Stable solutions of elliptic partial differential equations

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors H. Brezis, Université de Paris...

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CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics Main Editors

H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board

R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-6654-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

A mon père, à mon fils.

Contents Preface

xi

1 Defining stability 1.1 Stability and the variations of energy . . . . . . . . . . . 1.1.1 Potential wells . . . . . . . . . . . . . . . . . . . . 1.1.2 Examples of stable solutions . . . . . . . . . . . . 1.2 Linearized stability . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Principal eigenvalue of the linearized operator 1.2.2 New examples of stable solutions . . . . . . . . . 1.3 Elementary properties of stable solutions . . . . . . . . 1.3.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Nonuniqueness . . . . . . . . . . . . . . . . . . . . 1.3.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dynamical stability . . . . . . . . . . . . . . . . . . . . . . 1.5 Stability outside a compact set . . . . . . . . . . . . . . . 1.6 Resolving an ambiguity . . . . . . . . . . . . . . . . . . .

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1 1 1 5 9 9 11 15 15 16 18 20 24 26

2 The Gelfand problem 2.1 Motivation . . . . . . . . 2.2 Dimension N = 1 . . . . 2.3 Dimension N = 2 . . . . 2.4 Dimension N ≥ 3 . . . . 2.4.1 Stability analysis 2.5 Summary . . . . . . . . .

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29 29 30 34 35 39 44

3 Extremal solutions 3.1 Weak solutions . . . . . . . . . . . . . . . . . . . . 3.1.1 Defining weak solutions . . . . . . . . . 3.2 Stable weak solutions . . . . . . . . . . . . . . . 3.2.1 Uniqueness of stable weak solutions . . 3.2.2 Approximation of stable weak solutions

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3.3 The stable branch . . . . . . . . . . . . . . . . . . 3.3.1 When is λ∗ finite? . . . . . . . . . . . . . 3.3.2 What happens at λ = λ∗ ? . . . . . . . . . 3.3.3 Is the stable branch a (smooth) curve? 3.3.4 Is the extremal solution bounded? . . .

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58 61 62 69 73

4 Regularity theory of stable solutions 4.1 The radial case . . . . . . . . . . . . . . . . . . . . . . . 4.2 Back to the Gelfand problem . . . . . . . . . . . . . . 4.3 Dimensions N = 1, 2, 3 . . . . . . . . . . . . . . . . . 4.4 A geometric Poincaré formula . . . . . . . . . . . . . 4.5 Dimension N = 4 . . . . . . . . . . . . . . . . . . . . . 4.5.1 Interior estimates . . . . . . . . . . . . . . . . 4.5.2 Boundary estimates . . . . . . . . . . . . . . . 4.5.3 Proof of Theorem 4.5.1 and Corollary 4.5.1 4.6 Regularity of solutions of bounded Morse index . .

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75 75 80 82 85 88 88 94 95 96

5 Singular stable solutions 5.1 The Gelfand problem in the perturbed ball . . . . . . . . . 5.2 Flat domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Partial regularity of stable solutions in higher dimensions 5.3.1 Approximation of singular stable solutions . . . . . 5.3.2 Elliptic regularity in Morrey spaces . . . . . . . . . . 5.3.3 Measuring singular sets . . . . . . . . . . . . . . . . . 5.3.4 A monotonicity formula . . . . . . . . . . . . . . . . . 5.3.5 Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . .

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99 99 110 115 116 119 123 125 130

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137 . 137 . 141 . 141 . 143 . 145 . 147 . 147 . 154 . 158

6 Liouville theorems for stable solutions 6.1 Classifying radial stable entire solutions . . . . . . . . . . . 6.2 Classifying stable entire solutions . . . . . . . . . . . . . . . 6.2.1 The Liouville equation . . . . . . . . . . . . . . . . . 6.2.2 Dimension N = 2 . . . . . . . . . . . . . . . . . . . . . 6.2.3 Dimensions N = 3, 4 . . . . . . . . . . . . . . . . . . 6.3 Classifying solutions that are stable outside a compact set 6.3.1 The critical case . . . . . . . . . . . . . . . . . . . . . 6.3.2 The supercritical range . . . . . . . . . . . . . . . . . 6.3.3 Flat nonlinearities . . . . . . . . . . . . . . . . . . . .

