Ω)) = lim = dJ([I + f ](Ω); Vθf ) t0 t

Proof. By definition

for the family of transformations Ttf = [I + f + tθ] ◦ [I + f ]−1 , def

and from Theorem 4.1 in section 4.1 of Chapter 4, Ttf corresponds to the velocity field def

Vθf (t) =

∂Ttf ◦ (Ttf )−1 = θ ◦ [I + f + tθ]−1 . ∂t

Identity (3.34) now follows from the fact that J has a Hadamard semiderivative at Ωf . The other properties readily follow from the definitions.

3. First-Order Shape Semiderivatives and Derivatives

479

We have standard sufficient conditions for the Fr´echet differentiability from Theorem 2.3. Theorem 3.5. Let J be a real-valued shape functional. Let Ω be a subset of RN and Θ be C0k+1 (RN , RN ), C k+1 (RN , RN ), C k,1 (RN ), or Bk+1 (RN , RN ), k ≥ 0. If JΩ is Gateaux differentiable for all f in Bε and the map f → ∇JΩ (f ) : Bε → Θ is continuous in f = 0, then JΩ is Fr´echet differentiable in f = 0.

3.4

Shape Gradient and Structure Theorem

In view of the previous discussion we now specialize to autonomous vector fields V to further study the properties and the structure of dJ(Ω; V ). For simplicity we also specialize to the unconstrained case in RN . The constrained case yields similar ´results but is technically more involved (cf. M. C. Delfour and J.-P. Zole sio [14] for D open in RN ). The choice of a shape gradient depends on the choice of the topological vector subspace Θ of Lip (RN , RN ). We choose to work in the classical framework of the Theory of Distributions (cf. L. Schwartz [3]) with Θ = D(RN , RN ), the space of all infinitely differentiable transformations θ of RN with compact support. For these velocity fields V , conditions (3.1) are satisfied. Definition 3.4. Let J be a real-valued shape functional. Let Ω be a subset of RN . (i) The function J is said to be shape differentiable at Ω if it is differentiable at Ω for all θ in D(RN , RN ). (ii) The map (3.11) defines a vector distribution G(Ω) in D(RN , RN ) , which will be referred to as the shape gradient of J at Ω. (iii) When, for some finite k ≥ 0, G(Ω) is continuous for the Dk (RN , RN )-topology, we say that the shape gradient G(Ω) is of order k. The next theorem gives additional properties of shape differentiable functions. Notation 3.1. Associate with a subset A of RN and an integer k ≥ 0 the set  def

LkA = V ∈ Dk (RN , RN ) : ∀x ∈ A, V (x) ∈ LA (x) , where LA (x) = {−CA (x)} ∩ CA (x) and CA (x) is given by (5.27) in Chapter 4. Theorem 3.6 (Structure theorem). Let J be a real-valued shape functional. Assume that J has a shape gradient G(Ω) for some Ω ⊂ RN with boundary Γ. (i) The support of the shape gradient G(Ω) is contained in Γ.

480

Chapter 9. Shape and Tangential Differential Calculuses

(ii) If Ω is open or closed in RN and the shape gradient is of order k for some def k ≥ 0, then there exists [G(Ω)] in (Dk /LkΩ ) such that for all V in D k = Dk (RN , RN ) dJ(Ω; V ) = [G(Ω)], qL V Dk /Lk , Ω

(3.36)

where qL : Dk → Dk /LkΩ is the canonical quotient surjection. Moreover G(Ω) = (qL )∗ [G(Ω)],

(3.37)

where (qL )∗ denotes the transpose of the linear map qL . Proof. (i) Any V in D such that V = 0 on Γ satisfies assumptions (V 1Ω ) and (V 2Ω ) in (5.5) (with Ω in place of D) and V ∈ L∞ Ω . Since V = 0 on Γ, Ts (Γ) = Γ. ¯ →Ω ¯ is a homeomorphism and Moreover, by Theorem 5.1 (i) of Chapter 4, Ts : Ω ¯ ¯ ¯ Ts (Ω) = Ω and (Ω)s = Ts (Ω) = Ω since Ω = Ω ∪ Γ and Ω ∩ Γ = ∅. Thus when Ω is ¯ or open (Ω = int Ω), closed (Ω = Ω) ∀s ≥ 0,

Ts (Ω) = Ω

⇒

J(Ωs ) = J(Ω)

⇒

dJ(Ω; V ) = 0.

(ii) It is sufficient to prove that dJ(Ω; V ) = 0 for all V in LkΩ . The other statements follow by standard arguments and the fact that LkΩ is a closed linear subspace of D k . From part (i) we know that the result is true for all V in L∞ Ω and hence by a density argument for all V in LkΩ . Remark 3.1. When the boundary Γ of Ω is compact and J is shape differentiable at Ω, the distribution G(Ω) is of finite order. Once this is known, the conclusions of Theorem 3.6 (ii) apply with k equal to the order of G(Ω). Hence G(Ω) will belong to a Hilbert space H −s (RN ) for some s ≥ 0. The quotient space is very much related to a trace on the boundary Γ, and when the boundary Γ is sufficiently smooth we can indeed make that identification. Corollary 1. Assume that the assumptions of Theorem 3.6 are satisfied for an open domain Ω, that the order of G(Ω) is k ≥ 0, and that the boundary Γ of Ω is C k+1 . Then for all x in Γ, LΩ (x) is an (N − 1)-dimensional hyperplane to Ω at x and there exists a unique outward unit normal n(x) which belongs to C k (Γ; RN ). As a result, the kernel of the map V → γΓ (V ) · n : Dk (RN , RN ) → C k (Γ)

(3.38)

coincides with LkΩ , where γΓ : Dk (RN , RN ) → C k (Γ, RN ) is the trace of V on Γ. Moreover, the map pL (V )  def  qL (V ) → pL qL (V ) = γΓ (V ) · n : Dk /LkΩ → C k (Γ) (3.39) is a well-defined isomorphism. In particular, there exists a scalar distribution g(Γ) in RN with support in Γ such that g(Γ) ∈ C k (Γ) and for all V in Dk (RN , RN ) dJ(Ω; V ) = g(Γ), γΓ (V ) · nC k (Γ)

(3.40)

3. First-Order Shape Semiderivatives and Derivatives

481

and G(Ω) = ∗ (qL )[G(Ω)], When g(Γ) ∈ L1 (Γ)

 dJ(Ω; V ) =

[G(Ω)] = ∗ (pL )g(Γ).

(3.41)

g V · n dΓ and G = γΓ∗ (g n),

(3.42)

Γ

where γΓ is the trace operator on Γ. Proof. The surjectivity of (3.39) is a consequence of the fact that Γ is compact and that for a C k+1 boundary, k ≥ 0, it is always possible to construct an extension N of the unit normal n on Γ which belongs to Dk (RN , RN ) with support in a neighborhood of Γ. Then for any v in C k (Γ), there exists also an extension v˜ in Dk (RN ) with support in a neighborhood of Γ, and the vector V = v˜N belongs to D k (RN , RN ) and coincides with vn on Γ. Remark 3.2. In 1907, J. Hadamard [1] used displacements along the normal to the boundary Γ of a C ∞ -domain (as in section 3.3.1 of Chapter 4) to compute the derivative of the first eigenvalue of the clamped plate. Theorem 3.6 and its corollary are generalizations to arbitrary shape functionals of that property to open or closed domains with an arbitrary boundary. The structure theorem for shape functionals ´sio [12] in 1979 and on open domains with a C k+1 -boundary is due to J.-P. Zole not to Hadamard even if the formula (3.42) is often called the Hadamard formula. Example 3.2. For any measurable subset Ω of RN , consider the volume shape function (3.12) of Example 3.1:  dx. J(Ω) = Ω

For Ω with finite volume and V in D (R , RN ), we have seen in Example 3.1 that   dJ(Ω; V ) = div V dx = χΩ div V dx, (3.43) 1

N

RN

Ω

and this is formula (3.40) with k = 1. For an open domain Ω with a C 1 compact boundary Γ,  V · n dΓ, (3.44) dJ(Ω; V ) = Γ

which is also continuous with respect to V in D 0 (RN , RN ). Here the smoothness of the boundary decreases the order of the distribution G(Ω). This raises the question of the characterization of the family of all subsets Ω of RN for which the map  div V dx : D1 (RN , RN ) → R (3.45) V → Ω

can be continuously extended to D0 (RN , RN ). But this is the family of locally finite perimeter sets: sets Ω whose characteristic function belongs to BVloc (RN ).

482

4

Chapter 9. Shape and Tangential Differential Calculuses

Elements of Shape Calculus

´sio [7, 8] for In this section we recall a number of basic formulae from J.-P. Zole the derivative of domain and boundary integrals. The reader is also referred to the ´sio [9] for the computation of companion book of J. Soko
lowski and J.-P. Zole shape derivatives associated with a wide range of partial differential equations. In this section, a more modern treatment of the derivative of boundary integrals is given that emphasizes a new approach to the tangential calculus on a C 2 submanifold of RN of codimension 1. This development took place in the context of the theory of shells, where a simple and self-contained intrinsic differential calculus has been developed by using the oriented distance function of Chapter 7. It completely avoids the use of local maps and bases and Christoffel’s symbols. In this section a new, considerably simplified proof of the formula for the derivative of boundary integrals is presented by using the oriented distance function.

4.1

Basic Formula for Domain Integrals

The simplest examples of domain functions are given by volume integrals over a bounded open domain Ω in RN . They use a basic formula in connection with the family of transformations {Tt : 0 ≤ t ≤ τ }. Assume that condition (3.1) is satisfied by the velocity field {V (t) : 0 ≤ t ≤ τ }. Further assume that V ∈ 1 (RN , RN )) and that τ > 0 is such that the Jacobian Jt is strictly C 0 ([0, τ ]; Cloc positive: ∀t ∈ [0, τ ],

def

Jt (X) = det DTt (X) > 0,

(DTt )ij = ∂j Ti ,

where DTt (X) is the Jacobian matrix of the transformation Tt = Tt (V ) associated 1,1 (RN ), consider for with the velocity vector field V . Given a function ϕ in Wloc 0 ≤ t ≤ τ the volume integral  def J(Ωt (V )) = ϕ dx, (4.1) Ωt (V ) def

where Ωt (V ) = Tt (V )(Ω). By the change of variables formula   ϕ dx = ϕ ◦ Tt Jt dx, J(Ωt (V )) = Ωt (V )

(4.2)

Ω

and the following formulae and results are easy to check. 1,1 Theorem 4.1. Let ϕ be a function in Wloc (RN ). Assume that the vector field V = {V (t) : 0 ≤ t ≤ τ } satisfies condition (V ).

(i) For each t ∈ [0, τ ] the map 1,1 1,1 ϕ → ϕ ◦ Tt : Wloc (RN ) → Wloc (RN )

and its inverse are both locally Lipschitzian and ∇(ϕ ◦ Tt ) = ∗DTt ∇ϕ ◦ Tt .

4. Elements of Shape Calculus

483

1 (RN , RN )), then the map (ii) If V ∈ C 0 ([0, τ ]; Cloc 1,1 t → ϕ ◦ Tt : [0, τ ] → Wloc (RN )

is well-defined and for each t d ϕ ◦ Tt = (∇ϕ · V (t)) ◦ Tt ∈ L1loc (RN ). dt

(4.3)

Hence the function 1,1 t → ϕ ◦ Tt belongs to C 1 ([0, τ ]; L1loc (RN )) ∩ C 0 ([0, τ ]; Wloc (RN )). 1 (RN , RN )), then the map (iii) If V ∈ C 0 ([0, τ ]; Cloc 0 t → Jt : [0, τ ] → Cloc (RN )

is differentiable and dJt 0 = [ div V (t)] ◦ Tt Jt ∈ Cloc (RN ). dt

(4.4)

0 (RN )). Hence the map t → Jt belongs to C 1 ([0, τ ]; Cloc

Indeed it is easy to check that d DTt (X) = DV (t, Tt (X)) DTt (X), DT0 (X) = I, dt d det DTt (X) = tr DV (t, Tt (X)) det DTt (X) dt ⇒

d det DTt (X) = div V (t, Tt (X)) det DTt (X), dt

det DT0 (X) = 1,

and (4.4) follows directly by definition of Jt (X). From (4.2), (4.3), and (4.4)  d J(Ωt (V )) = ∇ϕ · V (0) + ϕ div V (0) dx dJ(Ω; V ) = dt t=0  Ω div (ϕ V (0)) dx. ⇒ dJ(Ω; V ) = Ω

If Ω has a Lipschitzian boundary, then by Stokes’s theorem  ϕ V (0) · n dΓ. dJ(Ω; V ) = Γ

Theorem 4.2. Assume that there exists τ > 0 such that the velocity field V (t) 1 (RN , RN )). Given a function ϕ ∈ satisfies conditions (V ) and V ∈ C 0 ([0, τ ]; Cloc

484

Chapter 9. Shape and Tangential Differential Calculuses

1,1 (RN )) ∩ C 1 (0, τ ; L1loc (RN )) and a bounded measurable domain Ω with C(0, τ ; Wloc boundary Γ, the semiderivative of the function def



JV (t) =

ϕ(t) dx

(4.5)

Ωt (V )

at t = 0 is given by 

ϕ (0) + div (ϕ(0) V (0)) dx,

dJV (0) =

(4.6)

Ω def

def

where ϕ(0)(x) = ϕ(0, x) and ϕ (0)(x) = ∂ϕ/dt(0, x). If, in addition, Ω is an open domain with a Lipschitzian boundary Γ, then 





dJV (0) =

4.2

ϕ(0) V (0) · n dx.

ϕ (0) dx + Ω

(4.7)

Γ

Basic Formula for Boundary Integrals

2 Given ψ in Hloc (RN ), consider for some bounded open Lipschitzian domain Ω in N R the shape functional  def J(Ω) = ψ dΓ. (4.8) Γ

This integral is invariant with respect to a homeomorphism which maps Ω onto itself (and hence Γ onto itself). Given the velocity field V and t ≥ 0, consider the expression  def J(Ωt (V )) = ψ dΓt . Γt (V )

Using the change of variables Tt (V ) and the material introduced in (3.13) of section 3.2 and (5.30) of section 5 in Chapter 2, this integral can be brought back from Γt to Γ:   def J(Ωt (V )) = ψ dΓt = ψ ◦ Tt ωt dΓ, (4.9) Γt

Γ

where the density ωt is given as ωt = |M (DTt )n|,

(4.10)

n is the outward normal field on Γ, and M (DTt ) is the cofactor matrix of DTt , that is, M (DTt ) = Jt ∗(DTt )−1

⇒ ωt = Jt | ∗(DTt )−1 n|.

(4.11)

4. Elements of Shape Calculus

485

It can easily be checked from (4.10) and (4.11) that t → ωt is differentiable in C 0 (Γ) and that the limit 1 ω  = lim (ωt − ω) = div V (0) − DV (0)n · n  t 0 t

(4.12)

1 (RN , RN ) in the C 0 (Γ)-norm is linear and continuous with respect to V (0) in the Cloc Fr´echet topology. Hence  dJ(Ω; V ) = ∇ψ · V (0) + ψ (div V (0) − DV (0)n · n) dΓ. (4.13) Γ

From Corollary 1 to the structure Theorem 3.6, dJ(Ω; V ) depends only on the normal component vn of the velocity field V (0) on Γ: def

vn = v · n,

def

v = V (0)|Γ

(4.14)

through Hadamard’s formula (3.40). In view of this property any other velocity field with the same smoothness and normal component on Γ will yield the same limit. Given k > 0 consider the tubular neighborhood  def

Sk (Γ) = x ∈ RN : |b(x)| < k

(4.15)

of Γ in RN for the oriented distance function b = bΩ associated with Ω. Assuming that Γ is compact and of class C 2 , there exists h > 0 such that b ∈ C 2 (S2h (Γ)). Let ϕ ∈ D(RN ) be such that ϕ = 1 in Sh (Γ) and ϕ = 0 outside of S2h (Γ). Consider the velocity field def

W (t) = (V (0) · ∇b) ∇b ϕ. Clearly, the normal component of W (0) on Γ coincides with vn . Moreover, in Sh (Γ) ∇ψ · W = ∇ψ · ∇b V (0) · ∇b

⇒ ∇ψ · W |Γ = ∇ψ · n V (0) · n =

∂ψ vn , ∂n

DW = V (0) · ∇b D2 b + ∇b ∗∇(V (0) · ∇b), div W = V (0) · ∇b ∆b + ∇b · ∇(V (0) · ∇b), DW ∇b · ∇b =V (0) · ∇b D2 b∇b · ∇b + ∇(V (0) · ∇b) · ∇b =∇(V (0) · ∇b) · ∇b, div W − DW ∇b · ∇b = V (0) · ∇b ∆b ⇒ div W − DW ∇b · ∇b|Γ = V (0) · n H = H vn since ∇b|Γ = n, D2 b∇b = 0, and H = ∆b is the additive curvature, that is, the sum of the N − 1 curvatures of Γ or N − 1 times the mean curvature H. Finally,    ∂ψ dJ(Ω; V ) = + ψ H vn dΓ. (4.16) ∂n Γ We have proven the following result.

486

Chapter 9. Shape and Tangential Differential Calculuses

Theorem 4.3. Let Γ be the boundary of a bounded open subset Ω of RN of class C 2 2 1 and ψ be an element of C 1 ([0, τ ]; Hloc (RN )). Assume that V ∈ C 0 ([0, τ ]; Cloc (RN , N R )). Consider the function def



JV (t) =

ψ(t) dΓt . Γt (V )

Then the derivative of JV (t) with respect to t in t = 0 is given by the expression 

ψ  (0) +

dJV (0) = 

Γ

=



∂ψ + Hψ ∂n

 V (0) · n dΓ (4.17)

ψ  (0) + ∇ψ · V (0) + ψ (div V (0) − DV (0)n · n) dΓ,

Γ def

where ψ  (0)(x) = ∂ψ/∂t(0, x). Note that, as in the case of the volume integral, we have two formulae. Hence we have the following identity:  

 ∂ψ + Hψ V (0) · n dΓ ∂n Γ  ∇ψ · V (0) + ψ (div V (0) − DV (0)n · n) dΓ. =

(4.18)

Γ

4.3

Examples of Shape Derivatives

4.3.1

Volume of Ω and Surface Area of Γ

Consider the volume shape functional

 dx.

J(Ω) = Ω

This shape functional is used as a constraint on the domain in several examples of shape optimization problems. We get  dJ(Ω; V ) = div V (0) dx (4.19) Ω

and, if Γ is Lipschitzian,

 V (0) · n dΓ.

dJ(Ω; V ) =

(4.20)

Γ

A sufficient condition on the field V (0) to preserve the volume is div V (0) = 0 in Ω and, if Γ is Lipschitzian, V (0) · n = 0 on Γ. Consider the (shape) area function  J(Ω) = dΓ. Γ

4. Elements of Shape Calculus

487

Assuming that Γ is of class C 2 we get from (4.17)  dJ(Ω; V ) = H V (0) · n dΓ,

(4.21)

Γ

where H = ∆bΩ is the additive curvature. The condition to keep the surface of Γ constant is that V (0) · n be orthogonal (in L2 (Γ)) to H. 4.3.2

H 1 (Ω)-Norm

2 Given φ and ψ in Hloc (RN ), consider the shape functional  ∇φ · ∇ψ dx. J(Ω) = Ω

By using the change of variables Tt (V ), Ωt = Ωt (V ) = Tt (V )(Ω) and   ∇φ · ∇ψ dx = [A(V )(t)∇(φ ◦ Tt )] · ∇(ψ ◦ Tt ) dx, Ωt

(4.22)

Ω

where A(V ) is the following matrix associated with the field V : A(V )(t) = J(t) (DTt )−1 ∗(DTt )−1

(4.23)

and J(t) = det(DTt ). Expression (4.23) is easily obtained from the identity (∇φ) ◦ Tt = ∗(DTt )−1 ∇(φ ◦ Tt ).

(4.24)

k k If V ∈ C 0 ([0, τ ]; Cloc (RN ; RN )), k ≥ 1, T and T −1 belong to C 1 ([0, τ ]; Cloc (RN , RN )) N2 k−1 1 N 2 N and A(V ) to C ([0, τ ], Cloc (R ; R )). Then if φ, ψ belongs to Hloc (R ), we get 1 (RN ), with ∂φ/∂t ◦ Tt |t=0 = ∇φ · V (0), t → φ ◦ Tt , which is differentiable in Hloc 1 N which belongs to Hloc (R ). Finally, we obtain  dJ(Ω; V ) = [A (V )∇φ] · ∇ψ dx  Ω (4.25) + {∇(∇φ · V (0)) · ∇ψ + ∇φ · ∇(∇ψ · V (0))} dx, Ω 2

k−1 (RN , RN )-norm where A (V ) is the derivative in the Cloc

A (V ) =

def

∂ A(V )(t)|t=0 = div V (0) I − 2ε(V (0)) ∂t

(4.26)

and ε(V (0)) is the symmetrized Jacobian matrix (the strain tensor associated with the field V (0) in elasticity) 1 (4.27) ε(V (0)) = [∗ DV (0) + DV (0)]. 2 We finally obtain the volume expression  dJ(Ω; V ) = [div V (0) I − 2ε(V (0))] ∇φ · ∇ψ Ω

+ [∇(∇φ · V (0)) · ∇ψ + ∇φ · ∇(∇ψ · V (0))] dx.