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7 A conjecture of De Giorgi 163 7.1 Statement of the conjecture . . . . . . . . . . . . . . . . . . . . . . . 163 7.2 Motivation for the conjecture . . . . . . . . . . . . . . . . . . . . . . 164 7.2.1 Phase transition phenomena . . . . . . . . . . . . . . . . . . 164 viii

7.2.2 Monotone solutions and global minimizers 7.2.3 From Bernstein to De Giorgi . . . . . . . . . 7.3 Dimension N = 2 . . . . . . . . . . . . . . . . . . . . 7.4 Dimension N = 3 . . . . . . . . . . . . . . . . . . . .

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166 172 173 174

8 Further readings 8.1 Stability versus geometry of the domain . . . . . . . . . 8.1.1 The half-space . . . . . . . . . . . . . . . . . . . . 8.1.2 Domains with controlled volume growth . . . . 8.1.3 Exterior domains . . . . . . . . . . . . . . . . . . . 8.2 Symmetry of stable solutions . . . . . . . . . . . . . . . . 8.2.1 Foliated Schwarz symmetry . . . . . . . . . . . . 8.2.2 Convex domains . . . . . . . . . . . . . . . . . . . 8.3 Beyond the stable branch . . . . . . . . . . . . . . . . . . 8.3.1 Turning point . . . . . . . . . . . . . . . . . . . . . 8.3.2 Mountain-pass solutions . . . . . . . . . . . . . . 8.3.3 Uniqueness for small λ . . . . . . . . . . . . . . . 8.3.4 Regularity of solutions of bounded Morse index 8.4 The parabolic equation . . . . . . . . . . . . . . . . . . . . 8.5 Other energy functionals . . . . . . . . . . . . . . . . . . . 8.5.1 The p-Laplacian . . . . . . . . . . . . . . . . . . . 8.5.2 The biharmonic operator . . . . . . . . . . . . . . 8.5.3 The fractional Laplacian . . . . . . . . . . . . . . 8.5.4 The area functional . . . . . . . . . . . . . . . . . 8.5.5 Stable solutions on manifolds . . . . . . . . . . .

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179 179 179 181 183 184 184 186 186 186 187 188 191 191 194 194 195 196 199 199

A Maximum principles A.1 Elementary properties of the Laplace operator . . . . A.2 The maximum principle . . . . . . . . . . . . . . . . . . A.3 Harnack’s inequality . . . . . . . . . . . . . . . . . . . . A.4 The boundary-point lemma . . . . . . . . . . . . . . . . A.5 Elliptic operators . . . . . . . . . . . . . . . . . . . . . . A.6 The Laplace operator with a potential . . . . . . . . . A.7 Thin domains and unbounded domains . . . . . . . . A.8 Nonlinear comparison principle . . . . . . . . . . . . . A.9 L 1 theory for the Laplace operator . . . . . . . . . . . . A.9.1 Linear theory and weak comparison principle A.9.2 The boundary-point lemma . . . . . . . . . . . A.9.3 Sub- and supersolutions in the L 1 setting . . .