(4.28)

488

Chapter 9. Shape and Tangential Differential Calculuses

2 (RN ), we obtain the simpler boundary When Γ is a C 1 -submanifold, φ, ψ ∈ Hloc expression by directly using formula (4.17):

 ∇φ · ∇ψ V (0) · n dΓ.

dJ(Ω; V ) =

(4.29)

Γ

This simple example nicely illustrates the notion of density gradient g. From expression (4.28) it was obvious that the mapping V → dJ(Ω; V ) was well-defined, 1 (RN ; RN )). By the structure theorem and linear, and continuous on C 1 ([0, τ ]; Cloc Hadamard’s formula we knew that dJ could be written in the form  g V (0) · n dΓ. Γ

But from (4.29) we know that g, which is an element of D 1 (Γ) , is an element of W 1/2,1 (Γ) given by g = ∇φ · ∇ψ (traces on Γ).

(4.30)

The direct calculation of g from expression (4.24) would have been very fastidious. 4.3.3

Normal Derivative

2 Let Γ be of class C 2 and φ ∈ Hloc (RN ) be given. Consider the following shape function:  2  ∂φ def 2 dΓ = |∇φ · n| dΓ. J(Ω) = ∂n Γ Γ

By the change of variables formula we get, with Ωt = Tt (V )(Ω) and Γt = Tt (V )(Γ),  

∗ 2 def 2 |∇φ · nt | dΓt = (DTt )−1 ∇(φ ◦ Tt ) · (nt ◦ Tt ) ωt dΓ, (4.31) J(Ωt ) = Γt

Γ

where nt ◦ Tt is the transported normal field nt from Γt onto Γ. The derivative can be obtained by using formula (4.17) of Theorem 4.3 and one of the above two expressions. However, the first expression first requires the construction of an extension Nt of the normal nt in a neighborhood of Γ. In both cases the following result will be useful. Theorem 4.4. Let k ≥ 1 be an integer. Given a velocity field V (t) satisfying k condition (V ) such that V ∈ C([0, τ ]; Cloc (RN , RN )), then nt ◦ Tt =

∗

(DTt )−1 n M (DTt )n = , ∗ −1 | (DTt ) n| |M (DTt )n|

(4.32)

where n and nt are the respective outward normals to Ω and Ωt on Γ and Γt and M (DTt ) is the cofactor’s matrix of DTt .

4. Elements of Shape Calculus

489

Proof. Go back to Definition 3.1 in section 3.1 of Chapter 2. We have shown in that section that for a domain Ω of class C k the unit outward normal at any point y ∈ Γx = Γ ∩ U (x) is given by expression (3.9) ∀y ∈ Γx ,

n(y) = −

∗

|

(Dhx )−1 eN −1 (hx (y)), ∗(Dh )−1 e | x N

where hx is the local diffeomorphism specified by (3.3):   gx ∈ C k U (x), B ,

  hx = gx−1 ∈ C k B, U (x) .

def

For Ωt = Ωt (V ) and xt = Tt (x), choose the following new neighborhood and local diffeomorphism: def

def

ht = Tt ◦ hx : B → Ut ,

Ut = Tt (U (x)),

gt = h−1 = gx ◦ Tt−1 : Ut → B. t def

On Γt (V )∩Ut the normal in given by the same expression but with ht in place of hx : nt = −

(Dht )−1 ◦ h−1 t eN . ∗ −1 | (Dht ) ◦ h−1 t eN | ∗

However, D(Tt ◦ hx ) = DTt ◦ hx Dhx , 

D(Tt ◦ hx ) ◦ (Tt ◦ hx )−1 = DTt Dhx ◦ h−1 ◦ Tt−1 , x 

−1 ∗ (DTt ◦ hx )−1 ◦ (Tt ◦ hx )−1 eN = ∗(DTt )−1 ∗(Dhx )−1 ◦ h−1 x e N ◦ Tt . Therefore, by using the previous expressions for n and nt , ∗

nt =

(DTt )−1 n ◦ Tt−1 | ∗(DTt )−1 n|

⇒ nt ◦ Tt =

∗

(DTt )−1 n . | ∗(DTt )−1 n|

(4.33)

The other expression for nt ◦ Tt in terms of the cofactor matrix M (DTt ) of DTt readily follows from the identity M (DTt ) = J(t) ∗(DTt )−1 . Recalling the expression for ωt ωt = |M (DTt )n| = J(t) | ∗(DTt )−1 n|, our boundary integral becomes 

 2

Γt

|[A(t)∇(φ ◦ Tt )] · n | ωt−1 dΓ. 2

|∇φ · nt | dΓt = Γ

(4.34)

490

Chapter 9. Shape and Tangential Differential Calculuses

Using expression (4.26) of A (V ) and expression (4.12) of ω  we get dJ(Ω; V ) 2  ∂φ ∂φ  =2 [A (V )∇φ · n + ∇(∇φ · V (0)) · n] − ω  dΓ ∂n ∂n Γ ∂φ { [div V (0) I − DV (0) − ∗DV (0)] ∇φ · n + ∇(∇φ · V (0)) · n} =2 ∂n Γ 2 ∂φ − (div V (0) − DV (0)n · n) dΓ ∂n    ∂φ ∂φ = 2 div V (0) − DV (0) ∇φ · n + D2 φ V (0) · n ∂n Γ ∂n 2 ∂φ − (div V (0) − DV (0)n · n) dΓ ∂n  2 ∂φ (div V (0) − DV (0)n · n) = Γ ∂n   ∂φ ∂φ DV (0)n · n − DV (0) ∇φ · n + D2 φ V (0) · n dΓ. +2 ∂n ∂n This formula can be somewhat simplified by using identity (4.18) with 2 ∂φ 2 ψ = |∇φ · ∇b| ⇒ ψ|Γ = , ∂n 

2 ∇ψ = 2 ∇φ · ∇b ∇(∇φ · ∇b) = 2 ∇φ · ∇b D φ∇b + D2 b∇ψ  ∂φ 2 ⇒ ∇ψ|Γ = 2 D φ n + D2 b ∇φ ∂n  ∂φ 2 D φ n + D2 b ∇φ · V (0) ⇒ ∇ψ · V (0)|Γ = 2 ∂n  ∂ψ ∂φ 2 ∂φ 2 = ∇ψ · ∇b|Γ = 2 D φ n + D2 b ∇φ · ∇b = 2 D φ n · n. ⇒ ∂n ∂n ∂n We obtain 2 1  0 ∂φ ∂φ 2 2 V (0) · n dΓ D φ n · n + H ∂n ∂n Γ (4.35) 2   ∂φ 2 ∂φ = D φ n + D2 b ∇φ · V (0) + (div V (0) − DV (0)n · n) dΓ, 2 ∂n Γ ∂n and hence dJ(Ω; V ) 2 1  0 ∂φ ∂φ 2 2 V (0) · n D φn · n + H = ∂n ∂n Γ     ∂φ ∂φ +2 DV (0)n · n − ∇φ · ∗DV (0) n + D2 b V (0) dΓ. ∂n ∂n

(4.36)

5. Elements of Tangential Calculus

491

This formula can be more readily obtained from the first expression (4.31) and the extension ∗

Nt =

|

(DTt )−1 ∇b ∗(DT )−1 ∇b| t

◦ Tt−1

(4.37)

of the normal nt . To compute N  , decompose Nt as follows: Nt = f (t) ◦ Tt−1 ,

f (t) = '

N  = f  − Df (0)V (0), 

g(t) , g(t) · g(t)

g(t) = ∗(DTt )−1 ∇b,

g  = − ∗DV (0)∇b,

f (0) = ∇b,



g |g(0)| − g(0) · g g(0)/|g(0)| |g(0)|2   = g − g · ∇b ∇b = DV (0)∇b · ∇b ∇b − ∗DV (0)∇b.

f =

So finally N  |Γ = (DV (0)n · n) n − ∗DV (0)n − D 2 b V (0) and

(4.38)

∂ ∂φ 2 |∇φ · Nt | ∇φ · N  =2 dt ∂n t=0 

∂φ =2 ∇φ · (DV (0)n · n)n − ∗DV (0) n − D2 b V (0) . ∂n

The first part of the integral (4.36) depends explicitly on the normal component of V (0). Yet we know from the structure theorem that for this function, the shape derivative depends only on the normal component of V (0). To make this explicit, it is necessary to introduce some elements of tangential calculus.

5

Elements of Tangential Calculus

In this section some basic elements of the differential calculus on a C 2 -submanifold of codimension 1 are introduced. The approach avoids local bases and coordinates by using the intrinsic tangential derivatives. In the classical theory of shells (see, for instance, Ph. D. Ciarlet [1]), the midsurface ω is defined as the image of a flat smooth bounded connected domain U in R2 via a C 2 -immersion ϕ : U → R3 . When U is sufficiently smooth and the thickness sufficiently small, the associated tubular neighborhood Sh (ω) of thickness 2h is a Lipschitzian domain that is identified with a thin shell of thickness 2h around ω. Local bases are generated by the derivatives of the immersion ϕ and tangential derivatives can be defined in the surface ω. Such hypersurfaces ω in RN can be viewed as a subset of the boundary Γ of an open subset Ω of RN . In that case, the gradient and Hessian matrix of the associated oriented distance function bΩ to the underlying set Ω completely describes the normal and the N fundamental forms of ω, and a fairly complete intrinsic theory

492

Chapter 9. Shape and Tangential Differential Calculuses

of Sobolev spaces on C 1,1 -hypersurfaces is available in M. C. Delfour [7]. M. C. Delfour [8] showed that all smooth hypersurfaces and the hypersurfaces with little regularity of S. Anicic, H. Le Dret, and A. Raoult [1] are C 1,1 hypersurfaces that can be described by the oriented distance function bΩ or the signed distance function bω . Such an approach has been used successfully in the theory of thin and asymptotic shells with a C 1,1 -midsurface. Let Ω be an open domain of class C 2 in RN with compact boundary Γ. Therefore, there exists h > 0 such that b = bΩ ∈ C 2 (S2h (Γ)). The projection of a point x onto Γ is given by def p(x) = x − b(x) ∇b(x), and the orthogonal projection operator of a vector onto the tangent plane Tp(x) Γ is given by def P (x) = I − ∇b(x) ∗∇b(x). Notice that, as a transformation of Tp(x) Γ, P (x) : Tp(x) Γ → Tp(x) Γ,

P (x) = I − ∇b(x) ∗∇b(x)

is the identity transformation on Tp(x) Γ. In fact P (x) coincides with the first fundamental form. Similarly D2 b can be considered as a transformation of Tp(x) since D2 b(x)n(x) = D 2 b(x)∇b(x) = 0. We shall show in section 5.6 that D2 b(x) : Tp(x) → Tp(x) coincides with the second fundamental form of Γ. Similarly D2 b(x)2 coincides with the third fundamental form. Finally Dp(x) = I − ∇b(x) ∗∇b(x) − b D 2 b(x)

5.1

and Dp|Γ = P.

Intrinsic Definition of the Tangential Gradient

The classical way to define the tangential gradient of a scalar function f : Γ → R is through an appropriately smooth extension F of f in a neighborhood of Γ using the fact that the resulting expression on Γ is independent of the choice of the extension F . In this section an equivalent direct intrinsic definition is given in terms of the extension f ◦ p of the function f . This is the basis of a simple differential calculus on Γ which uses the Euclidean differential calculus in the ambient neighborhood of Γ. Given f ∈ C 1 (Γ), let F ∈ C 1 (S2h (Γ)) be a C 1 -extension of f . Define def

g(F ) = ∇F |Γ −

∂F n ∂n

on Γ.

This is the orthogonal projection P (x)∇F (x) of ∇F (x) onto the tangent plane Tx Γ to Γ at x. To get something intrinsic, g(F ) must be independent of the choice of F . It is sufficient to show that g(F ) = 0 for f = 0. But F = f = 0 on Γ and the tangential component of ∇F is 0 on Γ, and ∇F |Γ =

∂F n ∂n

⇒ g(F ) = ∇F |Γ −

∂F n = 0. ∂n

5. Elements of Tangential Calculus

493

Definition 5.1 (General extension). Assume that Γ is compact and that there exists h > 0 such that bΩ ∈ C 2 (S2h (Γ)). Given an extension F ∈ C 1 (S2h (Γ)) of f ∈ C 1 (Γ), the tangential gradient of f in a point of Γ is defined as def

∇Γ f = ∇F |Γ −

∂F n. ∂n

The notation is quite natural. The subscript Γ of ∇Γ f indicates that the gradient is with respect to the variable x in the submanifold Γ. Theorem 5.1. Assume that Γ is compact and that there exists h > 0 such that bΩ ∈ C 2 (S2h (Γ)) and that f ∈ C 1 (Γ). Then (i) ∇Γ f = (P ∇F )|Γ and n · ∇Γ f = ∇b · ∇Γ f = 0; (ii) ∇(f ◦ p) = [I − b D 2 b] ∇Γ f ◦ p and ∇(f ◦ p)|Γ = ∇Γ f . Proof. (i) By definition P ∇F = (I − ∇b ∗∇b) ∇F = ∇F − ∇F · ∇b ∇b, ∂F (P ∇F )|Γ = (∇F )|Γ − n = ∇Γ f. ∂n Moreover ∇b · P ∇F = (I − ∇b ∗∇b) ∇b · ∇F = 0 ⇒ (∇b)|Γ · (P ∇F )|Γ = 0 ⇒ n · ∇Γ f = 0. (ii) F = f ◦ p is a C 1 -extension of f and ∇(f ◦ p) = ∇(f ◦ p ◦ p) = Dp ∇(f ◦ p) ◦ p = [I − ∇b ∗∇b − b D2 b] ∇(f ◦ p)|Γ ◦ p. But by definition of ∇Γ f ∂(f ◦ p) n   ∂n ∂(f ◦ p) ∗ 2 ⇒ ∇(f ◦ p) = [I − ∇b ∇b − b D b] ∇Γ f + n ◦ p. ∂n Recall, from Theorem 8.4 (i) in Chapter 7, that n ◦ p = ∇b so that   ∂(f ◦ p) ∗ 2 ∇(f ◦ p) = [I − ∇b ∇b − b D b] ∇Γ f ◦ p + ∇b ∂n ∇(f ◦ p)|Γ = ∇Γ f +

= [I − ∇b ∗∇b − b D2 b] ∇Γ f ◦ p. Again ∇b · ∇Γ f ◦ p = n ◦ p · ∇Γ f ◦ p = (n · ∇Γ f ) ◦ p = 0 from part (i) and ∇(f ◦ p) = [I − bD2 b] ∇Γ f ◦ p

⇒ ∇(f ◦ p)|Γ = ∇Γ f.

494

Chapter 9. Shape and Tangential Differential Calculuses

In view of part (ii) of the theorem f ◦ p plays the role of a canonical extension of a map f : Γ → R to a neighborhood S2h (Γ) of Γ and its gradient is tangent to the level sets of b. This suggests the use of the following definition of tangential gradient, which will be the clue to the tangential differential calculus. Definition 5.2 (Canonical extension). Under the assumptions of Definition 5.1 on bΩ , associate with f ∈ C 1 (Γ) def

∇Γ f = ∇(f ◦ p)|Γ .

(5.1)

Remark 5.1. This definition naturally extends to nonempty sets A such that d2A belongs to C 1,1 (S2h (A)), since the projection onto A, 1 pA (x) = x − ∇d2A (x), 2 is C 0,1 . They are the sets of positive reach introduced by Federer. They include convex sets and submanifolds of codimension larger than or equal to 1. Theorem 5.2. Under the assumption of Definition 5.1 on bΩ for f ∈ C 1 (Γ) the following hold: (i) ∇b · ∇(f ◦ p) = 0 in Sh (Γ) and n · ∇Γ f = 0 on Γ. (ii) ∇F |Γ −

∂F ∂n

n = (P ∇F )|Γ = ∇Γ f and ∇(f ◦ p) = [I − b D 2 b] ∇Γ f ◦ p in Γ.

Proof. (i) Consider ∇(f ◦ p) · ∇b = ∇(f ◦ p ◦ p) · ∇b = Dp ∇(f ◦ p) ◦ p · ∇b = ∇(f ◦ p) ◦ p · Dp ∇b. But

Dp ∇b = [I − ∇b ∗∇b − b D2 b] ∇b = 0

and ∇(f ◦ p) · ∇b = 0, (ii) By definition

∇Γ f · n = ∇(f ◦ p)|Γ · ∇b|Γ = 0.

P ∇F = (I − ∇b ∗∇b) ∇F.

However, since F ◦ p = f ◦ p on Γ ∇(F ◦ p) = ∇(F ◦ p ◦ p) = Dp ∇(F ◦ p) ◦ p = Dp ∇(f ◦ p) ◦ p = Dp ∇Γ f ◦ p and ∇(F ◦ p) = Dp ∇F ◦ p

⇒ Dp ∇F ◦ p = Dp ∇Γ f ◦ p.

By restricting to Γ Dp|Γ ∇F |Γ = Dp|Γ ∇Γ f

⇒ P ∇F |Γ = P ∇Γ f = ∇Γ f

5. Elements of Tangential Calculus

5.2

495

First-Order Derivatives

The tangential Jacobian matrix of a vector function v ∈ C 1 (Γ)M , M ≥ 1, is defined in the same way as the gradient: def

DΓ v = D(v ◦ p)|Γ or (DΓ v)ij = (∇Γ vi )j .

(5.2)

If ∗v = (v1 , . . . , vM ), then ∗

DΓ v = (∇Γ v1 , . . . , ∇Γ vM ),

where ∇Γ vi is a column vector. From the previous theorems about the tangential gradient we can recover the definition from an extension V ∈ C 1 (S2h (Γ))M of v: ∗

DΓ v = (P ∇V1 , . . . , P ∇VM )|Γ = (I − ∇b ∗∇b) ∗DV |Γ = ∗DV |Γ − ∇b ∗(DV ∇b)|Γ

and DΓ v = DV |Γ − DV n ∗n = (DV P )|Γ .

(5.3)

Also, ∗

D(v ◦ p) = [I − b D2 b] (∇Γ v1 , . . . , ∇Γ vM ) ◦ p = [I − b D 2 b] ∗DΓ v ◦ p,

and we have for the extension D(v ◦ p) = DΓ v ◦ p [I − b D2 b].

(5.4)

D(v ◦ p) ∇b = 0 and DΓ v n = 0.

(5.5)

Note that

For a vector function v ∈ C 1 (Γ)N define the tangential divergence as def

divΓ v = div (v ◦ p)|Γ ,

(5.6)

and it is easy to show that divΓ v = div (v ◦ p)|Γ = tr D(v ◦ p)|Γ = tr DΓ v, divΓ v = tr [DV |Γ − DV n ∗n] = div V |Γ − DV n · n. The tangential linear strain tensor of linear elasticity is given by def

εΓ (v) =

1 (DΓ v + ∗DΓ v), 2

1 (D(v ◦ p) + ∗D(v ◦ p)) 2 b = εΓ (v) ◦ p − [DΓ v ◦ p D2 b + D2 b ∗DΓ v ◦ p] 2 ⇒ εΓ (v) = ε(v ◦ p)|Γ .

(5.7)

ε(v ◦ p) =

(5.8)

(5.9)

496

Chapter 9. Shape and Tangential Differential Calculuses

The tangential Jacobian matrix of the normal n is especially interesting since, from Theorem 8.4 (i) in Chapter 7, n ◦ p = ∇b = ∇b ◦ p. As a result DΓ (n) = D 2 b|Γ = ∗DΓ (n)

⇒ εΓ (n) = D 2 b|Γ .

(5.10)

The tangential vectorial divergence of a matrix or tensor function A is defined as def

˜ Γ A) = div (Ai,· ) . (div Γ i

5.3

(5.11)

Second-Order Derivatives

Assume that Ω is of class C 3 . The simplest second-order derivative is the Laplace– Beltrami operator of a function f ∈ C 2 (Γ), which is defined as def

∆Γ f = div Γ (∇Γ f ).

(5.12)

Recall from Theorem 5.2 the identity ∇(f ◦ p) = [I − b D2 b] ∇Γ f ◦ p. Then div (∇Γ f ◦ p) = div (∇(f ◦ p)) + div (b D2 b ∇Γ f ◦ p), div (∇Γ f ◦ p) = ∆(f ◦ p) + b div (D2 b ∇Γ f ◦ p) + (D 2 b ∇Γ f ◦ p) · ∇b, div (∇Γ f ◦ p) = ∆(f ◦ p) + b div (D 2 b ∇Γ f ◦ p), and by taking restrictions to Γ, ∆Γ f = ∆(f ◦ p)|Γ . The tangential Hessian matrix of second-order derivatives is defined as def

DΓ2 f = DΓ (∇Γ f ).

(5.13)

Here the curvatures of the submanifold begin to appear. The Hessian matrix is not symmetrical and does not coincide with the restriction of the Hessian matrix of the canonical extension. Specifically, D2 (f ◦ p) = D(∇(f ◦ p)) = D([I − b D2 b] ∇Γ f ◦ p) = D(∇Γ f ◦ p) − D(b D2 b ∇Γ f ◦ p) = D(∇Γ f ◦ p) − b D(D 2 b ∇Γ f ◦ p) − D2 b ∇Γ f ◦ p ∗∇b, D2 (f ◦ p)|Γ = DΓ (∇Γ f ) − D 2 b ∇Γ f ∗∇b = DΓ2 f − D 2 b ∇Γ f ∗n.