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203 203 208 209 210 214 216 220 221 222 222 225 226

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B Regularity theory for elliptic operators B.1 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Interior regularity . . . . . . . . . . . . . . . . . . . . B.1.2 Solving the Dirichlet problem on the unit ball . . . B.1.3 Solving the Dirichlet problem on smooth domains B.2 Schauder estimates . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Poisson’s equation on the unit ball . . . . . . . . . . B.2.2 A priori estimates for C 2,α solutions . . . . . . . . . B.2.3 Existence of C 2,α solutions . . . . . . . . . . . . . . . B.3 Calderon-Zygmund estimates . . . . . . . . . . . . . . . . . . B.4 Moser iteration . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 The inverse-square potential . . . . . . . . . . . . . . . . . . B.5.1 The kernel of L = −∆ − |x|c 2 . . . . . . . . . . . . . . B.5.2 Functional setting . . . . . . . . . . . . . . . . . . . . B.5.3 The case ξ = 0 . . . . . . . . . . . . . . . . . . . . . . B.5.4 The case ξ 6= 0 . . . . . . . . . . . . . . . . . . . . . .

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233 233 233 235 237 240 240 247 249 252 253 257 258 259 260 268

C Geometric tools C.1 Functional inequalities . . . . . . . . . . . . . . . . . . . C.1.1 The isoperimetric inequality . . . . . . . . . . . C.1.2 The Sobolev inequality . . . . . . . . . . . . . . C.1.3 The Hardy inequality . . . . . . . . . . . . . . . C.2 Submanifolds of RN . . . . . . . . . . . . . . . . . . . . . C.2.1 Metric tensor, tangential gradient . . . . . . . . C.2.2 Surface area of a submanifold . . . . . . . . . . C.2.3 Curvature, Laplace-Beltrami operator . . . . . C.2.4 The Sobolev inequality on submanifolds . . . C.3 Geometry of level sets . . . . . . . . . . . . . . . . . . . C.3.1 Coarea formula . . . . . . . . . . . . . . . . . . . C.4 Spectral theory of the Laplace operator on the sphere

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273 273 273 275 276 278 279 281 282 287 294 295 297

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References

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Index

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x

Preface On the shelves of the library where you may have picked this book, there were probably many more volumes dealing with nonlinear elliptic partial differential equations (PDEs). Thank you for opening this one. What you will find here is a look at the subject through the loophole of stability. I have tried to present the main ideas on the most simple-looking equations. One could even argue that only one equation is treated, this one: −∆u = λeu

(P.1)

Chapter 1 What is a stable solution exactly? Oftentimes in physics, one is interested in finding the state of a system of lowest energy E. Such a state is stable because it is trapped at the bottom of an energy well. Through calculus, this leads to solving an equality, E 0 = 0 (critical points), and then an inequality, E 00 ≥ 0. Accordingly, thinking of solutions of PDEs as critical points of an energy functional, we say that a solution is stable when the second variation of energy is nonnegative. Equivalently, the operator associated with the linearized equation must have a nonnegative spectrum. Yet another way of thinking of stable solutions is through their dynamical properties (asymptotic stability). Using these different points of view, we give examples of several classes of solutions that are stable, such as local minimizers, minimal solutions, and monotone solutions, and we study their basic properties. This being done, we introduce stability outside a compact set, a notion general enough to encompass all (classical) solutions defined on, say, bounded domains. Chapter 2 The PDE (P.1), posed on the unit ball with a homogeneous Dirichlet boundary condition, is our model problem. We construct all radial solutions and classify them according to their stability properties (their Morse index). The resulting picture, Figure P.1, is the object of this chapter and will serve as a paradigm throughout the book. As you can see, the picture depends very much on the xi

Preface kukL∞ (B)

λ∗ 1≤N ≤2

kukL∞ (B)

kukL∞ (B)

λ

2(N − 2)

λ∗

3≤N ≤9

λ

λ∗

λ

N ≥ 10

Figure P.1: Bifurcation diagram for the Gelfand problem.