5. Elements of Tangential Calculus

497

Of course, since D2 (f ◦ p) is symmetrical we also have D2 (f ◦ p) = ∗D(∇(f ◦ p)) = ∗D([I − b D 2 b] ∇Γ f ◦ p) = ∗D(∇Γ f ◦ p) − ∗D(b D 2 b ∇Γ f ◦ p) = ∗D(∇Γ f ◦ p) − b ∗D(D2 b ∇Γ f ◦ p) − ∇b ∗(D2 b ∇Γ f ◦ p), D 2 (f ◦ p)|Γ = ∗DΓ (∇Γ f ) − ∇b ∗(D2 b ∇Γ f ) = ∗DΓ2 f − n ∗(D2 b ∇Γ f ). As a final result we have the following identity: DΓ2 f − (D2 b ∇Γ f ) ∗n = D2 (f ◦ p)|Γ = ∗DΓ2 f − n ∗(D2 b ∇Γ f ),

(5.14)

since by definition ∗DΓ2 f = ∗ (DΓ2 f ). So the Hessian and its transpose differ by terms that contain a first-order derivative as is well known in differential geometry. Note that D2 b ∇Γ f = −∗DΓ (∇Γ f )n = −∗DΓ2 f n and that we also can write DΓ2 f + ∗DΓ2 f [n ∗n] = D 2 (f ◦ p)|Γ = ∗DΓ2 f + [n ∗n] DΓ2 f ⇒ P DΓ2 f = ∗(P DΓ2 f ).

5.4

(5.15)

A Few Useful Formulae and the Chain Rule

Associate with F ∈ C 1 (S2h (Γ)) and V ∈ C 1 (S2h (Γ))N def

f = F |Γ ,

def

v = V |Γ ,

def

vn = v · n,

def

vΓ = v − vn n,

(5.16)

where vΓ and vn are the respective tangential part and the normal component of v. In view of the previous definitions the following identities are easy to check: ∂F n, ∂n DV |Γ = DΓ v + DV n ∗n, div V |Γ = divΓ v + DV n · n, ∇F |Γ = ∇Γ f +

DΓ v n = 0,

2

D b n = 0.

(5.17) (5.18) (5.19) (5.20)

Decomposing v into its tangential part and its normal component, DΓ v = DΓ vΓ + vn D 2 b + n ∗∇Γ vn , divΓ v = divΓ vΓ + ∆b vn = divΓ vΓ + H vn ,

(5.21) (5.22)

∇Γ vn = ∗DΓ v n + D2 b vΓ .

(5.23)

Given f ∈ C 1 (Γ) and g ∈ C 1 (Γ; Γ), consider the canonical extensions f ◦ p ∈ C (S2h (Γ)) and g ◦ p ∈ C 1 (S2h (Γ); RN ) and the gradient of the composition 1

∇(f ◦ p ◦ g ◦ p) = ∗D(g ◦ p) ∇(f ◦ p) ◦ g ◦ p ⇒ ∇Γ (f ◦ g) = ∗DΓ g ∇Γ f ◦ g,

(5.24)

and for a vector-valued function v ∈ C 1 (Γ; RN ), DΓ (v ◦ g) = DΓ v ◦ g DΓ g.

(5.25)

498

5.5

Chapter 9. Shape and Tangential Differential Calculuses

The Stokes and Green Formulae

One interesting application of the shape calculus in connection with the tangential calculus is the tangential Stokes formula. Given v ∈ C 1 (Γ)N , consider Stokes formula in RN for the vector function v ◦ p:   div (v ◦ p) dx = v · n dΓ. Ω

Γ

Given an autonomous velocity field V , differentiate both sides of Stokes’s formula with respect to t,    div (v ◦ p) dx = (v ◦ p) · nt = (v ◦ p) · Nt dΓt , Ωt (V )

Γt (V )

Γt (V )

where Nt is the extension (4.37) of nt . This gives the new identity     ∂ [(v ◦ p) · ∇b] + H v · n V · n dΓ. div (v ◦ p) V · n dΓ = v · N + ∂n Γ Γ Now choose the velocity field V = ∇b ψ, where ψ ∈ D(S2h (Γ)) is chosen in such a way that ψ = 1 on Sh (Γ). Using expression (4.38) for N  , V · n = n · n = 1, 

div (v ◦ p)|Γ = divΓ v, ∗

N |Γ = DV n · n n − DV n − D2 b V = D2 b∇b · ∇b ∇b − D 2 b∇b − D 2 b∇b = 0, ∇((v ◦ p) · ∇b) = D 2 b v ◦ p + ∗D(v ◦ p)∇b,

 ∂ [(v ◦ p) · ∇b] = D2 b v + ∗DΓ v n · n = v · D2 b∇b + n · DΓ v n = 0. ∂n Finally we get the tangential Stokes formula with H = ∆b: 

 H v · n dΓ.

divΓ v dΓ = Γ

(5.26)

Γ

For a function f ∈ C 1 (Γ) and a vector v ∈ C 1 (Γ)N , the above formula also yields the tangential Green’s formula 

 f divΓ v + ∇Γ f · v dΓ =

Γ

5.6

H f v · n dΓ.

(5.27)

Γ

Relation between Tangential and Covariant Derivatives

One of the simplest examples of a tangential derivative is when Ω is the half-space  def

H+ = ζ ∈ RN : ζ · eN > 0

5. Elements of Tangential Calculus

499

for some orthonormal basis {e1 , . . . , eN } in RN and Γ is the boundary of H+ denoted by  def

H = ζ ∈ RN : ζ · eN = 0 . def

If pH denotes the projection onto H, then pH (ζ) = pH (ζ  , ζN ) = ζ  = (ζ1 , . . . , ζN −1 ) ∈ H, and for any function ϕ ∈ C 1 (H) the tangential gradient ∇H ϕ = ∇(ϕ ◦ pH )|H coincides with the gradient of ϕ in H: ∇H ϕ · eα =

∂ϕ , ∂ζα

1 ≤ α ≤ N − 1.

With the notation of section 3.1 of Chapter 2, consider a set Ω locally of class C 2 in RN . Its boundary Γ = ∂Ω is an (N − 1)-dimensional submanifold of RN of class C 2 . At each point x ∈ Γ, there is a C 2 -diffeomorphism hx from the open unit ball B onto a neighborhood U (x) of x such that def

hx (B0 ) = Γx = Γ ∩ U (x),

hx (B+ ) = Ω ∩ U (x),

where B+ = H+ ∩ B and B0 = H ∩ B and H+ and H are as defined above for some appropriate orthonormal basis. def For a function f ∈ C 1 (Γ), f ◦ Φ ∈ C 1 (B0 ) with Φ = hx |B0 : B0 → Γ. Observing that f ◦ Φ ◦ pH = f ◦ p ◦ Φ ◦ pH , we have ∇(f ◦ Φ ◦ pH ) = ∇(f ◦ p ◦ Φ ◦ pH ), ∇(f ◦ Φ ◦ pH ) = ∗D(Φ ◦ pH )∇(f ◦ p) ◦ Φ ◦ pH , ∇H (f ◦ Φ) = ∗DH Φ ∇Γ f ◦ Φ. The covariant basis associated with Φ is defined as ∂Φ aα = , ∂ζα def

1 ≤ α ≤ N − 1,

aN

(Dhx )−1 eN = ∗ , | (Dhx )−1 eN | B0

def

∗

where from identity (3.9) of section 3.1 in Chapter 2 we know that aN ◦ Φ−1 is the inward unit normal to Ω, that is, n = −aN ◦ Φ−1 . The covariant partial derivatives of f are defined as def

fα =

∂(f ◦ Φ) , ∂ζα

1 ≤ α ≤ N − 1.

(5.28)

500

Chapter 9. Shape and Tangential Differential Calculuses

But from the previous observations aα =

∂Φ = DH Φeα , ∂ζα

fα =

∂(f ◦ Φ) = eα · ∇H (f ◦ Φ), ∂ζα

f ◦ Φ ◦ pH = f ◦ p ◦ Φ ◦ pH , and ∇(f ◦ Φ ◦ pH ) = ∇(f ◦ p ◦ Φ ◦ pH ) = ∗D(Φ ◦ pH )∇(f ◦ p) ◦ Φ ◦ pH , ∇H (f ◦ Φ) = ∗DH (Φ) ∇Γ f ◦ Φ, fα = eα · ∇H (f ◦ Φ) = eα · ∗DH Φ ∇Γ f ◦ Φ = DH Φ eα · ∇Γ f ◦ Φ = a α · ∇Γ f ◦ Φ ⇒ fα = aα · ∇Γ f ◦ Φ = eα · ∇H (f ◦ Φ). For a vector function v ∈ C 1 (Γ; RN ) def

v,α =

∂(v ◦ Φ) , ∂ζα

1 ≤ α ≤ N − 1.

Again, v ◦ Φ ◦ pH = v ◦ p ◦ Φ ◦ pH and D(v ◦ Φ ◦ pH ) = D(v ◦ p ◦ Φ ◦ pH ) = D(v ◦ p) ◦ Φ ◦ pH D(Φ ◦ pH ), DH (v ◦ Φ) = DΓ v ◦ Φ DH Φ

⇒ v,α = DH (v ◦ Φ)eα = DΓ v ◦ Φ aα .

The second fundamental form of Γ is defined from the inward normal aN , def

bαβ = −aα · aN,β , where from (5.28) aN = −n ◦ Φ, and from (5.10) DΓ n = D2 b|Γ , DH aN = −DΓ n ◦ Φ DH Φ = −D2 b ◦ Φ DH Φ, aN,β = −DH aN eβ = −D2 b ◦ Φ DH Φ eβ = −D2 b ◦ Φ aβ , bαβ = aα · D2 b ◦ Φ aβ . Recall that since |∇b| = 1, D2 b∇b = 0 and ∇b is an eigenvector for the eigenvalue 0. As a result the eigenvalues of D2 b at a point of Γ are the eigenvalues of the second fundamental form bαβ (the N − 1 principal curvatures) plus {0}. In particular, ∆b = tr D2 b = H = (N − 1)H, where H is the mean curvature of Γ and H the additive curvature (cf. section 3.3 of Chapter 2).

6. Second-Order Semiderivative and Shape Hessian

5.7

501

Back to the Example of Section 4.3.3

Coming back to formula (4.36) for dJ(Ω; V (0)) of the boundary integral of the square of the normal derivative in section 4.3.3, the tangential calculus is now used on the term    ∗  ∂φ ∂φ 2 DV (0)n · n − ∇φ · DV (0) n + D b V (0) 2 dΓ, ∂n Γ ∂n   def ∂φ DV (0)n · n − ∇φ · ∗DV (0) n + D2 b V (0) . T = ∂n From identity (5.18) with v = V (0)|Γ and identity (5.23), ∗

DV (0)n = ∗DΓ v n + DV (0)n · n n, ∂φ ∇φ · ∗DV (0)n = ∇φ · ∗DΓ v n + DV (0)n · n ∂n 

∗ ⇒ T = −∇φ · DΓ v n + D2 b vΓ = −∇φ · ∇Γ vn = −∇Γ φ · ∇Γ vn . Therefore by using the tangential Stokes formula (5.26),   ∂φ ∂φ T dΓ = − 2 ∇Γ φ · ∇Γ vn dΓ 2 Γ ∂n Γ ∂n      ∂φ ∂φ vn ∇Γ φ + 2 divΓ ∇Γ φ vn dΓ −2 divΓ = ∂n ∂n Γ    ∂φ ∂φ vn ∇Γ φ · n + 2 divΓ ∇Γ φ vn dΓ = −2H ∂n ∂n Γ       ∂φ ∂φ ∇Γ φ vn dΓ = ∇Γ φ V (0) · n dΓ. = 2 divΓ 2 divΓ ∂n ∂n Γ Γ Substituting into expression (4.36), we finally get the explicit formula in terms of V (0) · n as predicted by the structure theorem dJ(Ω; V ) 2  &  ∂φ ∂φ 2 ∂φ 2 D φ n · n + H + 2 divΓ ∇Γ φ V (0) · n dΓ. = ∂n ∂n ∂n Γ

6

(5.29)

Second-Order Semiderivative and Shape Hessian

The object of this section is to study second-order derivatives and semiderivatives by the velocity method for smooth and nonsmooth domains and discuss their relationship with the method of perturbations of the identity in the unconstrained case. The analysis of this section is motivated by first computing the second-order derivative of a domain integral by using the combined strengths of the shape and tangential calculuses in section 6.1. A basic formula for the second-order semiderivative of the domain integral is given in section 6.2, which also reveals the general structure

502

Chapter 9. Shape and Tangential Differential Calculuses

of this derivative. Structure theorems for the second-order Eulerian semiderivative d2 J(Ω; V ; W ) of a function J(Ω) for two vector fields V and W are given in section 6.3 and section 6.4. A first theorem shows that under some natural continuity assumptions,     d2 J(Ω; V ; W ) = d2 J Ω; V (0); W (0) + dJ Ω; V  (0) , where V  (0) is the time-partial derivative ∂t V (t, x) at t = 0. As in the study of first-order Eulerian semiderivatives, this first theorem reduces the study of secondorder Eulerian semiderivatives to the autonomous case. So we then specialize to N fields in Dk (D,  R ) and give  the equivalent of Hadamard’s structure theorem for the term d2 J Ω; V (0); W (0) . This bilinear term is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket [V (0), W (0)] in section 6.5. The symmetrical part is itself decomposed into a symmetrical part that depends only on the normal component of the velocity fields and a symmetrical term made up of the gradient acting on a generic group of terms, which occurs in all examples considered in this section.

6.1

Second-Order Derivative of the Domain Integral

2 Given f ∈ Cloc (RN ) and a domain Ω of class C 2 , consider the function  def J(Ωt,s (V, W )) = f dx,

(6.1)

Ωt,s (V,W )

where def

Ωt,s (V, W ) = Ts (W )(Ωt (V )) = Ts (W )(Tt (V )(Ω)), def

JV,W (t, s) = J(Ωt,s (V, W )) for some pair of autonomous velocity fields V and W satisfying condition (3.1) and additional smoothness conditions as necessary. The objective is to compute 2

d JV,W

∂ = ∂s

def



 ∂ . JV,W (t, s) ∂t t=0 s=0

From formula (4.7) in Theorem 4.2, we already know that   ∂ JV,W (t, s) = dJ(Ωs (W ); V ) = f V · ns dΓs = f V · Ns dΓs , ∂t Γs (W ) Γs (W ) t=0 where Ns = Ns (W ) is the extension (4.37) of the normal ns . So we can readily use formula (4.17) from Theorem 4.3: 

f V · N  (W ) +

d2 JV,W = Γ





∂ (f V · ∇b) + H f V · n ∂n

W · n dΓ,

(6.2)

6. Second-Order Semiderivative and Shape Hessian

503

where N  (W ) is the derivative of the extension Ns = Ns (W ) given by expression (4.38). This yields d2 JV,W 

 = f V · (DW n · n) n − ∗DW − D2 b W   Γ ∂ + (f V · ∇b) + H f V · n W · n dΓ ∂n   

 ∂ ∗ 2 = f V · (DW n · n) n − DW n − D b W + (V · ∇b) W · n ∂n Γ   ∂f + H f V · n W · n dΓ. + ∂n It remains to untangle the following term in the first part of the integral:

 ∂ def T = V · (DW n · n) n − ∗DW n − D2 b W + (V · ∇b) W · n. ∂n Using the notation v = V |Γ and w = W |Γ , ∇(V · ∇b) = ∗DV ∇b + D2 b V, ∇(V · ∇b) · ∇b = ∗DV ∇b · ∇b + D2 b V · ∇b = DV ∇b · ∇b ∂ (V · ∇b) = DV n · n, ⇒ ∂n V · ∗DW n = V · [ ∗DΓ w + n ∗(DW n)] n

 = V · ∇Γ wn − D2 b w + DW n · n V · n, 

V · ∗DW n + D2 b W = ∇Γ wn · vΓ + DW n · n vn . Finally, T = DW n · n vn − (vΓ · ∇Γ wn + DW n · n vn ) + DV n · n wn = DV n · n wn − vΓ · ∇Γ wn , and by using the tangential Stokes formula (5.26) 



f {DV n · n wn − vΓ · ∇Γ wn } dΓ

f T dΓ = Γ

Γ

= Γ = Γ

f DV n · n wn − divΓ (f wm vΓ ) + divΓ (f vΓ ) wn dΓ {f DV n · n + divΓ (f vΓ )} wn − f wn vΓ · n dΓ {f (DV n · n + divΓ vΓ ) + ∇Γ f · vΓ } wn dΓ.

= Γ

504

Chapter 9. Shape and Tangential Differential Calculuses

Finally, we get two equivalent expressions: d2 JV,W    ∂f = + H f vn wn + f (DV n · n wn − vΓ · ∇Γ wn ) dΓ ∂n Γ     ∂f + H f vn + f (DV n · n + divΓ vΓ ) + ∇Γ f · vΓ wn dΓ. = ∂n Γ

(6.3)

Since the above expressions involved the composition Ts (W ) ◦ Tt (V ), it is expected that the condition for the symmetry of expression (6.3) will involve the Lie bracket [V, W ] = DV W − DW V . Indeed, by using identity (5.23) DV W · n = (DΓ v + DV n ∗n} W · n = w · ∗DΓ v n + DV n · n wn   = wΓ · ∇Γ vn − D2 b vΓ + DV n · n wn and substituting in the first expression, we get a symmetrical term plus the first half of the Lie bracket: d2 JV,W      ∂f = + H f vn wn + f D2 b vΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn ∂n Γ + f DV W · n dΓ.

(6.4)

Thus  d2V,W

=

d2W,V

⇐⇒

f [V, W ] · n dΓ = 0,

(6.5)

Γ

from which either f [V, W ] · n = 0 on Γ or div (f [V, W ]) = 0 on Ω can be used as sufficient conditions. Example 6.1. Let Ω = {(x, y) : x2 + y 2 < 1} be the unit ball in R2 with boundary Γ = {(x, y) : x2 + y 2 = 1}. Choose ' V (x, y) = (1, 0), W (x, y) = (x2 /2, 0), n = (x, y)/ x2 + y 2 ' ⇒ DV W − DW V = (x, 0) ⇒ [DV W − DW V ] · n = x2 / x2 + y 2 = x2 .

6.2

Basic Formula for Domain Integrals

We have proved the following result. 2 Theorem 6.1. Let f ∈ C 2 ([0, τ ] × [0, τ ]; Hloc (RN )) and let Γ be the boundary of a 2 (RN , RN )) bounded open subset Ω of RN of class C 2 . Assume that V ∈ C 0 ([0, τ ]; Cloc

6. Second-Order Semiderivative and Shape Hessian

505

1 (RN , RN )). Consider the function and that W ∈ C 0 ([0, τ ]; Cloc def



JV,W (t, s) =

f (t, s) dx.

(6.6)

Ωt,s (V,W )

Then the partial derivative of JV,W (t, s) with respect to t in t = 0 is given by  ∂ ∂f = JV,W (t, s) (0, s) + div (f (0, s)V (0)) dx ∂t ∂t Ω (W ) t=0 s (6.7)   ∂f (0, s) dx + f (0, s) V (0) · ns dΓs = Ωs (W ) ∂t Γs (W ) and the second-order mixed derivative of JV,W (t, s) in (t, s) = (0, 0),   ∂ def ∂ 2 d JV,W = JV,W (t, s) , ∂s ∂t t=0 s=0

(6.8)

is given by the expression 

2

d JV,W

∂ = Ω ∂s 

∂ = Ω ∂s





∂f ∂t ∂f ∂t



 + div

 

 dx +

Γ



∂f ∂f V (0) + W (0) ∂s ∂t

+ div [div (f V (0)) W (0)] dx  ∂f ∂f V (0) + W (0) · n ∂s ∂t

(6.9)

+ div (f V (0)) W (0) · n dΓ. The last term in the second integral can be expressed in terms of v = V (0)|Γ and w = W (0)|Γ as follows:  div (f V (0)) W (0) · n dΓ     ∂f + H f vn + f (DV n · n + divΓ vΓ ) + ∇Γ f · vΓ wn dΓ = ∂n Γ      ∂f + H f vn wn + f D2 bvΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn = ∂n Γ + f DV W · n dΓ. Γ

6.3

(6.10)

Nonautonomous Case

The framework introduced in sections 4 and 5 of Chapter 4 has reduced the computation of the Eulerian semiderivative of J(Ω) to the computation of the derivative of the function  def  (6.11) j(t) = J Ωt (V )

506

Chapter 9. Shape and Tangential Differential Calculuses

k (RN ; RN )). In t ≥ 0 for a velocity field V ∈ C([0, τ ]; Cloc   j  (t) = dJ Ωt (V ); Vt

(6.12)

def

since Ts+t (V ) = Ts (Vt ) ◦ Tt (V ), where Vt (s) = V (t + s) and Vt (0) = V (t). This suggests the following definition. Definition 6.1. Let J be a real-valued shape functional. Let V and W satisfy conditions (V)  (conditions (4.2) of Chapter 4) and assume that for all t ∈ [0, τ ], dJ Ωt (W ); Vt exists for Ωt (W ) = Tt (W )(Ω). The function J is said to have a second-order Eulerian semiderivative at Ω in the directions (V, W ) if the following limit exists: lim

t0

dJ(Ωt (W ); Vt ) − dJ(Ω; V ) . t

(6.13)

When it exists, it is denoted d2 J(Ω; V ; W ). If, for all t, J has a Hadamard semiderivative at Ωt (W ), recall that dJ(Ωt (W ); Vt ) =dH J(Ωt (W ); Vt (0)) =dH J(Ωt (W ); V (t)) = dJ(Ωt (W ); V (t)), and the above definition reduces to     dJ Ωt (W ); V (t) − dJ Ω; V (0) d J(Ω; V ; W ) = lim . t0 t 2

Remark 6.1. This last definition is compatible with the second-order expansion of j(t) with respect to t around t = 0: j(t) ∼ = j(0) + tj  (0) +

t2  j (0), 2

(6.14)

j  (0) = d2 J(Ω; V ; V ).