space dimension N . Look, for example, at stable solutions: for 1 ≤ N ≤ 9, the stable branch is the lowest part of the curve, before it bends back to the left. In dimension N ≥ 10, the stable branch is the whole curve. Also observe this: stable solutions are uniformly bounded in dimension 1 ≤ N ≤ 9, while they are not for N ≥ 10. Chapter 3 In fact, there still exists a weak (unbounded) solution to the equation for N ≥ 10 and λ = λ∗ . In this chapter, we study such stable weak solutions to equations of the type −∆u = f (u) (P.2) Under fairly general assumptions, we show that they are unique and arise as limits of classical stable solutions, much as in Figure P.1. For this reason, they earn the name of extremal solutions. We further discuss qualitative properties of the stable branch, in the context of (P.2), except for one major issue— determining when the extremal solution is bounded—which we save for the next chapter. Chapter 4 We treat the issue of regularity under three different restrictive assumptions: the solution is known to be radial, the nonlinearity has some specific scaling features, such as f (u) = eu , or the space dimension is small. In the latter case, xii

Preface we introduce a restatement of stability in terms of the level sets of solutions, known as the geometric Poincaré formula. In the last section of the chapter, we address the question of regularity for unstable solutions. This is done through a blow-up argument and taking advantage of a Liouville-type theorem, the proof of which we postpone to Chapter 6. Chapter 5 In the previous chapter, we learned that in low dimensions, at least, stable solutions must be smooth. What happens in large dimensions? Figure P.1 shows that for N ≥ 10, there exists a singular stable solution to (P.1) on the unit ball. In this chapter, we prove that this remains true on any domain that is sufficiently close to the unit ball, for example, a round ellipsoid. On the contrary, if the ellipsoid is very flat, the extremal solution is smooth. We conclude this chapter by estimating the Hausdorff dimension of the singular set of stable solutions. Such a result is presented for the Lane-Emden equation, −∆u = u p , p > 1, taking advantage of a monotonicity formula for an appropriate rescaled energy functional. Chapter 6 In Chapter 4, we derived a regularity theory for solutions to (P.1) in terms of their Morse index. To do this, we used a Liouville-type theorem for solutions to the same equation, posed in the entire Euclidean space. In this chapter, we review the available classification results for stable solutions to (P.2), first in the radial case, then in the context of (P.1), and finally in the general context of (P.2). In the last part of the chapter, we classify solutions to (P.1) that are stable outside a compact set and show in an example that such a result cannot hold for general nonlinearities. Chapter 7 Chapter 7 is dedicated to a specific class of stable solutions, the monotone solutions, which arise naturally in the context of phase transition phenomena. We discuss the heuristics behind a celebrated conjecture of De Giorgi, its relation to the theory of minimal surfaces, and present the proof of the conjecture in dimensions N = 2 and N = 3. Chapter 8 The final chapter contains few proofs. I have tried instead to list some of the directions of current research on stable solutions: classification results in half-spaces and other geometries, partial symmetry of solutions of low Morse index, advanced topics on bifurcation diagrams, extensions to quasi-linear, higher-order and nonlocal equations, and stable solutions on manifolds.

xiii

Preface Appendices If you are new to the subject, Appendices A, B, and C1 are intended to walk you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities used in this book. Two additional topics, the inverse-square potential and some background material on submanifolds of Euclidean space, are also included. I wish to thank Haim Brezis for giving me the opportunity to write this book. I am indebted to Xavier Cabré, Juan Dávila, and Alberto Farina for the many discussions we have had on the subject. The material presented here shows in itself the great influence they have had on my research. Thanks also to Jean Dolbeault, Filippo Gazzola, and Enrico Valdinoci, for their sharp answers to my picky questions. I feel lucky to be a member of the Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées (LAMFA), Unité Mixte de Recherche Centre National de la Recherche Scientifique (UMR CNRS) 6140. Thanks to Olivier Goubet for creating such a great work environment and for revising this manuscript. Bergamote, pour le dire brièvement, merci de tes encouragements de chaque instant.