(6.15)

where j(0) = J(Ω),

j  (0) = dJ(Ω; V ),

The next theorem is the analogue of Theorem 3.2 and provides the canonical structure of the second-order Eulerian semiderivative (cf. (3.20) to (3.22) in → − section 3.1 for the definitions of V m,
and V
). Theorem 6.2. Let J be a real-valued shape functional, Ω be a subset of RN , and m ≥ 0 and  ≥ 0 be two integers. Assume that → − → − (i) ∀V ∈ V m+1,
, ∀W ∈ V m,
,

d2 J(Ω; V ; W ) exists;

6. Second-Order Semiderivative and Shape Hessian

507

→ − (ii) ∀W ∈ V m,
, ∀t ∈ [0, τ ], J has a shape gradient of order  and is Hadamard semidifferentiable at Ωt (W ); (iii) ∀U ∈ V
, the map → − W → d2 J(Ω; U ; W ) : V m,
→ R

(6.16)

is continuous. → − → − Then for all V in V m+1,
and all W in V m,
,     d2 J(Ω; V ; W ) = d2 J Ω; V (0); W (0) + dJ Ω; V  (0) ,

(6.17)

where V (t, x) − V (0, x) . t0 t

V  (0)(x) = lim def

(6.18)

Proof. The differential quotient (6.13) can be split into the sum of two terms:         dJ Ωt (W ); V (t) − dJ Ωt (W ); V (0) dJ Ωt (W ); V (0) − dJ Ω; V (0) + . (6.19) t t In view of (i) and (iii), for all U in V
,   d2 J(Ω; U ; W ) = d2 J Ω; U ; W (0) by the same argument as in the proof of Theorem 3.2 for the gradient. Hence the first term converges to     d2 J Ω; V (0); W = d2 J Ω; V (0); W (0) . → − For the second term recall that V belongs to V m+1,
and observe that the vector field V (t) − V (0) V= (t) = t → − belongs to V m,
and that V= (0) = V  (0). Thus by linearity of dJ(Ω; V ), the second term in (6.19) can be written as     V (t) − V (0) dJ Ωt (W ); = dJ Ωt (W ); V= (t) , t .     /    1 dJ Ωt (W ); V= (t) − dJ Ω; V= (0) + dJ Ω; V= (0) . dJ Ωt (W ); V= (t) = t t

508

Chapter 9. Shape and Tangential Differential Calculuses

→ − → − But for any V in V m+2,
, V= belongs to V m+1,
. Then by assumption (i),     dJ Ωt (W ); V= (t) − dJ Ω; V= (0) lim = d2 J(Ω; V= ; W ), t0 t which implies that       lim dJ Ωt (W ); V= (t) = dJ Ω; V= (0) = dJ Ω; V  (0) .

t0

Now by assumption (ii), the map U → dJ(Ω; U ) is linear and continuous on D
(RN , RN ), and the map   − → V → V  (0) → dJ Ω; V  (0) : V m+2,
→ V
→ R → − is linear and continuous (hence uniformly continuous) for the topology V m+1,
for → − all V in the dense subspace V m+2,
. Hence it uniquely and continuously extends →m+1,
− . This completes the proof of the theorem. to all elements of V This important theorem gives the canonical structure of the second-order Eulerian semiderivative: a first depends on V (0) and W (0) and a sec  term that the ond term that is equal to dJ Ω; V  (0) . When V is autonomous   second term disappears and the semiderivative coincides with d2 J Ω; V ; W (0) , which can be separately studied for autonomous vector fields in V
. We conclude this section with the explicit computation of the second-order Eulerian semiderivative for a shape function J(Ω) with respect to two velocity fields V and W satisfying the conditions of Theorem 6.2 and such that the shape gradient at any t is of the form  dJ(Ωt (W ); V (t)) = g(t) V (t) · nt dΓt (6.20) Γt (W )

for some function g(t) ∈ C(Γt (W )). Further assume that the family of functions g(t) k has an extension Q ∈ C 1 ([0, τ ]; Cloc (N (Γ); RN )) to an open neighborhood N (Γ) of Γ such that ∪ {Γt (W ) : 0 ≤ t ≤ τ } ⊂ N (Γ). Therefore using the extension Nt (W ) of the normal nt on Γt (W ), it amounts to differentiating the expression  Q(t) V (t) · Nt (W ) dΓt . j(t) = Γt (W )

Apply the first formula (4.17) of Theorem 4.3 to get  (QW (0) V (0) + Q(0) V  (0)) · n + Q(0) V (0) · N  (W ) j  (0) =   Γ ∂ + (Q(0) V (0) · ∇b) + H Q(0) V (0) · n W (0) · n dΓ, ∂n where QW (0) depends only on W . But the last three terms have already been computed in several forms. They constitute expression (6.2) in section 6.1, which

6. Second-Order Semiderivative and Shape Hessian

509

yields (6.3) and (6.4) with f = Q(0), V = V (0), and W = W (0). This yields with the notation v = V (0)|Γ and w = W (0)|Γ   ∂Q(0) + H Q(0) vn wn QW (0) vn + Q(0) V  (0) · n + ∂n Γ   2 + Q(0) D b vΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn + Q(0) DV W · n dΓ    ∂Q(0) = QW (0) vn + Q(0) V  (0) · n + + H Q(0) vn wn ∂n Γ 

d2 J(V ; W ) =

(6.21)

+ {Q(0) (DV n · n + divΓ vΓ ) + ∇Γ Q(0) · vΓ } wn dΓ. Remark 6.2. When V is autonomous the term in V  (0) disappears. In that case the first half of the Lie bracket can be eliminated by restarting the computation with V(t) = V ◦Tt−1 (W ) in place of V since V(0) = V and  V  (0) = −DV W ⇒ Q(0) V  (0) · n + Q(0) DV W · n dΓ = 0 Γ

⇒ d2 J(V, W ) = d2 J(V, W ) + dJ(Ω; DV W ). Remark 6.3. Except for the terms that contain the first half of the Lie bracket DV W and V  (0), the only term that might not be symmetrical in the first expression (6.21) is the one in QW (0). In fact, according to the second expression and our theorem, QW (0) = QW (0) depends only on W (0). Now choose autonomous velocity fields V and W . Furthermore, assume that W is of the form W = wΓ ◦ p. Since W · n = 0 on Γ, dJ(Ωt (W ); V (0)) = dJ(Ω; V (0)) and necessarily d2 J(Ω; V ; wΓ ◦ p) = 0. Therefore, d2 J(Ω; V ; W ) depends only on wn and hence QW (0) = Qwn (0) and the integral  Qwn (0) vn dΓ Γ

depends only on wn and vn . The above expressions give valuable information on the structure of the secondorder Eulerian derivative. Other expressions can also be obtained. For instance, if j(t) is transformed into the volume integral   Q(t) V (t) · nt dΓt = div (Q(t) V (t)) dx, j(t) = dJ(Ωt (W ); V (t)) = Γt (W )

Ωt (W )

we get from formula (4.6) in Theorem 4.2 the equivalent volume expression d2 J(Ω; V ; W )    = div QW (0) (0) V (0) + Q(0) V  (0) + div (Q(0) V (0)) W (0) dx, Ω

which can obviously be transformed into a boundary expression.

510

6.4

Chapter 9. Shape and Tangential Differential Calculuses

Autonomous Case

Definition 6.2. Let J be a real-valued shape functional. Let Ω be a subset of RN . (i) The function J(Ω) is said to be twice shape differentiable at Ω if ∀V, ∀W ∈ D(RN , RN ),

d2 J(Ω; V ; W ) exists

(6.22)

and the map (V, W ) → d2 J(Ω; V ; W ) : D(RN , RN ) × D(RN , RN ) → R

(6.23)

is bilinear and continuous. We denote by h the map (6.23). (ii) Denote by H(Ω) the vector distribution in (D(RN , RN ) ⊗ D(RN , RN )) associated with h: d2 J(Ω; V ; W ) = H(Ω), V ⊗ W  = h(V, W ),

(6.24)

where V ⊗ W is the tensor product of V and W defined as (V ⊗ W )ij (x, y) = Vi (x)Wj (y),

1 ≤ i, j ≤ N,

(6.25)

and Vi (x) (resp., Wj (y)) is the ith (resp., jth) component of the vector V (resp., W ) (cf. L. Schwartz’s [1] kernel theorem and M. Gelfand and N. Y. Vilenkin [1]). H(Ω) will be called the shape Hessian of J at Ω. (iii) When there exists a finite integer  ≥ 0 such that H(Ω) is continuous for the D
(RN , RN ) ⊗ D
(RN , RN )-topology, we say that H(Ω) is of order . In what follows, the compact notation D
will be used in place of D
(RN , RN ). Theorem 6.3. Let J be a real-valued shape functional and Ω be a subset of RN with boundary Γ. Assume that J is twice shape differentiable. (i) The vector distribution H(Ω) has support in Γ × Γ. (ii) If Ω is an open or closed domain in RN and H(Ω) is of order  ≥ 0, then there exists a continuous bilinear form     (6.26) [h] : D
/DΓ
× D
/L
Ω → R such that for all [V ] in D
/DΓ
and [W ] in D
/L
Ω ,   d2 J(Ω; V ; W ) = [h] qD (V ), qL (W ) ,

(6.27)

where qD : D
→ D
/DΓ
and qL : D
→ D
/L
Ω are the canonical quotient surjections and

   DΓ
= V ∈ D
RN , RN : ∂ α V = 0 on Γ, ∀α, |α| ≤  .

(6.28)

6. Second-Order Semiderivative and Shape Hessian

511

Proof. (i) It is sufficient to prove the following two properties: (a) For all V, W ∈ D such that W = 0 in a neighborhood of Γ, d2 J(Ω; V ; W ) = 0. (b) For all V, W ∈ D such that V = 0 in a neighborhood of Γ, d2 J(Ω; V ; W ) = 0. In case (a) the proof is similar to the one in Theorem 3.6 for the gradient and we prove the stronger result that for W such that W = 0 on Γ,   Ωt (W ) = Ω, ∀t ≥ 0 =⇒ dJ Ωt (W ); V = dJ(Ω; V ) =⇒ d2 J(Ω; V ; W ) = 0. In case (b) V = 0 in a neighborhood N of Γ and in K, the complement of the compact support K of V . So U = K is a neighborhood of Γ where V = 0. By construction U ∩ K = ∅ and there exists a bounded neighborhood U of K such that U ∩ Γ = ∅. Since U is compact and Γ is closed, the minimum distance d from U to Γ is finite and nonzero. Let 

N (Γ) = y ∈ RN : dΓ (y) < d/2 , where dΓ (y) = inf{|y − x| : x ∈ Γ}. For all X in Γ



t

Tt (X) − X =

  W Ts (X) ds = tW (X) +

0

 t.

/   W Ts (X) − W (X) ds,

0

and by condition (3.1) on W , |Tt (X) − X| ≤ t|W (X)| + ct max |Ts (X) − X|, [0,t]

and it can easily be shown that for t < 1/c max |Ts (X) − X| < [0,t]

Thus sup max |Ts (X) − X| ≤

X∈Γ [0,t]

t |W (X)|. 1 − ct t sup |W (X)|. 1 − ct X∈Γ

But W is continuous with compact support. Therefore, sup |W (X)| ≤

X∈Γ

sup X∈supp W

|W (X)| = W C(RN ;RN ) < ∞

and there exists τ > 0 such that ∀s ∈ [0, τ ],

d s W C < . 1 − cs 2

By definition and the previous inequalities   d dΓ Ts (X) = inf |Ts (X) − Y | ≤ |Ts (X) − X| < Y ∈Γ 2

512

Chapter 9. Shape and Tangential Differential Calculuses

for all s in [0, τ ] and all X ∈ Γ. This implies that Γs (W ) = Ts (W )(Γ) ⊂ N (Γ).

∀s ∈ [0, τ ], ∀X ∈ Γ,

By construction, V = 0 in N (Γ) since the distance from K to Γ is greater than or equal to d. Therefore, ∀s ∈ [0, τ ], V ∈ L∞ Ωs (W ) and as in the proof of Theorem 3.6, dJ(Ωs (W ); V ) = 0 and necessarily d2 J(Ω; V ; W ) = 0. (ii) We have already established in (i) that the bilinear form (V, W ) → h(V, W ) : D × D → R is zero for all V ∈ D and W ∈ D such that W = 0 on Γ and also zero for all W ∈ D and V ∈ D for which V = 0 in a neighborhood of Γ. By density all this is still true in D
, and now by the same argument as in the proof of Theorem 3.6 for all V in D
, [W ] → h(V, W ) : D
/L
→ R is well-defined, linear, and continuous. For the first component it is necessary to show that for all W in

 DΓ
= V ∈ D
(RN , RN ) : ∂ α V = 0 on Γ, ∀α, |α| ≤  the bilinear form h(V, W ) = 0. We first prove the result for the subspace ¯ RN ). A = D(Ω; RN ) ⊕ D(Ω; Then by density and continuity the result holds for the D
(RN , RN )-closure A of A. Finally, we prove that A = DΓ
. For any V in A, there exist V1 ∈ D(Ω; RN ) and ¯ RN ) such that V = V1 + V2 . Moreover, V2 ∈ D(Ω; K1 = supp V1 ⊂ Ω

¯ and K2 = supp V2 ⊂ Ω

¯ respectively. Hence V1 = 0 (resp., are compact subsets of the open sets Ω and Ω, V2 = 0) in the open neighborhood K1 (resp., K2 ) of Γ and necessarily V = V1 + V2 = 0 in the neighborhood U = (K1 ∪ K2 ) of Γ. Hence from part (i) ¯ RN ) ⊕ D
(Ω; RN ). Now A ⊂ D
, h(V, W ) = 0. By definition of DΓ
, DΓ
⊂ D
(Ω; Γ ¯ RN ), A = D(Ω; RN ) ⊕ D(Ω; and ¯ RN ), D(Ω; RN ) = D
(Ω;

¯ RN ) = D
(Ω; RN ). D(Ω;

By construction each V in A¯ is of the form V = V1 + V2 for ¯ RN ), K1 = supp V1 compact in Ω, ¯ V1 ∈ D
(Ω; ∀|α| ≤ ,

∂ α V1 = 0 on Γ,

V2 ∈ D
(Ω; RN ), K2 = supp V2 compact in Ω, ∀|α| ≤ , ∂ α V2 = 0 on Γ.

6. Second-Order Semiderivative and Shape Hessian Hence

513

 N  ¯ ∪ R \Ω = RN supp V = K1 ∪ K2 compact in Ω

and V ∈ D
(RN , RN ). Moreover, ∀α, |α| ≤ ,

∂ α V = ∂ α V1 + ∂ α V2 = 0 on Γ.

This proves that A ⊂ DΓ
and hence A = DΓ
. To complete the proof notice that by continuity of V → h(V, W ), for all W in D
the map [V ] → h(V, W, ) : D
/DΓ
→ R is well-defined, linear, and continuous. Finally the map   [V ], [W ] → h(V, , W ) : (D
/L
) × (D
/DΓ
) → R is well-defined, bilinear, and continuous. The next and last result is the extension of the structure Theorem 3.6 to second-order Eulerian semiderivatives. We need the result established in the corollary to Theorem 3.6. For a domain Ω with a boundary Γ which is C
+1 ,  ≥ 0, the map  
/L
Ω → C
(Γ) (6.29) qL (W ) → pL qL (W ) = γΓ (W ) · n : DΩ is a well-defined isomorphism. This will be used for the V -component. For the W -component we need the following lemma. Lemma 6.1. Assume that the boundary Γ of Ω is C
+1 ,  ≥ 0. Then the map   qD (V ) → pD qD (V ) = γΓ (V ) : D
/DΓ
→ C
(Γ, RN ) (6.30) is a well-defined isomorphism, where pD : D
→ D
/DΓ


(6.31)

is the canonical surjection and DΓ
is given by (6.28). Proof. The proof follows by standard arguments. Theorem 6.4. Let J be a real-valued shape functional. Assume that the conditions of Theorem 6.3 (ii) are satisfied and that the boundary Γ of the open domain Ω is C
+1 for  ≥ 0. (i) The map −1 (v, w) → hD×L (v, w) = [h](p−1 D v, pL w) : C
(Γ, RN ) × C
(Γ) → R

is bilinear and continuous, and for all V and W in D
(RN , RN )    d2 J(Ω; V ; W ) = hD×L γΓ P (V ), (γΓ W ) · n , where P (V ) is a linear combination of derivatives of V up to order .

(6.32)

(6.33)

514

Chapter 9. Shape and Tangential Differential Calculuses

(ii) This induces a vector distribution h(Γ ⊗ Γ) on C
(Γ, RN ) ⊗ C
(Γ) of order , h(Γ ⊗ Γ) : C
(Γ, RN ) ⊗ C
(Γ) → R,

(6.34)

such that for all V and W in D
(RN , RN )   (6.35) h(Γ ⊗ Γ), (γΓ V ) ⊗ (γΓ W ) · n  = d2 J(Ω; V ; W ),   where (γΓ V ) ⊗ (γΓ W ) · n is defined as the tensor product      (γΓ V ) ⊗ (γΓ W ) · n (x, y) = (γΓ Vi )(x) (γΓ W ) · n (y), x, y ∈ Γ, (6.36) i

Vi (x) is the ith component of V (x), and   ∀y ∈ Γ, γΓ (W ) · n (y) = (γΓ W )(y) · n(y).

(6.37)

´sio In the regular case the reader is referred to the early papers of J.-P. Zole ´ [24, p. 434] and D. Bucur and J.-P. Zolesio [12]. Remark 6.4. Finally, under the assumptions of Theorems 6.3 and 6.4 d2 J(Ω; V ; W ) = h(Γ ⊗ Γ), (γΓ P (V (0))) ⊗ ((γΓ W (0)) · n) + (g(Γ), (γΓ V  (0)) · n)

(6.38)

→ − → − for all V in V m+1,
and W in V m,
. Example 6.2. Go back to the example of the domain integral in section 6.1,  def f dx. J(Ω) = Ω

For V ∈ D 1 (RN , RN )



 f V · n dΓ =

dJ(Ω; V ) =

div (f V ) dx.

Γ

Ω

Using the domain expression with V in D2 (RN , RN ) and W in D 1 (RN , RN )  div (f V ) dx dJ(Ωs (W ); V ) = we readily get

Ωs (W )



 div (f V ) W · n dΓ =

2

d J(Ω; V ; W ) = Γ

and if Γ is C 1 ,

 div div (f V ) W dx

(6.39)

Ω

 div V W · n dΓ

d2 J(Ω; V ; W ) = Γ

is continuous for pairs (V, W ) ∈ D 1 (RN , RN ) × D0 (RN , RN ) or C 1 (Γ, RN ) × C 0 (Γ, RN ).

(6.40)

6. Second-Order Semiderivative and Shape Hessian

6.5

515

Decomposition of d2 J (Ω; V (0), W (0))

One important observation in the explicit computation of the second-order Eulerian semiderivative of the domain integral (6.1) in section 6.1 was the lack of symmetry and the appearance of the first half of the Lie bracket in (6.4). The same phenomenon was observed in the final form of the basic formula (6.10) for domain integrals (6.6) in Theorem 6.1, and also in section 6.3 for the derivative of the shape gradient (6.21) when it can be represented in the integral form (6.20). In this section perturbations of the identity will be used to show that d2 J(Ω; V (0), W (0)) can be further decomposed into a symmetric term plus the gradient applied to the velocity DV (0)W (0): d2 J(Ω; V (0), W (0)) = d2 JΩ (0)V (0), W (0) + dJ(Ω; DV (0)W (0)). Furthermore the symmetrical term can be obtained by the velocity method  d  −1 2 d JΩ (0; V (0), W (0)) = J Ωt (W (0); V (0) ◦ Tt (W (0) . dt t=0 Theorem 6.5. Let J be a real-valued shape functional and Θ be a Banach subspace of Lip (RN ; RN ). (i) Given f , θ, and ξ in Bε , assume that there exists τ > 0 such that ∀t ∈ [0, τ ],

d2 JΩ (f + tξ; θ) exists.