If you find mistakes or typos, please notify me at [email protected].

xiv

Chapter 1 Defining stability 1.1

Stability and the variations of energy

Intuitively, a system is in a stable state if it can recover from perturbations; a small change will not prevent the system from returning to equilibrium. Place a marble at the center of a smooth bowl and tap it slighty. After some rolling back and forth, the marble will return to its stable position. If instead you turned the bowl over and put the marble carefully on top at the center, then it would be in a rather unstable equilibrium; a slight breeze would suffice to make it fall. Such phenomena can be understood by looking at the variations of the energy of the system.

1.1.1

Potential wells

Consider a smooth function E : R → R and think of it as the potential energy of a system, a quantity that varies with respect to a relevant physical parameter, a quantity that the system tends to minimize. A potential well is a neighborhood of a local minimum of E: potential energy captured in it is unable to convert to another type of energy. As such, a potential well is a stable state of this system.

1

Stable solutions of elliptic PDEs

Figure 1.1: A potential well.

From the mathematical point of view, it is clear that t 0 ∈ R is a point of local minimum of E as soon as E 0 (t 0 ) = 0 and E 00 (t 0 ) > 0. Relaxing the inequality, we obtain the following definition: Definition 1.1.1 Let I denote an open interval of R and E : I → R a function of class C 2 . t 0 ∈ I is a stable 1 critical point of E if E 0 (t 0 ) = 0 and E 00 (t 0 ) ≥ 0. Exercise 1.1.1 Prove that a point t 0 of local minimum of E is a stable critical point. Is the converse true? For the marble rolling inside a bowl, the potential energy of the system can be effectively described as a function of one parameter, the distance of the marble to the center of the bowl. The graph of E is then a parabola pointing up, as in Figure 1.1 (or pointing down, in the case where the bowl is turned over). In most situations, however, the state of a system cannot be described 1

A point t 0 satisfying Definition 1.1.1 is usually called semi-stable in the literature, while it is said to be stable if the strict inequality E 00 (t 0 ) > 0 holds. Since the theory we are about to develop does not use the strict inequality, we prefer to make no distinction between semistable and stable solutions.

2

Chapter 1. Defining stability by a single parameter t ∈ R. Instead, we shall work in a context where the energy is defined on a suitable functional space X . Consider for example a bounded region Ω of the Euclidean space RN , with smooth boundary ∂ Ω, and let Y = C 2 (Ω). Among the functions in Y , we select those that vanish on the boundary ∂ Ω and denote by X = C02 (Ω) the subspace thus formed. Now consider the energy functional EΩ : X → R defined for u ∈ X by ˆ ˆ 1 2 EΩ (u) = |∇u| d x − F (u) d x, (1.1) 2 Ω Ω where F : R → R is a given function of class C 2 . Fixing u ∈ X , we study the variations of the energy along a given direction2 ϕ ∈ X \ {0}, that is, we consider the function E : R → R defined for t ∈ R by E(t) = EΩ (u + tϕ).

(1.2)

When is 0 a stable critical point of E? To find out, compute the difference quotient of E at 0: ˆ ˆ ˆ t F (u + tϕ) − F (u) E(t) − E(0) 2 = ∇u∇ϕ d x + |∇ϕ| d x − d x. t 2 Ω t Ω Ω Taking |t| ≤ 1 and writing a = kuk L ∞ (Ω) + kϕk L ∞ (Ω) , the mean-value theorem implies that (F (u + tϕ) − F (u))/t ≤ k f k L ∞ ([−a,a]) , where f = F 0 . By the dominated convergence theorem, we may pass to the limit as t → 0: E 0 (0) = 0 if and only if ˆ ˆ f (u)ϕ d x. ∇u∇ϕ d x = Ω

Ω

Integrating by parts, we see that ˆ (−∆u − f (u))ϕ d x = 0. Ω

If u belongs to X and the above equality holds for arbitrary ϕ ∈ X , then we readily deduce (using, for example, [25, Lemma IV.2]) that u solves the following semilinear partial differential equation (PDE), also known as the EulerLagrange equation for the energy functional (1.1): −∆u = f (u) in Ω, (1.3) u=0 on ∂ Ω. Given λ > 0, the variations of Eλ defined by Eλ (t) := E(λt) can be readily deduced from those of E. So, we may always restrict to the case where ϕ belongs to the unit sphere of X (we then say that ϕ is a direction). 2