Then d2 JΩ (f ; θ; ξ) exists ⇐⇒ d2 J(Ωf ; V; Wξ ) exists

(6.41)

for Ωf = [I + f ](Ω) and the velocity fields Wξ (t) = ξ ◦ [I + f + tξ]−1 and V(t) = θ ◦ [I + f + tξ]−1 . def

def

(6.42)

(ii) If f , θ belong to V
+1 and ξ to V
, and V and Wξ satisfy the conditions of Theorem 6.2, then d2 JΩ (f ; θ; ξ) = d2 J(Ωf ; θ ◦ [I + f ]−1 ; ξ ◦ [I + f ]−1 ) − dJ(Ωf ; D(θ ◦ [I + f ]−1 ) ξ ◦ [I + f ]−1 ), d2 J(Ωf ; θ; ξ) = d2 JΩ (f ; θ ◦ [I + f ]; ξ ◦ [I + f ]) + dJ(Ωf ; Dθ ξ).

(6.43) (6.44)

→ − (iii) If V and W satisfy the conditions of Theorem 6.2 and V belongs to V m+1,
+1 , then d2 J(Ω; V (0); W (0)) = d2 JΩ (0; V (0); W (0)) + dJ(Ω; DV (0) W (0)), d2 J(Ω; V ; W ) = d2 JΩ (0; V (0); W (0)) + dJ(Ω; V  (0) + DV (0) W (0)),

(6.45)

(6.46)

516

Chapter 9. Shape and Tangential Differential Calculuses and

d2 JΩ (0; V (0); W (0))

d −1 . = dJ(Ωt (W (0)); V (0) ◦ Tt (W (0))) dt t=0

(6.47)

Proof. (i) Assume that d2 JΩ (f ; θ; ξ) exists and consider the differential quotient def 1 q(t) = [dJΩ (f + tξ; θ) − dJΩ (f ; θ)] → d2 JΩ (f ; θ; ξ). t From Theorem 3.4 (ii), dJΩ (f + tξ; θ) = dJ([I + f + tξ](Ω); θ ◦ [I + f + tξ]−1 ). Define Tt = [I + f + tξ] ◦ [I + f ]−1 , def

V(t) = θ ◦ [I + f + tξ]−1 . def

From Theorem 4.1 in Chapter 4, Tt = Tt (Wξ ) for the velocity field def

Wξ (t) =

∂Tt ◦ Tt−1 = ξ ◦ [I + f + tξ]−1 . ∂t

Therefore dJΩ (f + tξ; θ) = dJ(Tt (Wξ )(Ωf ); V(t)), 1 q(t) = [dJ(Tt (Wξ )(Ωf ); V(t)) − dJ(Ωf ; V(0))] → d2 (Ωf ; V; Wξ ) t since q(t) converges as t goes to 0. The converse is obvious. (ii) This is now a direct consequence of Theorem 6.2 and the fact that V(t) = θ ◦ [I + f ]−1 ◦ Tt−1 (Wξ ) ⇒ V  (0) = −D(θ ◦ [I + f ]−1 ) ξ ◦ [I + f ]−1 . (iii) From Theorem 6.2 and part (ii) with f = 0, θ = V (0), and ξ = W (0). If D2 JΩ (f ) exists in a neighborhood of f = 0 and if it is continuous at f = 0, then D2 JΩ (0) is symmetrical and this completes the decomposition of the shape gradient into a symmetrical operator and the gradient applied to the first half of the Lie bracket [V (0), W (0)]: d2 J(Ω; V (0); W (0)) = D2 JΩ (0) V (0), W (0) + G(Ω), DV (0) W (0). But this is not the end of the story. From the computation of the Hessian of the domain integral (6.4) in section 6.1, d2 JV,W      ∂f + H f vn wn + f D2 b vΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn = ∂n Γ + f DV W · n dΓ,

6. Second-Order Semiderivative and Shape Hessian

517

the result of the computation (6.10) in section 6.2,  div (f V (0)) W (0) · n dΓ  Γ     ∂f = + H f vn wn + f D2 bvΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn ∂n Γ + f DV W · n dΓ, and expression (6.21) of the derivative of (6.20) in section 6.3,    ∂Q(0) 2   + H Q(0) vn wn d J(V ; W ) = Qwn (0) vn + Q(0) V (0) · n + ∂n Γ   + Q(0) D2 b vΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn + Q(0) DV W · n dΓ, it is readily seen that the symmetrical term D 2 JΩ (0) V (0), W (0) further decomposes into a symmetrical term which depends only on the normal components vn and wn of V (0) and W (0) and another symmetrical term, which is boxed in the above expressions. This last term is the same in all expressions and depends only on the trace of G(0) (here of f and Q(0)) on Γ and the group of terms D 2 b vΓ · wΓ − vΓ · ∇Γ wn − wΓ · ∇Γ vn

(6.48)

involving vΓ , wΓ , and tangential derivatives of vn and wn on Γ. It would be interesting to further investigate this structure. In the context of a domain optimization problem, if the shape gradient is zero, both the term (6.48) and the first half of the Lie bracket will be multiplied by zero. Thus they will not contribute to the Hessian which will reduce to the symmetrical part that depends only on vn and wn ; that is, for autonomous velocity fields V and W ,    ∂f + H f vn wn dΓ, ∂n Γ    ∂Q(0)  Qwn (0) vn + + H Q(0) vn wn dΓ, ∂n Γ and the term

 Γ

Qwn (0) vn

 dΓ = Γ

Qvn (0) wn dΓ

is symmetrical. For earlier results on the structure of the second-order shape de´sio [24, p. 434] and D. Bucur and rivative the reader is referred to J.-P. Zole ´ J.-P. Zolesio [12]. Of course, the task of establishing similar results in the constrained case (D = RN ) remains to be done.

Chapter 10

Shape Gradients under a State Equation Constraint 1

Introduction

When a shape functional depends on the solution of a boundary value problem defined on the underlying domain, it is said to be constrained. This special type of constraint is to be distinguished from constraints on the geometry such as the volume, perimeter, and curvatures. Using the generic terminology of control theory the solution of the boundary value problem will be called the state and its corresponding equation or inequality the state equation or inequality. The domains will be identified with the controls and the constraints on the domains with the control constraints. Additional constraints on the solution of the state equation are usually called state constraints. Such problems have received a considerable amount of attention over the last century, and a rich and abundant literature can be found in mechanics, control theory, and optimization. Treating the state equation as an equality constraint and using Lagrange multipliers naturally yields a dual variable which is solution of the adjoint equation of the linearized state equation. This dual variable is called the adjoint state. Of course under appropriate differentiability assumptions with respect to the state and the control variables, necessary conditions can be obtained within the classical framework of the calculus of variations. Probably one of the most influential contributions of the last half of the 20th century to optimal control theory was made by L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mishchenko [1] in the 1960s when they showed that the differentiability with respect to the control could be relaxed and replaced by the pointwise maximization with respect to the control variable of the Hamiltonian constructed from the objective function and the coupled state and adjoint state equations, the so-called Maximum Principle. One of the important technical advantages of the control theory approach is to avoid the differentiation of the state with respect to the control. This is true not only for the characterization of optimal controls but also to obtain explicit expressions of the derivative of an objective function constrained by a state equation with respect to the control. Using a Lagrangian or Hamiltonian formulation, the derivative of 519

520

Chapter 10. Shape Gradients under a State Equation Constraint

the constrained objective function with respect to the control is typically equal to the “partial” derivative of the Lagrangian or the Hamiltonian with respect to the control, where the adjoint variable is a solution of an appropriate adjoint state equation that is coupled with the state equation. This chapter will concentrate on two generic examples often encountered in shape optimization. The first one is associated with the so-called compliance problems, where the shape functional is equal to the minimum of a domain-dependent energy functional. The special feature of such functionals is that the adjoint state coincides with the state. This obviously leads to considerable simplifications in the analysis. In that case it will be shown that theorems on the differentiability of the minimum of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the minimizer is not unique. The second one will deal with shape functionals that can be expressed as the saddle point of some appropriate Lagrangian. As in the first example, theorems on the differentiability of the saddle point of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the differentiation of the state equation with respect to the domain is particularly advantageous in shape problems. Here the state variable (the solution of the boundary value problem) lives in a function space (Banach, Hilbert, Sobolev) which depends on the control (underlying variable domain)! Thus the notion of derivative of the state with respect to the domain is more delicate. In this chapter two techniques will be presented to get around this difficulty: function space parametrization and function space embedding. Function space parametrization consists in transporting the functions on variable domains back onto the initial domain, where they can be compared. It is related to the notion of material derivative in continuum and structural mechanics. This will be illustrated by computing the volume and boundary integral expressions of the Eulerian semiderivative of a simple shape functional for the homogeneous Dirichlet and Neumann boundary value problems by the theorems on the differentiation of a minimum and a saddle point. Function space embedding consists in constructing extensions of the domaindependent boundary value problems to a larger fixed domain. For homogeneous Dirichlet boundary conditions, it is sufficient to consider extensions by zero to RN . The nonhomogeneous case requires the introduction of a multiplier to take into account the boundary condition. This technique will be illustrated by computing the boundary integral expression of the Eulerian semiderivative of a simple shape functional for the nonhomogeneous Dirichlet boundary value problem by the theorems on the differentiation of a saddle point. In addition to the generic example, the theorem on the differentiation of an infimum with respect to a parameter will be applied to the example of the buckling of columns considered in section 5 of Chapter 5. In section 3 an explicit expression of the semiderivative of Euler’s buckling load with respect to the cross-sectional area will be given. A necessary and sufficient condition will also be given to characterize the maximum Euler’s buckling load with respect to a family of cross-sectional areas. The theory is further illustrated in section 4 by providing the semiderivative

2. Min Formulation

521

of the first eigenvalue of several boundary value problems over a bounded open domain: Laplace equation, bi-Laplace equation, linear elasticity. In general the first eigenvalue is not simple over an arbitrary bounded open domain and the eigenvalue is not differentiable; yet the main theorem provides explicit domain expressions for a bounded open domain and boundary expressions of the semiderivatives for a sufficiently smooth bounded open domain.

2 2.1

Min Formulation An Illustrative Example and a Shape Variational Principle

Let Ω be a bounded open domain in RN with a smooth boundary Γ. Let y = y(Ω) be the solution of the Dirichlet problem −y = f in Ω, y = 0 on Γ,

(2.1)

where f is a fixed function in H 1 (RN ). The solution of (2.1) is the minimizing element in H01 (Ω) of the energy functional    1 2 |∇ϕ| − f ϕ dx. (2.2) E(Ω, ϕ) = Ω 2 Introduce the shape function def

J(Ω) =

 inf

ϕ∈H01 (Ω)

E(Ω, ϕ) = − Ω

We want to show that dJ(Ω; V ) = −

1 2

1 |∇y|2 dx. 2

 2 ∂y V · n dΓ. Γ ∂n

(2.3)

(2.4)

This example is the prototype of a free boundary problem1 which can be obtained from the following shape variational principle: ∀V,

dJ(Ω; V ) = 0.

(2.5)

It yields the extra boundary condition2 ∂y = 0 on Γ. ∂n

(2.6)

Equations (2.1) and (2.6) characterize a free boundary problem. It is the simplest example of a large family of problems where the shape function is an extremal of a natural internal energy (cf. section 7 of Chapter 5). They correspond to the socalled compliance problems in elasticity theory. They also occur in fracture theory and image segmentation. The first-order variation of this shape functional yields the extra boundary condition which is characteristic of a free boundary problem. 1 This usually comes with other constraints, such as a volume constraint on Ω and/or an area constraint on Γ. 2 With a volume equality constraint we would get that |∂y/∂n|2 is equal to a constant on Γ.

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2.2

Chapter 10. Shape Gradients under a State Equation Constraint

Function Space Parametrization

To compute the first-order derivative of J(Ω) we perturb the bounded open domain Ω by a velocity field V , which generates the family of transformations {Tt : 0 ≤ t ≤ τ } of RN and the family of domains {Ωt = Tt (Ω) : 0 ≤ t ≤ τ }. At t, J(Ωt ) =

inf

ϕ∈H01 (Ωt )

E(Ωt , ϕ)

(2.7)

and the minimizing element yt = y(Ωt ) is the solution of the Dirichlet problem −yt = f in Ωt ,

yt = 0 on Γt ,

(2.8)

where Γt is the boundary of Ωt . We want to compute the derivative dj(0) = lim

t0

of the function

j(t) − j(0) t

def

j(t) = J(Ωt ).

(2.9) (2.10)

We need a theorem that would give the derivative of an infimum with respect to a parameter t ≥ 0 at t = 0. The difficulty here is that the function space H 1 (Ωt ) depends on the parameter t. To get around this and obtain an infimum with respect to a function space that is independent of t, we introduce the following parametrization:

 (2.11) H01 (Ωt ) = ϕ ◦ Tt−1 : ϕ ∈ H01 (Ω) . Notice that since Tt is a homeomorphism, it transforms the open domain Ω into the open domain Ωt and sends the boundary Γ of Ω onto the boundary Γt of Ωt . In particular, when V is sufficiently smooth for all ϕ in H01 (Ω), ϕ ◦ Tt−1 ∈ H01 (Ωt ), and conversely, for all ψ in H01 (Ωt ), ψ ◦ Tt ∈ H01 (Ω). This parametrization does not affect the value of the minimum J(Ωt ) but changes the functional E J(Ωt ) =

inf

ϕ∈H01 (Ω)

E(Tt (Ω), ϕ ◦ Tt−1 ).

(2.12)

This parametrization is typical in shape analysis. It amounts to introducing the new energy functional
ϕ) = E(Tt (Ω), ϕ ◦ Tt−1 ), E(t,

ϕ ∈ H01 (Ω).

(2.13)

Our objective in the next section is to compute the limit (2.9) for j(t) =

inf

ϕ∈H01 (Ω)


ϕ). E(t,

(2.14)

This will be done in section 2.3. Before closing, it is interesting to characterize the minimizing element y t in H01 (Ω) of    1
ϕ) = E(t, (2.15) |∇(ϕ ◦ Tt−1 )|2 − f (ϕ ◦ Tt−1 ) dx, Ωt 2

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which is the solution of the following variational equation: find y t ∈ H01 (Ω) such that for all ϕ ∈ H01 (Ω) 

 ∇(y t ◦ Tt−1 ) · ∇(ϕ ◦ Tt−1 ) − f (ϕ ◦ Tt−1 ) dx = 0. (2.16) Ωt

Compare this expression with the characterization of the minimizing element yt of E(Ωt , ϕ) on H01 (Ωt ): find yt ∈ H01 (Ωt ) such that for all ϕ ∈ H01 (Ωt )  {∇yt · ∇ϕ − f ϕ} dx = 0. (2.17) Ωt

It is easy to verify that yt = y t ◦ Tt−1

and y t = yt ◦ Tt .

(2.18)

So y t is the solution yt of (2.8) transported back onto the fixed domain Ω by the change of variables induced by Tt . In view of this expression, (2.15) can be rewritten on the fixed domain Ω as  1
(A(t)∇ϕ) · ∇ϕ − (f ◦ Tt ) ϕ Jt dx, E(t, ϕ) = (2.19) Ω 2 where for t in [0, τ ] small DTt = Jacobian matrix of Tt , Jt = | det DTt | = det DTt for t ≥ 0 small,

(2.20) (2.21)

A(t) = Jt [DTt ]−1 ∗[DTt ]−1 .

(2.22)

With this change of variables y t is now characterized by the variational equation  t 1 1  y ∈ H0 (Ω) and ∀ϕ ∈ H0 (Ω),  (2.23)  A(t)∇yt · ∇ϕ − Jt (f ◦ Tt )ϕ dx = 0. Ω

2.3

Differentiability of a Minimum with Respect to a Parameter

Consider a functional G : [0, τ ] × X → R

(2.24)

for some τ > 0 and some set X. For each t in [0, τ ] define def

g(t) = inf{G(t, x) : x ∈ X}, def

X(t) = {x ∈ X : G(t, x) = g(t)}.

(2.25) (2.26)

The objective is to characterize the limit def

g(t) − g(0) t0 t

dg(0) = lim when X(t) is not empty for 0 ≤ t ≤ τ .

(2.27)

524

Chapter 10. Shape Gradients under a State Equation Constraint When X(t) = {xt } is a singleton, 0 ≤ t ≤ τ , and the derivative xt − x0 t0 t

x˙ = lim

(2.28)

of x is known, then it is easy to obtain dg(0) under appropriate differentiability of the functional G with respect to t and x. When x˙ is not readily available or when the sets X(t) are not singletons, this direct approach fails or becomes very intricate. In this section we present a theorem that gives an explicit expression for dg(0), the derivative of the min of the functional G with respect to t at t = 0. Its originality is that the differentiability of xt is replaced by a continuity assumption on the setvalued function and the existence of the partial derivative of the functional G with respect to the parameter t. In other words this technique does not require a priori knowledge of the derivative x˙ of the minimizing elements xt with respect to t. Theorem 2.1. Let X be an arbitrary set, let τ > 0, and let G : [0, τ ] × X → R be a well-defined functional. Assume that the following conditions are satisfied: (H1) for all t ∈ [0, τ ], X(t) = ∅; > (H2) for all x in t∈[0,τ ] X(t), ∂t G(t, x) exists everywhere in [0, τ ]; (H3) there exists a topology TX on X such that for any sequence {tn } ⊂ ]0, τ ], tn → t0 = 0, there exist x0 ∈ X(0) and a subsequence {tnk } of {tn }, and for each k ≥ 1, there exists xnk ∈ X(tnk ) such that (i) xnk → x0 in the TX -topology and (ii) lim inf ∂t G(t, xnk ) ≥ ∂t G(0, x0 ); k→∞ t0

(H4) for all x in X(0), the map t → ∂t G(t, x) is upper semicontinuous at t = 0. Then there exists x0 ∈ X(0) such that g(t) − g(0) = inf ∂t G(0, x) = ∂t G(0, x0 ). t0 t x∈X(0)

dg(0) = lim

(2.29)

Remark 2.1. In the literature condition (H3) (i) is known as sequential semicontinuity for set-valued functions. When X(0) is a singleton {x0 } we readily get dg(0) = ∂t G(0, x0 ). Remark 2.2. This theorem, and in particular the last part of property (2.29), extends a former result by B. Lemaire [1, Thm. 2.1, p. 38], where sequential compactness of the set ´sio [7] X was assumed. It also completes and extends Theorem 1 in J.-P. Zole ´sio [3]. and M. C. Delfour and J.-P. Zole

2. Min Formulation

525

Proof of Theorem 2.1. (i) We first establish upper and lower bounds to the differential quotient ∆(t) , t

def

∆(t) = g(t) − g(0).

Choose arbitrary x0 in X(0) and xt in X(t). Then, by definition, g(t) ≤ G(t, x0 ), −G(0, xt ) ≤ −g(0) = −G(0, x0 ). G(t, xt ) =

Add the above two inequalities to obtain G(t, xt ) − G(0, xt ) ≤ ∆(t) ≤ G(t, x0 ) − G(0, x0 ). By assumption (H2), there exist θt , 0 < θt < 1, and αt , 0 < αt < 1, such that G(t, xt ) − G(0, xt ) = t∂t G(θt t, xt ), G(t, x0 ) − G(0, x0 ) = t∂t G(αt t, x0 ), and by dividing by t > 0 ∂t G(θt t, xt ) ≤

∆(t) ≤ ∂t G(αt t, x0 ). t

(2.30)

(ii) Define dg(0) = lim inf t0

∆(t) , t

dg(0) = lim sup t0

∆(t) . t

There exists a sequence {tn : 0 < tn ≤ τ }, tn → 0, such that ∆(tn ) = dg(0). n→∞ tn lim

By assumption (H3), there exist x0 ∈ X(0) and a subsequence {tnk } of {tn } and for each k ≥ 1, there exists xnk ∈ X(tnk ) such that xnk → x0 in TX and lim inf ∂t G(t, xnk ) ≥ ∂t G(0, x0 ). t0 k→∞

So, from the first part of the estimate (2.30) for t = tnk , ∂t G(θtnk tnk , xnk ) ≤

∆(tnk ) tnk

and ∂t G(0, x0 ) ≤ lim inf ∂t G(θtnk tnk , xnk ) ≤ lim k→∞

k→∞

∆(tnk ) = dg(0). tnk

Therefore, ∃x0 ∈ X(0),

∂t G(0, x0 ) ≤ dg(0),

526

Chapter 10. Shape Gradients under a State Equation Constraint

and inf ∂t G(0, x) ≤ ∂t G(0, x0 ) ≤ dg(0).

x∈X(0)

From the second part of (2.30) and assumption (H4) we also obtain ∀x ∈ X(0), dg(0) ≤

∂t G(0, x) ≥ dg(0), inf ∂t G(0, x),

x∈X(0)

(2.31)

and necessarily inf ∂t G(0, x) = dg(0) = dg(0) =

x∈X(0)

inf ∂t G(0, x).

x∈X(0)

In particular, from (2.3) and (2.31) ∂t G(0, x0 ) = dg(0) =

inf ∂t G(0, x)

x∈X(0)

and x0 is a minimizing point of ∂t G(0, ·).