3

Stable solutions of elliptic PDEs This holds in particular if u is a critical point of EΩ , that is if EΩ is differentiable at u and its Fréchet derivative satisfies DEΩ (u) = 0 (see, for example, [51] for Fréchet derivatives and calculus in Banach spaces). Exercise 1.1.2 Prove that EΩ defined by (1.1) is differentiable in X and prove that for every u ∈ X , the Fréchet derivative of EΩ at u is given by ˆ ˆ f (u)ϕ d x, for every ϕ ∈ X . DEΩ (u).ϕ = ∇u∇ϕ d x − Ω

Ω

We have just identified solutions u ∈ X of (1.3) with critical points of EΩ . We summarize this result in the following proposition. Proposition 1.1.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1. Let f : R → R denote a function of class C 1 and F an antiderivative of f . Let X = C02 (Ω) = {u ∈ C 2 (Ω) : u(x) = 0 for all x ∈ ∂ Ω} endowed with its natural norm k · kC 2 (Ω) . Let EΩ : X → R defined by (1.1). Then, u ∈ C 2 (Ω) is a solution to (1.3) if and only if u is a critical point of EΩ . Which of these solutions should be called stable? To find out, we compute E 00 (0): E 0 (t) − E 0 (0)

=

t

E 0 (t)

DEΩ (u + tϕ).ϕ

t ˆ ˆ 1 2 ∇u∇ϕ d x + |∇ϕ| d x − f (u + tϕ)ϕ d x. = t Ω Ω t Ω ˆ ˆ f (u + tϕ) − f (u) 2 = |∇ϕ| d x − ϕ d x. t Ω Ω 1

ˆt

=

By a standard dominated convergence argument, ˆ ˆ 00 2 f 0 (u)ϕ 2 d x. E (0) = |∇ϕ| d x − Ω

Ω

Recalling Definition 1.1.1, it is natural to define stability as follows. Definition 1.1.2 Let f ∈ C 1 (R) and let Ω denote an open set of RN , N ≥ 1. A solution u ∈ C 2 (Ω) of −∆u = f (u) in Ω (1.4) is stable if Q u (ϕ) :=

ˆ

ˆ 2

Ω

|∇ϕ| d x −

Ω

f 0 (u)ϕ 2 d x ≥ 0, 4

∀ ϕ ∈ Cc1 (Ω).

(1.5)

Chapter 1. Defining stability Remark 1.1.1 • The quadratic form Q u is called the second variation of the energy EΩ . • u is stable in Ω if and only if u is stable in every subdomain ω ⊂⊂ Ω. In particular, the notion of stability is relevant also for unbounded and/or nonsmooth domains Ω. • However, if u is stable in two given domains ω1 , ω2 , u need not be stable in ω1 ∪ ω2 . See Proposition 1.5.1. • Using a standard approximation argument, if Ω is bounded or merely if f 0 (u)− is bounded in Ω, one can take ϕ ∈ H01 (Ω) in the above definition3 .

1.1.2

Examples of stable solutions

In the following list of examples, we assume again that Ω is smoothly bounded. Example 1.1.1 Local minimizers of the energy are stable. Definition 1.1.3 Let X denote a Banach space of functions, with norm k · k. Assume X contains Cc1 (Ω) and let EΩ : X → R, a functional. u ∈ X is a local minimizer of EΩ if there exists t 0 > 0 such that EΩ (u) ≤ EΩ (u + ϕ)

for all ϕ ∈ Cc1 (Ω) s.t. kϕk < t 0 .