2.4

Application of the Theorem

Our example has a unique minimizing point yt for t ≥ 0 small. Here X = H01 (Ω), X(t) = {y t }, and it is sufficient to establish the continuity of the map t → y t at t = 0 for an appropriate topology on H01 (Ω).  We now check assumptions (H1) to (H4). Assume that V ∈ C 0 [0, τ ]; D2 (RN ,  RN ) and that f ∈ H 1 (RN ). Choose τ > 0 small enough such that Jt = |Jt |,

0 ≤ t ≤ τ,

(2.32)

and that there exist constants 0 < α < β such that ∀ξ ∈ RN ,

α|ξ|2 ≤ A(t) ξ · ξ ≤ β|ξ|2 , and α ≤ Jt ≤ β.

(2.33)

Since the bilinear form associated with (2.23) is coercive, there exists a unique solution y t to (2.23) and ∀t ∈ [0, τ ],

X(t) = {y t } = ∅.

(2.34)

So assumption (H1) is satisfied. To check (H2) use expression (2.19) and compute    1 
ϕ) = ∂t E(t, [A (t)∇ϕ] · ∇ϕ − [div Vt (f ◦ Tt ) + Jt ∇f · Vt ] ϕ dx, (2.35) 2 Ω where Vt (X) = V (t, Tt (X)), DVt (X) = DV (t, Tt (X)), A (t) = (div Vt )I − (DVt )∗ − DVt .

(2.36) (2.37)


ϕ) exists everywhere in [0, τ ] for all ϕ in H 1 (Ω) By assumptions on V and f , ∂t E(t, 0 and assumption (H2) is satisfied.

2. Min Formulation

527

To check assumption (H3)(i) we first show that {y t } is bounded in H01 (Ω). From (2.33)  t 2 α∇y L2 (Ω) ≤ A(t)∇y t · ∇y t dx Ω (2.38)  t t = Jt (f ◦ Tt ) y dx ≤ Jt f ◦ Tt L2 (Ω) y L2 (Ω) . Ω

By using the norm ϕH01 (Ω) = ∇ϕL2 (Ω) and the continuous injection of H01 (Ω) into L2 (Ω), ∃c > 0,

ϕL2 (Ω) ≤ cϕH01 (Ω) .

So from (2.38) y t H01 (Ω) ≤

c Jt f ◦ Tt L2 (Ω) . α

(2.39)

But Jt → 1 as t → 0 and f ◦ Tt → f in L2 (Ω) by the following lemma.   Lemma 2.1. Assume that V ∈ C 0 [0, τ ]; D1 (RN , RN ) satisfies assumptions (V) of Chapter 4 and that f ∈ L2 (RN ). Then lim f ◦ Tt = f and lim f ◦ Tt−1 = f in L2 (RN ).

t0

t0

(2.40)

So by (2.39) y t is bounded: ∃c > 0,

sup y t H01 (Ω) ≤ c.

(2.41)

t∈[0,τ ]

The next step is to prove the continuity by subtracting (2.23) at t > 0 from (2.23) at t = 0:   t ∇y · ∇ϕ dx + (A(t) − I)∇y t · ∇ϕ dx  Ω Ω f ϕ dx + [Jt (f ◦ Tt ) − f ]ϕ dx, = Ω Ω   ∇y · ∇ϕ dx = f ϕ dx. Ω

Ω

Subtract and set ϕ = y t − y:  |∇(y t − y)|2 dx  Ω = − (A(t) − I)∇y t · ∇(y t − y) + [Jt (f ◦ Tt ) − f ](yt − y) dx Ω

≤ |A(t) − I|∇y t L2 (Ω) ∇(y t − y)L2 (Ω) + Jt f ◦ Tt − f L2 (Ω) y t − yL2 (Ω)

528

Chapter 10. Shape Gradients under a State Equation Constraint

and y t − yH01 (Ω) ≤ c {|A(t) − I| + Jt f ◦ Tt − f L2 (Ω) }. But A(t) − I → 0 and Jt f ◦ Tt → f in L2 (Ω), and finally y t → y in H01 (Ω). So assumption (H3)(i) is satisfied for H01 (Ω)-strong. For assumption (H3)(ii) we have    1  ∂t G(t, ϕ) = [A (t)∇ϕ] · ∇ϕ − [div Vt (f ◦ Tt ) + Jt ∇f ◦ Vt ] ϕ dx 2 Ω and for ϕ in H01 (Ω), f in H 1 (RN ), and V in C 0 ([0, τ ]; D2 (RN , RN )) ∂t G(t, ϕ) − ∂t G(0, y)   1  (A (t) − A (0))∇ϕ · ∇ϕ = 2 Ω

 − [div Vt (f ◦ Tt ) − Jt ∇f · Vt − div V (0)f + ∇f · V (0)] ϕ dx  1 + {A (0)∇ϕ · ∇ϕ − A (0)∇y · ∇y} dx. Ω 2

As ϕ goes to y in H01 (Ω) and t → 0, the first term converges to zero since ϕ is bounded and A (t) → A (0), div Vt (f ◦ Tt ) → div V (0)f in L2 (Ω), Jt ∇f · Vt → ∇f · V (0) in L2 (Ω). The second term is continuous with respect to ϕ in H01 (Ω) and goes to zero as ϕ → y in H01 (Ω). So assumption (H3)(ii) is satisfied. Finally, (H4) is satisfied since    1 
A (t)∇y · ∇y − [div Vt f + ∇f · Vt ] y dx (2.42) t → ∂t E(t, y) = 2 Ω is continuous in [0, τ ]. So all the assumptions of Theorem 2.1 are satisfied and for V in C 0 ([0, τ ]; D2 (RN , RN )) and f in H 1 (RN ),    1  (2.43) dJ(Ω; V ) = A (0)∇y · ∇y − [div V (0)f + ∇f · V (0)] y dx. 2 Ω For V autonomous, (2.43) is continuous with respect to the space D1 (RN , RN ) and the shape gradient is of order 1. We know by the structure Theorem 3.6 of Chapter 9 that for Ω open and a C 2 -boundary Γ, ∃g(Γ) ∈ D1 (Γ) ,

dJ(Ω; V ) = g(Γ), V D1 (Γ) .

The characterization of g(Γ) will be given in the next section. We complete this section with the proof of Lemma 2.1.

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529

Proof of Lemma 2.1. (i) By density of D1 (RN ) in L2 (RN ) for all ε > 0, there exists fε such that f − fε L2 <

ε max{Jt−1 : 0 ≤ t ≤ τ }

(≤ εα ≤ ε).

Hence f ◦ Tt − f  ≤ fε ◦ Tt − fε  + f ◦ Tt − fε ◦ Tt  + f − fε . The last term is less than ε and the middle term can be rewritten after a change of variables as   |f ◦ Tt − fε ◦ Tt |2 dx = |f − fε |2 Jt−1 dx ≤ ε2 . RN

RN

A function fε in D1 (RN ) has a compact support and is uniformly Lipschitz continuous, that is, ∃c > 0, ∀x, y ∈ RN ,

|fε (y) − fε (x)| ≤ c|y − x|.

Thus for all X in RN |fε (Tt (X)) − fε (X)| ≤ c|Tt (X) − X|. Now

 Tt (X) = X +



t

t

[V (s, Ts (X)) − V (s, X)] ds.

V (s, X) ds + 0

(2.44)

0

Since V ∈ C 0 ([0, τ ]; D1 (RN , RN )) is also uniformly Lipschitz continuous by assumption (V ) ∃c > 0, ∀(X, Y ), ∀s ∈ [0, τ ],

|V (s, Y ) − V (s, X)| ≤ c |Y − X|,

and for all t in [0, τ ]  |Tt (X) − X| ≤ tV (·, X)C 0 ([0,τ ];RN ) +

t

c|Ts (X) − X| ds. 0

It is now easy to verify that there exists a constant c > 0 such that max |Ts (X) − X| ≤ ctV (·, X)C 0 ([0,τ ];RN ) .

s∈[0,t]

Finally, in view of (2.44) and (2.45) the term   |fε (Tt (X)) − fε (X)|2 dX ≤ ct2 V (·, X)2C 0 dX, N N R R   2 |fε (Tt (X)) − fε (X)| dX = |fε (Tt (X)) − fε (X)|2 dX RN Kε  V (·, X)2C 0 dX ≤ c t2 . ≤ c t2 Kε

(2.45)

530

Chapter 10. Shape Gradients under a State Equation Constraint

So for t small enough the right-hand side of (2.4) is less than 3ε and this completes the proof of the first part of (2.41). (ii) The second part of (2.41) can be obtained by the change of variables   |f ◦ Tt−1 − f |2 dx = |f − f ◦ Tt |2 Jt−1 dX RN

RN

and the fact that β −1 < Jt−1 < α−1 . This completes the proof of the lemma.

2.5

Domain and Boundary Integral Expressions of the Shape Gradient

Expression (2.43) for the shape gradient is a volume (or domain) integral, and it is easy to check that the map → − V → dJ(Ω; V ) : V 0,1 → R

(2.46)

is linear and continuous (cf. section 3.4 in Chapter 9). So by Corollary 1 to the structure Theorem 3.6 in Chapter 9, we know that for a domain Ω with a C 2 boundary Γ there exists a scalar distribution g(Γ) in D(Γ) such that dJ(Ω; V ) = g(Γ), V (0) · nD1 (Γ) .

(2.47)

The next objective is to further characterize the boundary expression. Recall that we have assumed that f ∈ H 1 (RN ). So for a C 2 boundary Γ the solution y = y(Ω) of the problem −y = f in Ω,

y = 0 on Γ

belongs to H 2 (Ω). For velocity fields V in C 0 ([0, τ ]; D1 (RN , RN )) satisfying assumption (V ) the transported solution yt in H 2 (Ω) is the solution of the system −div (A(t)∇y t ) = Jt f ◦ Tt in Ω,

y t = 0 on Γ.

(2.48)

Knowing that for all t in [0, τ ], y t ∈ H 2 (Ω) ∩ H01 (Ω), we can repeat the computation
ϕ) for ϕ in H 2 (Ω) 2 H 1 (Ω) instead of H 1 (Ω). With this extra smoothness of ∂t E(t, 0 0 we can use the formula d dt

 Ωt

F (t, x) dx

 F (0, x)V (0) · n dΓ +

=

t=0



Γ

Ω

∂F (0, x) dx ∂t

(2.49)

for a sufficiently smooth function F : [0, τ ]×RN → R. Prior to applying this formula to expression (2.15) in section 2.2, notice that for ϕ in H 2 (Ω), ϕ˙ =

d ϕ ◦ Tt−1 t=0 = −∇ϕ · V (0) ∈ H 1 (Ω), dt

(2.50)

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531

but it generally does not belong to H01 (Ω). Then     1 2
∂t E(t, ϕ) t=0 = |∇ϕ| − f ϕ V (0) · n dΓ + {∇ϕ · ∇ϕ˙ − f ϕ} ˙ dx. 2 Γ Ω (2.51) Substitute ϕ = y in (2.51):    1 2
|∇y| − f y V (0) · n dΓ ∂t E(t, y) t=0 = Γ 2 {∇y · ∇(∇y · V (0)) − f (∇y · V (0))} dx. −

(2.52)

Ω

2

But y ∈ H (Ω) and    ∂y ∇y · V (0) dΓ ∇y · ∇(∇y · V (0)) dx = − y ∇y · V (0) dx + Ω Ω Γ ∂n 2

H01 (Ω)

and
y) = ∂t E(t, t=0

  Γ

  ∂y 1 2 |∇y| − f y V (0) · n − ∇y · V (0) dΓ. 2 ∂n

But

∂y n on Γ, ∂n 2  1 ∂y =− V (0) · n dΓ. Γ 2 ∂n

(2.53)

y = 0 on Γ ⇒ ∇y = and finally


y) ∂t E(t, t=0

(2.54)

This is the boundary expression that is continuous for V (0) in the space D 0 (RN , RN ). It has been obtained via a parametrization of the function space appearing in the min formulation. Thus 2  1 ∂y dJ(Ω; V ) = − (2.55) V (0) · n dΓ, Ω 2 ∂n as predicted in section 2.1. Remark 2.3. An original application of the computations made for the generic example can be ´sio [3]. In that paper the found in M. C. Delfour, G. Payre, and J.-P. Zole derivative of the energy function with respect to the nodes in a P 1 finite element approximation is obtained from the volume expression of the shape semiderivative of the continuous problem. This technique also applies to a broad class of boundary value problems and to mixed hybrid finite element approximations. As observed by several authors the boundary integral expression is not suitable since the finite element solution does not have the appropriate smoothness under which the boundary integral formula is obtained. This technique is used to obtain a triangularization which minimizes the approximation error of the solution. Other formulae of the same type have been obtained for mixed finite element approximations in

532

Chapter 10. Shape Gradients under a State Equation Constraint

´sio [1] and several other boundary M. C. Delfour, Z. Mghzali, and J.-P. Zole value problems. The reader is also referred to M. C. Delfour, G. Payre, and ´sio [1, 2, 4, 5] for application of those techniques to thermal problems J.-P. Zole such as diffusers and space radiators. The last paper combines the above techniques with the systematic handling of parametrized geometries.

3

Buckling of Columns

Auchmuty’s dual principle was used for the functional associated with the optimal design of a column against buckling in section 5 of Chapter 5. In this section we first use this construction to compute the directional semiderivative with respect to the cross-sectional area and then use it to give a necessary and sufficient condition that characterizes the maximizers. Indeed, recall the identity µ(A) = −

1 2λ(A)

and notice that A is a maximizer of λ over A if and only if A is a maximizer of µ over A. We have seen in Theorem 5.3 of section 5 of Chapter 5 that the concave upper semicontinuous functional

 def (3.1) µ(A) = inf L(A, v) : v ∈ H02 (0, 1) ,  1 1/2  1 def 1 L(A, v) = A |v  |2 dx − |v |2 dx (3.2) 2 0 0 has maximizers over the weakly compact convex set A. Therefore, if µ(A) has a directional semiderivative dµ(A; B), a maximizer is completely characterized by ∃A ∈ A, ∀B ∈ A,

dµ(A; B − A) ≤ 0.

This directional semiderivative always exists for concave continuous functions, and Theorem 2.1 can be used to get an explicit expression of dµ(A; B). Given A ∈ A, B ∈ L∞ (0, 1), and t ≥ 0, let def

At = A + tB. In view of the fact that A ≥ A0 > 0, there exists τ > 0 such that for all 0 ≤ t ≤ τ ∀t, 0 ≤ t ≤ τ, Define

At ≥ A0 /2.

 1 1/2  1 def 1  2  2 ˜ At |v | dx − |v | dx L(t, v) = 2 0  0  ˜ v) : v ∈ H 2 (0, 1) , X(t) def ⇒ µ(At ) = inf L(t, = E(At ). 0

3. Buckling of Columns

533

Theorem 3.1. Let B be an arbitrary function in L∞ (0, 1) and A be an element of A. (i) The directional semiderivative at A in the direction B is given by the following expression: dµ(A; B) =

1 v∈E(A) 2



1

inf

B |v  |2 dx,

(3.3)

0

and since µ is concave and continuous, dH µ(A; B) also exists. (ii) The maximizing elements of µ(A) or λ(A) in A are completely characterized by the variational inequality ∃A ∈ A, ∀B ∈ A,

1 v∈E(A) 2



1

inf

(B − A) |v  |2 dx ≤ 0

(3.4)

0

or equivalently ∃A ∈ A, ∀B ∈ A,

1 inf v∈E(A) 2



1

B |v  |2 dx ≤

0

1 = −µ(A). 2λ(A)

(3.5)

If we replace E(A) by

 1   |u |2 dx = 1 and ∀v ∈ H02 (0, 1)    0 def ˜  1 E(A) = u ∈ H02 (0, 1) :  1 ,    A u v  dx = λ(A) u v dx  0

0

then the maximizing elements of µ(A) or λ(A) in A are completely characterized by the variational inequality  ∃A ∈ A, ∀B ∈ A,

1

inf

˜ v∈E(A)

B |v  |2 dx ≤ λ(A).

(3.6)

0

Proof. (i) From Theorem 5.2 in Chapter 5, X(t) = ∅ and assumption (H1) is ˜ v) with respect to t is given by the expression satisfied. The partial derivative of L(t, ˜ v) = 1 ∂t L(t, 2



1

B |v  |2 dx,

(3.7)

0

which is independent of t and exists for all B ∈ L∞ (0, 1). Hence assumptions (H2) and (H4) are trivially satisfied. From (5.7) in Theorem 5.1 of Chapter 5  1  1 At ut v  dx = λ(At ) ut v  dx ∃ut ∈ H02 (0, 1), ∀v ∈ H02 (0, 1), 0 0   1  1 A0 1  2 1 ⇒ . |ut | dx ≤ At |ut |2 dx = λ(At ) |ut |2 dx = 2 0 λ(At ) 0 0

(3.8) (3.9)

534

Chapter 10. Shape Gradients under a State Equation Constraint

But 0<

A0 def ≤ At ≤ AB = A1 + τ BL∞ 2

1 µ(A0 ) ≤ µ(At ) ≤ µ(AB ) and 2λ(A0 ) ≤ λ(At ) ≤ λ(AB ). 2 Therefore for any sequence tn  0, there exist a subsequence {tnk }, µ, and u ∈ H02 (0, 1) such that ⇒

def

1 1 > 0, →λ=− 2µk 2µ ⇒ uk → u in H01 (0, 1).

def

µk = µ(Atnk ) → µ < 0,

λk = λ(Atnk ) = −

uk  u in H02 (0, 1)-weak

Going to the limit in (3.8)  1  ∀v ∈ H02 (0, 1), Atnk uk v  dx = λ(Atnk ) 0

1

uk v  dx,

uk L2 =

0

1 , λk

we get the following variational equation for u ∈ H02 (0, 1):  1  1 2   A u v dx = λ u v dx, ∀v ∈ H0 (0, 1), 0 

u L2 = lim

k→∞

0

uk L2

1 1 = lim = > 0. k→∞ λk λ

To show that u ∈ X(0) = E(A), let u0 be any element of E(A). By definition,   1 1 1 1 Atnk |uk |2 dx − uk L2 ≤ Atnk |u0 |2 dx − u0 L2 . (3.10) 2 0 2 0 But Atnk → A in L∞ (0, 1)-strong and  1  1  Atnk |uk |2 dx = lim inf (Atnk − A) |uk |2 dx + lim inf k→∞

k→∞

0

0

≥ lim inf −Atnk − AL∞ k→∞



≥

1



1

1

A |uk |2 dx

0

|uk |2 dx +

0



1

A |uk |2 dx

0

A |u |2 dx

0

since uk L2 is bounded. Since uk L2 → u L2 , by going to the limit in (3.10), we get  1 1 A |u |2 dx − u L2 L(A, u) = 2 0  1 1 ≤ lim inf Atnk |uk |2 dx − uk L2 k→∞ 2 0   1 1 1 1 ≤ lim inf Atnk |u0 |2 dx − u0 L2 = A |u0 |2 dx − u0 L2 k→∞ 2 0 2 0 = µ(A) = inf L(A, v). v∈V

4. Eigenvalue Problems

535

Therefore, by definition of the minimum u ∈ E(A) = X(0) and assumption (H3) (i) is satisfied. To check the second part of (H3) we first show that {uk } converges not only in H02 (0, 1)-weak but also in H02 (0, 1)-strong. Then assumption (H4) directly follows from that property  1  1 1 ˜ ˜ u). ∂ L(t, uk ) = B |uk |2 dx → B |u |2 dx = ∂ L(t, 2 0 0 The strong convergence follows from the following chain of inequalities: A0 2



1

|uk − u |2 dx ≤

0



1

A0  2 |u | dx + 2 k

0



1

Ak |uk |2 dx +

≤ 0



= λk 0 1

 →λ

 

1

0 1

A0  2 |u | dx − 2



A0 |u |2 dx − 2

0

1

|uk |2



0



0 1

A0 uk u dx

0



1

A0 uk u dx

0

1

 2



A0 |u | dx − 2

dx +

|u |2 dx −

1

1

A0 uk u dx

0

A0 |u |2 dx = 0

0

since λ = λ(A). This complete the proof of part (i). (ii) The first characterization now follows directly from the fact that µ is concave and continuous and A is closed and convex. The second characterization uses the following identity, which comes from (3.8) for the eigenvectors and their normalization (5.11) in Theorem 5.2 of Chapter 5 as elements of E(A): for all v ∈ E(A) 1 2

4



1

A |v  |2 dx = λ(A)

0

1 2

 0

1

|v |2 dx =

1 = −µ(A). 2λ(A)

Eigenvalue Problems

The first eigenvalue λ(Ω) of a linear boundary value problem defined on a bounded open subset Ω of RN is a classical example of a shape functional which comes in the form of an infimum. It is generally not differentiable since the eigenvalue can be repeated, but Theorem 2.1 of section 2.3 can be used to compute its Eulerian semiderivative. In this section we give the volume expression of the Eulerian semiderivative for the Laplacian and the bi-Laplacian with a homogeneous Dirichlet boundary condition for a general bounded open domain and its boundary expression when the domain is sufficiently smooth. In the case of the Laplacian over a smooth domain, the first eigenvalue is simple and differentiable. As in section 3 it is technically advantageous to work with G. Auchmuty [1]’s dual principle rather than with the Rayleigh quotient. It is also technically advantageous to embed the problem of the computation of the Eulerian semiderivative for the domain Ω into a larger and

536

Chapter 10. Shape Gradients under a State Equation Constraint

sufficiently smooth holdall D which will contain all the perturbations Ωt = Tt (V )(Ω) of Ω for t sufficiently small. The smoothness conditions that would normally occur on Ω will then occur on D. Once the expression for the semiderivative is obtained, D can be thrown away, leaving a final expression that does not require any smoothness assumption on Ω.

4.1

Transport of H0k (Ω) by W k,∞ -Transformations of RN

Recall Theorem 2.3 in Chapter 8 on the embedding of H0k (Ω) into H0k (D). As a consequence, the homogeneous Dirichlet boundary value problem in Ω is the following: find y ∈ H01 (Ω) such that   ∇y · ∇ϕ dx = f ϕ dx ∀ϕ ∈ H01 (Ω), Ω

Ω

is completely equivalent to the variational problem, find Y ∈ H01 (Ω; D) such that   ∇Y · ∇Φ dx = f Φ dx ∀Φ ∈ H01 (Ω; D), D

D

and Y |Ω = y. Let k ≥ 1 and let T be a transformation of RN such that T, T −1 ∈ W k,∞ (RN , RN ). Associate with T the map ϕ → T (ϕ) = ϕ ◦ T −1 : D(RN ) → H0k (RN ). def

By assumption on T , the norms ϕH k and T (ϕ)H k = ϕ◦T −1 H k are equivalent and by density T extends to a linear bijection ϕ → T (ϕ) = ϕ ◦ T −1 : H0k (RN ) → H0k (RN ) def

such that both T and T −1 are uniformly Lipschitzian. Given any bounded open subset Ω of RN , the map def

ϕ → TΩ (ϕ) = T (ϕ)|Ω : H0k (Ω) → H0k (T (Ω)) is again a linear bijection such that both TΩ and TΩ−1 are uniformly Lipschitzian. Given a bounded open subset D of RN , further assume that T is such that T (D) = D. As a result T (RN \D) = RN \D and T (∂D) = ∂D and TD : H0k (D) → H0k (D).

4. Eigenvalue Problems

537

Let Ω be an open subset of RN such that Ω ⊂ D. Then T (Ω) ⊂ D, T −1 (Ω) ⊂ D, and H01 (Ω) and H01 (T (Ω)) can be identified with the subspaces H01 (Ω; D) and H01 (T (Ω); D) of H01 (D). Moreover, ∀ϕ ∈ D(Ω),

e0 (ϕ) ◦ T −1 = e0 (ϕ ◦ T −1 )

and the following diagram commutes: H0k (Ω)

e

→0 H0k (D) ↓ TD

↓ TΩ e

H0k (T (Ω)) →0 H0k (D) ⇒ TD (H0k (Ω; D)) = H0k (T (Ω); D).

4.2

Laplacian and Bi-Laplacian

For an arbitrary bounded open subset Ω of RN the bounded open holdall D can be arbitrarily chosen such that Ω ⊂ D. For instance, D can be a large enough open ball. This will make it possible to throw on D any smoothness assumption that would normally occur on Ω and work with an arbitrary Ω. In this setup, D is used as an intermediate step and can be thrown away once the semiderivative has been computed. Given a velocity field V ∈ C([0, τ ]; W0k,∞ (D; RN )),

(4.1)

the flow mapping Tt = Tt (V ) maps D onto itself and ∂D onto itself, and is equal to the identity on RN \D. So it transports H01 (D) onto itself. Theorem 4.1. For k = 1, 2 and 0 ≤ t < τ ϕ ∈ H0k (D) ⇐⇒ ϕ ◦ Tt (V ) ∈ H0k (D), ϕ ∈ H0k (Ω; D) ⇐⇒ ϕ ◦ Tt (V ) ∈ H0k (Ω; D). Hence for Ωt = Tt (Ω)

 H0k (Ωt ; D) = ϕ ◦ Tt−1 (V ) : ∀ϕ ∈ H0k (Ω; D) .

(4.2)

Proof. The second statement follows from the fact that Tt and its inverse Tt−1 are both Lipschitz continuous and that, by the previous considerations or Lemma 6.2 in Chapter 8, they transport sets of zero capacity onto sets of zero capacity: Tt (D\Ω) = D\Tt (Ω) = D\Ωt . As Ω is compact and Ω ⊂ D, there exists τV > 0 such that for all t, 0 ≤ t < τV , Ωt (V ) ⊂ D. In both cases the first eigenvalue is given by the Rayleigh quotient: & akΩt (ϕ, ϕ) k λ(Ωt (V )) = inf  : ∀ϕ ∈ H0 (Ωt ), ϕ = 0 , k = 1, 2, ϕ2 dx Ωt

538

Chapter 10. Shape Gradients under a State Equation Constraint

where def

a1Ωt (ϕ, ψ) =

 ∇ϕ · ∇ψ dx Ωt



def

and a2Ωt (ϕ, ψ) =

∆ϕ ∆ψ dx. Ωt

In view of the previous constructions H0k (Ωt ) can be replaced by H0k (Ωt ; D):  k  aD (ϕ, ϕ) k : ∀ϕ ∈ H0 (Ωt ; D), ϕ = 0 , k = 1, 2. (4.3) λ(Ωt (V )) = inf  ϕ2 dx D Note that if ϕ = 0 is a minimizer, then for any real number α = 0, αϕ is also a minimizer. Theorem 4.2. Given k = 1, 2, let Ω be a bounded open subset of RN . Assume that V ∈ C([0, τ ]; W0k,∞ (D; RN )) for some bounded open domain D in RN such that Ω ⊂ D. There exists at least one nonzero solution ϕ ∈ H0k (Ωt ; D) to the minimization problem (4.3), λ(Ωt (V )) ≥ λ(D) > 0, and   ∀ϕ ∈ H01 (D), ϕ2 dx ≤ λ(D)−1 |∇ϕ|2 dx, D D 2 2 −1 ϕ dx ≤ λ(D) |∆ϕ|2 dx. ∀ϕ ∈ H0 (D), D

D

The solutions are completely characterized by the following variational equation: there exists ϕ ∈ H0k (Ωt ; D) such that  k k ϕ ψ dx (4.4) ∀ψ ∈ H0 (Ωt ; D), aD (ϕ, ψ) = λ(Ωt (V )) D

or equivalently

 ∀ψ ∈ H0k (Ωt ),

akΩt (ϕ, ψ) = λ(Ωt (V ))

ϕ ψ dx.

(4.5)

Ωt

Proof. Same technique as in the proof of Theorem 2.1 in Chapter 8. The minimization problem can be rewritten on the unit sphere in L2 (D) by normalization:   λ(Ωt (V )) = inf |∇ϕ|2 dx : ∀ϕ ∈ H01 (Ωt ; D), ϕL2 (D) = 1 . (4.6) D

The minimizing elements of (4.6) cannot be zero since the injection of H01 (D) into L2 (D) is compact (cf. Theorem 2.3 in Chapter 8). Remark 4.1. One of the consequences of the use of the embedding of H01 (Ω) into H01 (D) is that the characterization of the first eigenvalue on Ω requires only that Ω be a bounded open subset of RN . It is not necessary to assume that Ω is Lipschitzian in order to use Rellich’s theorem, since D can be chosen sufficiently large (Ω ⊂ D) and smooth in order to contain all variations Ωt ⊂ D as t goes to zero. This technique will be exploited in the computation of the Eulerian semiderivative of λ(Ω).

4. Eigenvalue Problems

539

G. Auchmuty [1]’s dual variational for this eigenvalue problem can be chosen as

 def (4.7) µ(Ωt ) = inf Lk (Ωt , ϕ) : ϕ ∈ H0k (Ωt ) ,  1/2 def 1 Lk (Ωt , ϕ) = akΩt (ϕ, ϕ) − ϕ2 dx . (4.8) 2 Ωt By using the embedding of H01 (Ωt ) into H01 (D), this problem can be rewritten as

 def µ(Ωt ) = inf Lk (D, ϕ) : ϕ ∈ H0k (Ωt ; D) ,  1/2 def 1 Lk (D, ϕ) = akD (ϕ, ϕ) − ϕ2 dx . 2 D

(4.9) (4.10)

Theorem 4.3. Given k = 1, 2, let Ω be a bounded open subset of RN . Assume that V ∈ C 0 ([0, τ [ ; W0k,∞ (D; RN )) for some bounded open domain D in RN such that Ω ⊂ D. Then for 0 ≤ t < τ 1 µ(Ωt ) = − (4.11) 2λ(Ωt ) and the set of minimizers of (4.7) is given by   ϕ is solution of (4.4) and     def 1/2 . E k (Ωt ) = ϕ ∈ H0k (Ωt ; D) :    |ϕ|2 dx = 1/λ(Ωt )  

(4.12)

D

Proof. From the previous theorem the set E(Ωt ) is not empty, and for any ϕ ∈ E(Ωt ) 1 µ(Ωt ) ≤ L(t, ϕ) = − < 0. 2λ(Ωt ) Therefore the minimizers of (4.8) are different from the zero functions. For ϕ = 0, the functional L(t, ϕ) is differentiable and its directional derivative is given by  1 ϕ ψ dx, (4.13) dL(t, ϕ; ψ) = akD (ϕ, ψ) − ϕL2 (D) D and any minimizer of L(t, ϕ) is a stationary point of dL(t, ϕ; ψ); that is,  1 ∀ψ ∈ H0k (Ωt ; D), akD (ϕ, ψ) − ϕ ψ dx = 0. ϕL2 (D) D

(4.14)

Therefore, ϕ is a solution of the eigenvalue problem with λ =

1 ϕL2 (D)

⇒ −

1 1 . = µ(Ωt ) ≤ − 2λ 2λ(Ωt )

By minimality of λ(Ωt ), we necessarily have λ = λ(Ωt ) and this concludes the proof of the theorem.

540

Chapter 10. Shape Gradients under a State Equation Constraint

We now turn to the computation of the Eulerian semiderivative dλ(Ω; V ) via dµ(Ω; V ) from the identity λ(Ωt ) = −

1 , 2µ(Ωt )

since whenever dµ(Ω; V ) exists dλ(Ω; V ) =

1 dµ(Ω; V ) = 2 λ(Ω)2 dµ(Ω; V ). 2µ(Ω)2

Assuming that V satisfies assumption (4.1), we use the function space parametrization of section 2.4 in conjunction with Theorem 2.1. From the characterization (4.2) of H0k (Ωt ; D), define the following new functional: for each ϕ ∈ H01 (D), ˜ ϕ) def L(t, = L(D, ϕ ◦ Tt−1 (V )) 1 = akD (ϕ ◦ Tt−1 (V ), ϕ ◦ Tt−1 (V )) − 2

 |ϕ ◦

Tt−1 (V

1/2 )| dx . 2

D

After a change of variables for k = 1 and ϕ ∈ H01 (D),  1/2  ˜ ϕ) = 1 A(t)∇ϕ · ∇ϕ dx − Jt |ϕ|2 dx , L(t, 2 D D A(t) = Jt DTt−1 ∗DTt−1 , and for k = 2 and ϕ ∈

Jt = det(DTt ),

H02 (D)

˜ ϕ) = 1 L(t, 2





1/2 Jt |ϕ|2 dx ,

|div (B(t)∇ϕ)|2 Jt dx − D

D

B(t) =

DTt−1 ∗DTt−1 .

To apply Theorem 2.1, choose   def def X(t) = ϕt = ϕt ◦ Tt : ∀ϕt ∈ E k (Ωt ) , endowed with the weak topology of H01 (D). Assumption (H1) is clearly satisfied. ˜ ϕ) is differentiable. For k = 1 and 0 = ϕ ∈ H01 (D), For all ϕ = 0, L(t,   1 ˜ ϕ) = 1 A (t)∇ϕ · ∇ϕ dx −  |ϕ|2 Jt dx, ∂t L(t, 1/2 2 D 2 D Jt |ϕ| dx D

and for k = 2 and 0 = ϕ ∈ H02 (D)  ˜ ϕ) = div (B  (t)∇ϕ) div (B(t)∇ϕ) Jt dx ∂t L(t, D  1 |div (B(t)∇ϕ)|2 Jt dx + 2 D  1 |ϕ|2 Jt dx. −  1/2 2 D J |ϕ| dx D t

4. Eigenvalue Problems

541

Hence assumptions (H2) and (H4) are satisfied. In t = 0 the above expressions simplify. For k = 1 and ϕ ∈ E 1 (Ω),   ˜ ϕ) = 1 ∂t L(0, A (0)∇ϕ · ∇ϕ dx − λ(Ω) |ϕ|2 div V (0) dx 2 D D     1 2 2 |∇ϕ| − λ(Ω)|ϕ| div V (0) dx, = − ε(V (0))∇ϕ · ∇ϕ dx + D D 2 def 1 A (0) = div V (0)I − DV (0) − ∗DV (0), ε(U ) = [DU + ∗DU ] , 2 and for k = 2 and ϕ ∈ E 2 (ω),  ˜ ϕ) =2 ∂t L(0, div (ε(V (0))∇ϕ) ∆ϕ dx D    1 |∆ϕ|2 − λ(Ω)|ϕ|2 div V (0) dx, + D 2 B  (0) = −DV (0) − ∗DV (0) = −2ε(V (0)). For volume-preserving velocities, div V (0) = 0, and the above expressions reduce to the first integral term. In order to apply Theorem 2.1 it remains to check assumption (H3). This is the first example where the set of minimizers is not necessarily unique and for which we have a complete description of the Eulerian semiderivative. Theorem 4.4. Given k = 1, 2, let Ω be a bounded open subset of RN . Assume that V ∈ C 0 ([0, τ [ ; W0k,∞ (D; RN )) for some bounded open domain D in RN such that Ω ⊂ D. Then for k = 1  1 dλ(Ω; V ) = inf − ε(V (0))∇ϕ · ∇ϕ dx ˜ 1 (Ω) 2 ϕ∈E Ω    1 2 2 + |∇ϕ| − λ(Ω)|ϕ| div V (0) dx, Ω 2 def 1 ε(U ) = [DU + ∗DU ] , 2 & − ∆ϕ = λ(Ω)ϕ in D(Ω) def 1 1  ˜ E (Ω) = ϕ ∈ H0 (Ω) : ; and |ϕ|2 dx = 1 Ω

for k = 2



1 dλ(Ω; V ) = inf ˜ 2 (Ω) 2 ϕ∈E

2 div (ε(V (0))∇ϕ) ∆ϕ dx   1 |∆ϕ)|2 − λ(Ω)|ϕ|2 div V (0) dx, + Ω 2   ∆(∆ϕ) = λ(Ω)ϕ in D(Ω)    ˜ 2 (Ωt ) def E . = ϕ ∈ H02 (Ω) :   and |ϕ|2 dx = 1 Ω

Ω

542

Chapter 10. Shape Gradients under a State Equation Constraint

In both cases λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) in Chapter 9, which is continuous with respect to V (0) ∈ W0k,∞ (D; RN ): dλ(Ω; V ) = dH λ(Ω; V (0)). Proof. As Ω is compact and Ω ⊂ D, there exists τV > 0 such that for all t, 0 ≤ t < τV , Ωt (V ) ⊂ D. Observe that µ(D) =

inf

ϕ∈H01 (D)

L(ϕ) ≤

inf

ϕ∈H01 (Ωt ;D)

L(ϕ) = µ(Ωt ).

˜ ϕ), for any ϕt ∈ X(t) and ϕ0 ∈ X(0), By continuity of t → L(t, ˜ ϕ0 ) ⇒ lim sup L(t, ˜ ϕ0 ) = µ(Ω) ˜ ϕt ) ≤ L(0, ˜ ϕt ) ≤ L(t, L(t, t0

⇒ µ(D) ≤ lim sup µ(Ωt ) ≤ µ(Ω). t0

Therefore there exists τ  , 0 < τ  ≤ τV , such that for all t, 0 ≤ t ≤ τ  , µ(D) ≤ µ(Ωt ) ≤

1 µ(Ω) < 0 2

⇒ 0 < λ(D) ≤ λ(Ωt ) ≤ 2 λ(Ω).

Since A(t) → I and Jt → 1 as t goes to zero, there exist c > 0 and 0 < τ  ≤ τ such that for all 0 ≤ t ≤ τ  , ˜ ϕ) = 1 L(t, 2



 A(t)∇ϕ · ∇ϕ dx −

1/2 Jt |ϕ|2 dx

D D 2 ≥c ∇ϕL2 (D) − ϕL2 (D) ≥ c∇ϕ2L2 (D) − c ∇ϕL2 (D)

for some c > 0 by Poincar´e’s inequality. Therefore for all ϕt ∈ X(t), ˜ ϕt ) ≤ 0 c ∇ϕt 2L2 (D) − c ∇ϕt L2 (D) ≤ L(t,

⇒ ∇ϕt L2 (D) ≤ c /c.

For any sequence tn  0, there exist subsequences µ, and ϕ0 ∈ H01 (Ω; D) such that µn = µ(Ωtn ) → µ and ϕn = ϕtn  ϕ0 in H01 (Ω; D)-weak, 1 λn = λ(Ωtn ) → λ = − > 0. 2µ Therefore, since A(tn ) → I, Jtn → 1, and {ϕn } converges in H 1 -weak,   1 1 ∀ψ ∈ H0 (Ω; D), A(tn )∇ϕn · ∇ψ dx = λn ϕn ψ dx 2 D  D  1 A(0)∇ϕ0 · ∇ψ dx = λ ϕ0 ψ dx, ⇒ ∀ψ ∈ H01 (Ω; D), 2 D D    Jtn |ϕn |2 dx → |ϕ0 |2 dx ⇒ |ϕ0 |2 dx = λ > 0, λn = D

D

D

4. Eigenvalue Problems

543

and λ is an eigenvalue for the problem on Ω. But we know that 1 1 1 , µ(D) ≤ − ≤− , 2λ 2λ 2λ(Ω) n→∞ λ(D) ≤ λ ≤ λ(Ω) ⇒ λ = λ(Ω),

µ(D) ≤ lim sup µ(Ωtn ) = −

since λ(Ω) is minimal and hence ϕ0 ∈ X(0). This proves condition (H3)(i). To prove condition (H3)(ii) and complete the proof we first prove that the weakly convergent subsequence strongly converges in H01 (Ω; D). By compactness of the injection of H01 (D) into L2 (D) (cf. Theorem 2.3 in Chapter 8) ϕn  ϕ0 in H01 (D)-weak

⇒ ϕn → ϕ0 in L2 (D)-strong,

(4.15)

and for some α > 0 independent of n,   α |∇(ϕn − ϕ0 )|2 dx ≤ A(tn )∇(ϕn − ϕ0 ) · ∇(ϕn − ϕ0 ) dx D D = A(tn )∇ϕn · ∇ϕn − 2A(tn )∇ϕ0 · ∇ϕn D + A(tn )∇ϕ0 · ∇ϕ0 dx  = λn Jtn ϕn ϕn − 2A(tn )∇ϕ0 · ∇ϕn D + A(tn )∇ϕ0 · ∇ϕ0 dx  → λ ϕ0 ϕ0 − 2∇ϕ0 · ∇ϕ0 D

+ ∇ϕ0 · ∇ϕ0 dx = 0.

By the same technique as in section 2.4 when n → ∞ and t  0  1 ˜ ∂t L(t, ϕn ) = A (t)∇ϕn · ∇ϕn dx 2 D  1 |ϕn |2 Jt dx −  1/2 2 dx D J |ϕ | t n  D 1 A (0)∇ϕ0 · ∇ϕ0 dx → 2 D  1 ˜ ϕ0 ). −  |ϕ0 |2 div V (0) dx = ∂t L(0, 1/2 2 D |ϕ0 | dx D The case k = 2 is analogous. If Ω is assumed to be of class C 2k , the eigenvector functions belong to H0k (Ω)∩ H (Ω) and the volume expressions of the previous theorem can be expressed as boundary integrals as in section 2.5. In that case we can use formulae (2.49) and (2.50) to compute the partial derivative of Lt = L(Ωt , ϕ ◦ Tt−1 ). For k = 1, 2k

1 Lt = 2

 |∇(ϕ ◦ Ωt

Tt−1 )|2

 dx −

|ϕ ◦ Ωt

Tt−1 |2

1/2 dx .