In particular, assume u is a local minimizer of EΩ defined by (1.1). Then 0 is a point of local minimum of E defined by (1.2). We deduce that E 0 (0) = 0 and E 00 (0) ≥ 0, and so u is a stable solution to (1.3). Example 1.1.2 ([35]) Critical points that minimize the energy from one side are stable. Definition 1.1.4 Let X denote a Banach space of functions containing Cc1 (Ω) and EΩ : X → R, a functional. u ∈ X is a one-sided local minimizer of EΩ if there exists t 0 > 0 and " ∈ {−1, +1} such that EΩ (u) ≤ EΩ (u + "ϕ) 3

for all ϕ ∈ Cc1 (Ω) s.t. ϕ ≥ 0 and kϕk < t 0 .

See, for example, [25] for an introduction to Sobolev spaces such as H01 (Ω).

5

Stable solutions of elliptic PDEs Assume that u is both a critical point and a one-sided minimizer of EΩ defined by (1.1) and assume for example that " = +1. Then, given ϕ ≥ 0, 0 is a point of local minimum of E|[0,t 0 ) , where E is defined by (1.2). We deduce that E 0 (0) = 0 and E 00 (0) ≥ 0, and so (1.5) holds for ϕ ≥ 0. Now take an arbitrary test function ϕ ∈ Cc1 (Ω) and split it into its positive and negative parts: ϕ = ϕ + − ϕ − . Since Ω is bounded, by a standard approximation argument, (1.5) remains valid for the test functions ϕ + , ϕ − . Then, Q u (ϕ) = Q u (ϕ + )+Q u (ϕ − ) ≥ 0 and so u is stable. Example 1.1.3 ([193]) Minimal solutions are stable. A function u ∈ C 2 (Ω) satisfying −∆u ≤ f (u) u≤0

in Ω, on ∂ Ω,

is called a subsolution to (1.3). Similarly, a function u ∈ C 2 (Ω) satisfying the reverse set of inequalities is called a supersolution to (1.3). Assume such a sub- and a supersolution exist and assume in addition that they are distinct and ordered: u ≤ u in Ω. By the strong maximum principle (Proposition A.8.1), it turns out that the inequality is always strict: u
in Ω.

(1.6)

Lemma 1.1.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let f ∈ C 1 (R). Assume that there exists u, u ∈ C 2 (Ω) a sub- and a supersolution to (1.3). In addition, assume (1.6). Then, there exists a unique solution u of (1.3) having the following properties: Given any supersolution u2 ∈ C 2 (Ω) of (1.3) such that u2 ≥ u, 1. u ≤ u ≤ u2 and 2. u minimizes EΩ among all functions v ∈ C 2 (Ω) such that u ≤ v ≤ u2 . u is called the minimal solution to (1.3) relative to u. Assume temporarily that Lemma 1.1.1 is established. Let u denote the minimal solution relative to u. The strong maximum principle (Proposition A.8.1) implies that either u ∈ {u, u} or u 0 such that u ≤ u + tϕ ≤ u for 0 ≤ t < t 0 . By item 2 of the lemma, we deduce that 0 is a point of minimum of E|[0,t 0 ) . Since u solves (1.3), E 0 (0) = 0. So, we must have E 00 (0) ≥ 0 and (1.5) holds for ϕ ≥ 0. Decomposing an arbitrary test 6

Chapter 1. Defining stability function ϕ into its positive and negative part as in Example 1.1.2, we deduce that u is stable. It remains to prove Lemma 1.1.1. Proof. The proof we present here is not standard and a bit technical (we follow this strategy to prove point 2 of the lemma). If you are unfamiliar with the method of sub- and supersolutions, we recommend you first read [193] or simply do Exercise 1.2.1. Define the truncated nonlinearity g(x, u) for x ∈ Ω and u ∈ R by  if u < u(x),  f (u(x)) f (u) if u(x) ≤ u ≤ u(x), (1.7) g(x, u) =  if u > u(x). f (u(x)) Let G(x, u) =