544

Chapter 10. Shape Gradients under a State Equation Constraint

From formula (2.49)  def

L = ∂t Lt |t=0

 1 Γ |ϕ|2 V (0) · n dΓ |∇ϕ| V (0) · n dΓ − 2  |ϕ|2 dx1/2 Γ Ω   ϕ ϕ ˙ dx + ∇ϕ · ∇ϕ˙ dx −  Ω 1/2 . Ω |ϕ|2 dx Ω

1 = 2



2

For ϕ ∈ E 1 (Ω), ϕ = 0 on Ω, −∆ϕ = λ(Ω)ϕ in Ω, and λ(Ω)ϕL2 (Ω) = 1,   1 |∇ϕ|2 V (0) · n dΓ + ∇ϕ · ∇ϕ˙ dx − λ(Ω) ϕ ϕ˙ dx 2 Ω Ω Γ 1 ∂ϕ |∇ϕ|2 V (0) · n dΓ + ϕ˙ dΓ. = 2 Γ Γ ∂n

L =



But ϕ = 0 on Γ implies that ∇ϕ = ∂ϕ/∂n n on Γ, and from identity (2.50) ∂ϕ V (0) · n on Γ ⇒ ϕ˙ = − ∂n 2  1 ∂ϕ ⇒ L = − V (0) · n dΓ. Γ 2 ∂n

ϕ˙ = −∇ϕ · V (0)

For k = 2 the computation of the partial derivative of Lt = L(Ωt , ϕ ◦ Tt−1 ) is similar, with obvious changes:  1/2  1 −1 2 −1 2 Lt = |∆(ϕ ◦ Tt )| dx − |ϕ ◦ Tt | dx . 2 Ωt Ωt From formula (2.49)  def

L = ∂t Lt |t=0

 1 Γ |ϕ|2 V (0) · n dΓ |∆ϕ| V (0) · n dΓ − 2  |ϕ|2 dx1/2 Γ Ω   ϕ ϕ ˙ dx + ∆ϕ ∆ϕ˙ dx −  Ω  . 2 dx 1/2 Ω |ϕ| Ω

1 = 2



2

For ϕ ∈ E 2 (Ω), ϕ = 0 on Ω, ∆(∆ϕ) = λ(Ω)ϕ in Ω, and λ(Ω)ϕL2 (Ω) = 1,   1 2 |∆ϕ| V (0) · n dΓ + ∆ϕ ∆ϕ˙ dx − λ(Ω) ϕ ϕ˙ dx L = 2 Ω Ω Γ 1 ∂∆ϕ ∂ ϕ˙ = |∆ϕ|2 V (0) · n dΓ + ϕ˙ − ∆ϕ dΓ. 2 ∂n ∂n Γ Γ 



But ϕ = 0 and ∂ϕ/∂n = 0 on Γ imply that ∇ϕ = 0 on Γ and D2 ϕ = (D2 ϕ n) ∗n on Γ

⇒ ∆ϕ = (D2 ϕ n) · n on Γ.

4. Eigenvalue Problems

545

From identity (2.50) we have on Γ ϕ˙ = −∇ϕ · V (0)

⇒ ϕ˙ = 0 on Γ

⇒ ∇ϕ˙ = −∇(∇ϕ · V (0)) = −D ϕ V (0) − ∗DV (0)∇ϕ = −D2 ϕ V (0) ∂ ϕ˙ ⇒ = −D 2 ϕ V (0) · n = −D2 ϕ n · n V (0) · n = −∆ϕ V (0) · n, ∂n  1 2  |∆ϕ| V (0) · n dΓ. L =− 2 Γ 2

The quantity D 2 ϕ n · n is equal to ∂ 2 ϕ/∂n2 . Let bΩ be the oriented function associated with the domain Ω of class C 4 . It is C 4 in a neighborhood of Γ. Therefore, in that neighborhood   ∂ϕ ∂ϕ ∂ ∂ψ def , = = ∇ψ · ∇bΩ |Γ , ψ = ∇ϕ · ∇bΩ , ψ|Γ = ∂n ∂n ∂n ∂n ∇ψ = ∇(∇ϕ · ∇bΩ ) = D2 bΩ ∇ϕ + D2 ϕ∇bΩ , ∇ψ · ∇bΩ = D 2 bΩ ∇ϕ · ∇bΩ + D2 ϕ∇bΩ · ∇bΩ = D 2 ϕ∇bΩ · ∇bΩ ,   ∂ 2ϕ ∂ϕ ∂ = D2 ϕ∇bΩ · ∇bΩ Γ = D2 ϕ Γ n · n = ∆ϕ|Γ . = ∂n2 ∂n ∂n We summarize the results in the next theorem. Theorem 4.5. Given k = 1, 2, let Ω be a bounded open subset of RN of class C 2k . Assume that V ∈ C 0 ([0, τ [ ; W0k,∞ (D; RN )) for some bounded open domain D in RN such that Ω ⊂ D. Then for k = 1  2 ∂ϕ dλ(Ω; V ) = inf − V (0) · n dΓ, ˜ 1 (Ω) ϕ∈E Γ ∂n & − ∆ϕ = λ(Ω)ϕ in Ω def  ˜ 1 (Ω) = ϕ ∈ H01 (Ω) ∩ H 2 (Ω) : ; E and |ϕ|2 dx = 1 Ω

for k = 2

 2 2 ∂ ϕ dλ(Ω; V ) = inf − 2 V (0) · n dΓ, ˜ 2 (Ω) ϕ∈E Γ ∂n   ∆(∆ϕ) = λ(Ω)ϕ in Ω   def ˜ 2 (Ωt ) = ϕ ∈ H02 (Ω) ∩ H 4 (Ω) : . E  |ϕ|2 dx = 1  and Ω

In both cases λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) in Chapter 9 which is continuous with respect to V (0) ∈ C0 (D; RN ): dλ(Ω; V ) = dH λ(Ω; V (0)).

546

4.3

Chapter 10. Shape Gradients under a State Equation Constraint

Linear Elasticity

The constructions and results of the previous section readily extend to the vectorial case of linear elasticity: find U ∈ H01 (Ω)3 such that   Cε(U )·· ε(W ) dx = F · W dx (4.16) ∀W ∈ H01 (Ω)3 , Ω

Ω

for some distributed loading F ∈ L (Ω) and a constitutive law C which is a bilinear symmetric transformation of  def

Sym = τ ∈ L(R3 ; R3 ) : ∗ τ = τ 2

3

(L(R3 ; R3 ) is the space of all linear transformations of R3 or 3 × 3-matrices) under the following assumption. Assumption 4.1. The constitutive law is a linear bijective and symmetric transformation C : Sym → Sym for which there exists a constant α > 0 such that Cτ ·· τ ≥ α τ ·· τ for all τ ∈ Sym. For instance, for the Lam´e constants µ > 0 and λ ≥ 0, the special constitutive law Cτ = 2µ τ + λ tr τ I satisfies Assumption 4.1 with α = 2µ. The associated bilinear form is  def Cε(U )·· ε(W ) dx, aΩ (U, W ) = Ω

where the unknown is now a vector function. To make sense of (4.16) we shall use Korn’s inequality on a larger bounded open Lipschitzian domain D, Ω ⊂ D:   ∃cD > 0, ∀W ∈ H01 (D)3 , |W |2 dx ≤ cD |ε(W )|2 dx. D

D

In view of the considerations of the previous section, for a bounded open domain Ω and the velocity fields V ∈ C 0 ([0, τ [ ; W01,∞ (D; R3 )), for all 0 ≤ t < τ   1 3 2 |W | dx ≤ cD |ε(W )|2 dx. ∃cD > 0, ∀W ∈ H0 (Ωt (V )) , Ωt (V )

Ωt (V )

So for any F ∈ L (D) , the variational equation (4.16) with Ωt in place of Ω has a unique solution Ut in H01 (Ωt (V ))3 . With the same assumptions the first eigenvalue is given by the Rayleigh quotient & aΩt (U, U ) : ∀U ∈ H01 (Ωt )3 , U = 0 . λ(Ωt (V )) = inf  |U |2 dx Ωt 2

3

In view of the previous constructions H01 (Ωt ) can be replaced by H01 (Ωt ; D): & aΩt (U, U ) 1 3 λ(Ωt (V )) = inf  : ∀U ∈ H0 (Ωt ; D) , U = 0 . (4.17) |U |2 dx Ωt

4. Eigenvalue Problems

547

Theorem 4.6. Let Ω be a bounded open subset of R3 . Assume that V ∈ C 0 ([0, τ [ ; W01,∞ (D; R3 )) for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. There exists at least one nonzero solution U ∈ H01 (Ωt ; D)3 to the minimization problem (4.17), λ(Ωt (V )) ≥ λ(D) > 0, and   1 3 2 −1 |W | dx ≤ λ(D) ε(W )2 dx. ∀W ∈ H0 (D) , D

D

The solutions are completely characterized by the following variational equation: there exists U ∈ H01 (Ωt ; D)3 such that  1 3 ∀W ∈ H0 (Ωt ; D) , aD (U, W ) = λ(Ωt (V )) U · W dx (4.18) D

or equivalently



∀W ∈

H01 (Ωt )3 ,

U · W dx.

aΩt (U, W ) = λ(Ωt (V ))

(4.19)

Ωt

G. Auchmuty [1]’s dual variational principle for this eigenvalue problem can be chosen as 

def µ(Ωt ) = inf L(Ωt , U ) : U ∈ H01 (Ωt )3 , (4.20)  1/2 def 1 |U |2 dx . (4.21) L(Ωt , U ) = aΩt (U, U ) − 2 Ωt By using the embedding of H01 (Ωt ) into H01 (D), this problem can be rewritten as

 def µ(Ωt ) = inf L(D, U ) : U ∈ H01 (Ωt ; D)3 ,  1/2 def 1 2 |U | dx . L(D, U ) = aD (U, U ) − 2 D

(4.22) (4.23)

Theorem 4.7. Let Ω be a bounded open subset of R3 . Assume that V ∈ C 0 ([0, τ [ ; W01,∞ (D; R3 )) for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. Then for 0 ≤ t < τ, 1 (4.24) µ(Ωt ) = − 2λ(Ωt ) and the set of minimizers of (4.20) is given by   U is solution of (4.18) and     def 1/2 E(Ωt ) = U ∈ H01 (Ωt ; D)3 :   .  |U |2 dx = 1/λ(Ωt )   D

(4.25)

548

Chapter 10. Shape Gradients under a State Equation Constraint From the identity λ(Ωt ) = −

1 , 2µ(Ωt )

if dµ(Ω; V ) exists, then dλ(Ω; V ) exists and dλ(Ω; V ) =

1 dµ(Ω; V ) = 2 λ(Ω)2 dµ(Ω; V ). 2µ(Ω)2

We again use the function space parametrization of section 2.4 in conjunction with Theorem 2.1. From the characterization (4.2) of H01 (Ωt ; D), define the new functional: for each U ∈ H01 (D)3 ˜ U ) def = L(D, U ◦ Tt−1 (V )) L(t, =

1 aD (U ◦ Tt−1 (V ), U ◦ Tt−1 (V )) − 2

After a change of variables for U ∈ ˜ U) = 1 L(t, 2

 Cε(U ◦

Tt−1 )



1/2 |U ◦ Tt−1 (V )|2 dx . D

H01 (D)3 ,

◦ Tt ·· ε(U ◦

Tt−1 )

 ◦ Tt Jt dx −

D

1/2 Jt |U | dx , 2

D

where Jt = det(DTt ) and D(U ◦ Tt−1 ) ◦ Tt = D(U )(DTt )−1 , 2 ε(U ◦ Tt−1 ) ◦ Tt = D(U )(DTt )−1 + ∗(DTt )−1 ∗D(U ). Defining the transformation




τ ·· σ def = C τ (DTt )−1 + ∗(DTt )−1 ∗τ ·· σ(DTt )−1 + ∗(DTt )−1 ∗σ , 4 C(t) the previous expression can be written in the following more compact form:  1/2  1 2
˜ C(t)D(U )·· D(U ) Jt dx − L(t, U ) = Jt |U | dx . 2 D D To apply Theorem 2.1, choose   def def X(t) = U t = Ut ◦ Tt : ∀Ut ∈ E(Ωt ) endowed with the weak topology of H01 (D)3 . Assumption (H1) is clearly satisfied. ˜ U ) is differentiable. For 0 = U ∈ H01 (D)3 , For all U = 0, L(t,  1
˜
 (t)D(U )·· D(U ) Jt + C(t)D(U ∂t L(t, U ) = )·· D(U ) Jt dx C 2 D  1 |U |2 Jt dx, −  1/2 2 D J |U | dx D t

4. Eigenvalue Problems

549

where


 (t) τ ·· σ = C {τ T  (t) + ∗T  (t) ∗τ } ·· σ(DTt )−1 + ∗(DTt )−1 ∗σ 4C

 + C τ (DTt )−1 + ∗(DTt )−1 ∗τ ·· {σT  (t) + ∗T  (t) ∗σ} , T (t) = (DTt )−1 , def

T  (t) = −(DTt )−1 DV (t) ◦ Tt .

Hence assumptions (H2) and (H4) are satisfied. In t = 0 the above expressions simplify. For U ∈ E(Ω)  1 ˜
 (0)D(U )·· D(U ) + C ε(U )·· ε(U ) div V (0) dx C ∂t L(0, U ) = 2 D  1 −  |U |2 div V (0) dx 1/2 2 dx D |U | D and for all U and W
 (0) D(U )·· D(W ) = − C {D(U ) DV (0) + ∗DV (0) ∗D(U )} ·· ε(W ) 2C − C ε(U )·· {D(W ) DV (0) + ∗DV (0) ∗D(W )} . This is another example of the application of Theorem 2.1 to a case where the set of minimizers is not a singleton and for which we have a complete description of the Eulerian semiderivative. Theorem 4.8. Let Ω be a bounded open subset of R3 . Assume that V ∈ C 0 ([0, τ [ ; W01,∞ (D; R3 )) for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. Then  1
 (0)D(U )·· D(U ) dx dλ(Ω; V ) = inf − C ˜ 2 U ∈E(Ω) Ω    1 C ε(U )·· ε(U ) − λ(Ω)|U |2 div V (0) dx + 2 Ω = inf − C ε(U )·· {D(U ) DV (0) + ∗DV (0) ∗D(U )} dx ˜ U ∈E(Ω)

Ω

 

 1 C ε(U )·· ε(U ) − λ(Ω)|U |2 div V (0) dx, Ω 2   −→   − div (ε(U )) = λ(Ω)U in D(Ω) def ˜  = U ∈ H01 (Ω)3 : . E(Ω)   and |U |2 dx = 1 +

Ω

λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) in Chapter 9 which is continuous with respect to V (0) ∈ W01,∞ (D; R3 ): dλ(Ω; V ) = dH λ(Ω; V (0)).

550

Chapter 10. Shape Gradients under a State Equation Constraint

If Ω is assumed to be of class C 2 , the eigenvector functions belong to H01 (Ω)3 ∩ H (Ω)3 and the volume expressions of the previous theorem can be expressed as boundary integrals as in section 2.5. In that case we can use formulae (2.49) and (2.50) to compute the partial derivative of Lt = L(Ωt , U ◦ Tt−1 ):  1/2  1 Lt = C ε(U ◦ Tt−1 )·· ε(U ◦ Tt−1 ) dx − |U ◦ Tt−1 |2 dx . 2 Ωt Ωt 2

From formula (2.49)  def

L = ∂t Lt |t=0

 1 Γ |U |2 V (0) · n dΓ C ε(U )·· ε(U ) V (0) · n dΓ − 2  |U |2 dx1/2 Γ Ω   ˙ dx U · U + C ε(U )·· ε(U˙ ) dx −  Ω 1/2 . Ω |U |2 dx Ω

1 = 2



−→

For U ∈ E(Ω), U = 0 on Γ, λ(Ω)U L2 (Ω) = 1, and −div (ε(U )) = λ(Ω)U in Ω,   1  L = C ε(U )·· ε(U ) V (0) · n dΓ + C ε(U )·· ε(U˙ ) − λ(Ω) U · U˙ dx 2 Γ Ω  1 C ε(U )·· ε(U ) V (0) · n + [C ε(U )]n · U˙ dΓ. = Γ 2 But U = 0 on Γ implies that DΓ (U ) = 0, D(U ) = [D(U )]n ∗n on Γ, and from identity (2.50) U˙ = −D(U )V (0) ∈ H01 (Ω)3 ⇒ U˙ = −[D(U )]n V (0) · n on Γ   ⇒ L = − [C ε(U )]n · [D(U )]n V (0) · n dΓ. Γ

This expression can be rewritten only in terms of ε(U ) as follows: 2 ε(U ) = D(U )n ∗n + n ∗(D(U )n),

2 ε(U )n = D(U )n + (D(U )n) · n n, 2 ε(U )n · n = 2 D(U )n · n, [C ε(U )]n · D(U )n = 2 [C ε(U )]n · ε(U )n − [C ε(U )]n · n (ε(U )n) · n.

We summarize the results in the next theorem. Theorem 4.9. Let Ω be a bounded open subset of R3 of class C 2 . Assume that V ∈ C 0 ([0, τ [ ; W01,∞ (D; R3 )) for some bounded open Lipschitziandomain D in R3 such that Ω ⊂ D. Then dλ(Ω; V ) = inf − [C ε(U )]n · [D(U )]n V (0) · n dΓ, ˜  U ∈E(Ω) Γ dλ(Ω; V ) = inf − 2 [C ε(U )]n · ε(U )n ˜ U ∈E(Ω) Γ − [C ε(U )]n · n (ε(U )n) · n V (0) · n dΓ, −→   −div (ε(U )) = λ(Ω) U in Ω   def ˜ . E(Ω) = U ∈ H01 (Ω)3 ∩ H 2 (Ω)3 :   |U |2 dx = 1 and Ω

5. Saddle Point Formulation and Function Space Parametrization

551

For the special constitutive law Cτ = 2µτ + λtr τ I, 

 4 µ|[ε(U )]n|2 + (λ − 2µ) |tr ε(U )|2 V (0) · n dΓ. dλ(Ω; V ) = inf − ˜ U ∈E(Ω)

Γ

λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) in Chapter 9 which is continuous with respect to V (0) ∈ C0 (D; R3 ): dλ(Ω; V ) = dH λ(Ω; V (0)).

5 5.1

Saddle Point Formulation and Function Space Parametrization An Illustrative Example

Let Ω be a bounded open domain in RN with a smooth boundary Γ. Let y = y(Ω) be the solution of the Neumann problem −y + y = f in Ω,

∂y = 0 on Γ, ∂n

(5.1)

where f is a fixed function in H 1 (RN ). Associate with y(Ω) the objective function  1 J(Ω) = |y(Ω) − yd |2 dx, (5.2) 2 Ω where yd is a fixed function in H 1 (RN ). The solution of (5.1) coincides with the minimizing element of the following variational problem:

 inf E(Ω, ϕ) : ϕ ∈ H 1 (Ω) , (5.3)  

def 1 (5.4) E(Ω, ϕ) = |∇ϕ|2 + ϕ2 − 2 f ϕ dx. 2 Ω The minimizing element y of (5.4) is the solution in H 1 (Ω) of Euler’s equation dE(Ω, y; ϕ) = 0, ∀ϕ ∈ H 1 (Ω),  [∇y · ∇ϕ + yϕ − f ϕ] dx, dE(Ω, y; ϕ) =

(5.5) (5.6)

Ω

which is the variational equation for y. The objective function J(Ω) is a shape functional, and the solution of (5.1) will be called the state. It is convenient to introduce the objective function  def 1 |ϕ − yd |2 dx, (5.7) F (Ω, ϕ) = 2 Ω which clearly expresses the dependence on Ω and ϕ. To sum up, we consider the objective function   J(Ω) = F Ω, y(Ω) , (5.8)

552

Chapter 10. Shape Gradients under a State Equation Constraint

where y = y(Ω) is the solution of y ∈ H 1 (Ω),

∀ϕ ∈ H 1 (Ω),

dE(Ω, y; ϕ) = 0.

(5.9)

We wish to find an expression for the shape derivative dJ(Ω; V ).

5.2

Saddle Point Formulation

The basic approach is the one of control theory. Equation (5.1) (or in its variational form (5.9)) is considered as a state constraint in the minimization problem. We construct a Lagrangian functional by introducing a Lagrange multiplier function or the so-called adjoint state ψ: G(Ω, ϕ, ψ) = F (Ω, ϕ) + dE(Ω, ϕ; ψ).

(5.10)

Then the objective function is given by J(Ω) =

min

sup

G(Ω, ϕ, ψ)

(5.11)

F (Ω, y(Ω)), if ϕ = y(Ω), +∞, if ϕ = y(Ω).

(5.12)

ϕ∈H 1 (Ω) ψ∈H 1 (Ω)

since G(Ω, ϕ, ψ) =

sup ψ∈H 1 (Ω)

In our example the Lagrangian G is convex and continuous with respect to the variable ϕ and concave and continuous with respect to the variable ψ. Moreover, the space H 1 (Ω) is convex and closed. So the functional G has a saddle point if and only if the saddle point equations have a solution (y, p) (cf. I. Ekeland and R. Temam [1]): p ∈ H 1 (Ω),

dG(Ω, y, p; 0, ψ) = 0, ∀ψ ∈ H 1 (Ω),

(5.13)

y ∈ H (Ω),

dG(Ω, y, p; ϕ, 0) = 0, ∀ϕ ∈ H (Ω).

(5.14)

1

1

They are completely equivalent to y ∈ H 1 (Ω), p ∈ H (Ω), 1

dE(Ω, y; ψ) = 0, ∀ψ ∈ H 1 (Ω),

(5.15)

dF (Ω, y; ϕ) + d E(Ω, y; p; ϕ) = 0, ∀ϕ ∈ H (Ω), 2

1

(5.16)

or ∂y = 0 on Γ, ∂n ∂p −p + p + y − yd = 0 in Ω, = 0 on Γ. ∂n −y + y = f in Ω,

(5.17) (5.18)

System (5.15)–(5.16) has a unique solution in H 1 (Ω) × H 1 (Ω) which coincides with the unique saddle point of G(Ω, ϕ, ψ) in H 1 (Ω) × H 1 (Ω).

5. Saddle Point Formulation and Function Space Parametrization

5.3

553

Function Space Parametrization

We have shown that the objective function J