´u 0

g(x, t) d t and define E b : H01 (Ω) → R by ˆ ˆ 1 2 E b (u) = |∇u| d x − G(x, u(x)) d x. 2 Ω Ω

(1.8)

Then, E b is bounded below (see Exercise 1.1.3). So, there exists a minimizing sequence (un ) of E b . Letting a = minΩ u and b = maxΩ u, there exists constants C1 , C2 , C3 > 0 such that ˆ ˆ 2 1 ∇u d x ≤ E (u ) + G(x, u (x)) d x n n b n 2 Ω Ω ˆ ≤ C1 + k f k L ∞ (a,b) un d x Ω

≤ C1 + C2 kun k L 2 (Ω) ≤ C1 + C3 k∇un k L 2 (Ω) , where we used the mean-value theorem in the second inequality, Hölder’s inequality in the third, and Poincaré’s inequality in the last. Solving the previous (quadratic) inequality for k∇un k L 2 (Ω) , we deduce that (un ) is bounded in H01 (Ω). It follows that a subsequence (ukn ) converges to a function u ∈ H01 (Ω) weakly in H01 (Ω), in L 2 (Ω) and almost everywhere (a.e.) in strongly ´ Ω. Since G(x, ukn (x)) ≤ C ukn (x) , we deduce that Ω G(x, ukn (x)) d x → ´ ´ Since u 7→ Ω |∇u|2 d x is weakly lowerΩ G(x, u(x)) d x as n → ∞. semicontinuous in H 1 (Ω), it follows that E b (u) ≤ lim inf E b (ukn ) = inf E b (u). n→∞

u∈H01 (Ω)

7

Stable solutions of elliptic PDEs So, u minimizes E b . In particular, u is a weak solution to −∆u = g(x, u) in Ω, u=0 on ∂ Ω,

(1.9)

that is, u ∈ H01 (Ω) and for all ϕ ∈ Cc1 (Ω), ˆ ˆ g(x, u)ϕ d x. ∇u∇ϕ d x = Ω

Ω

In addition, u ∈ C 1,α (Ω) (see Exercise 1.1.4). Observe that any solution to (1.9) satisfies u ≤ u ≤ u and solves (1.3). Indeed, assume by contradiction that ω = {x ∈ Ω : u(x) < u(x)} is nonempty. Then, v = u − u is harmonic in ω and v ≥ 0 on ∂ ω. By the maximum principle (Proposition A.2.2), v ≥ 0 in ω, which is absurd by definition of ω. So u ≥ u. Similarly, u ≤ u. By (1.7) and (1.9), u is a solution to (1.3). Let u2 denote another supersolution to (1.3) such that u2 ≥ u. Define the truncation g 2 of g by   g(x, u(x)) if u < u(x), 2 g(x, u) if u(x) ≤ u ≤ u2 (x), g (x, u) =  g(x, u2 (x)) if u > u2 (x). Working as above, we can construct a solution u2 of (1.9), satisfying u ≤ u2 ≤ u2 . As observed earlier, u2 then solves (1.3) and u2 ≤ u. Next, take a finite family of supersolutions i = {u, u2 , . . . , un } such that uk ≥ u for k = 2, . . . , n. Also let I denote the set of all such families, ordered by inclusion. Repeating the truncation process inductively, we obtain a solution ui of (1.3) such that u ≤ ui ≤ u, u2 , . . . , un . Note that given i ∈ I, there may be more than one such solution ui . Nevertheless, using the axiom of choice on the set of all such solutions, we can construct a well-defined generalized sequence (ui )i∈I , contained in the set K of all solutions u satisfying u ≤ u ≤ u. By standard elliptic estimates, K is a compact subset of C 2 (Ω) so there exists a generalized subsequence (uφ( j) ) j∈J converging to a solution u of (1.3). Now choose an arbitrary supersolution v ≥ u and let i1 := {v, u} ∈ I. Given ε > 0, let j0 ∈ J such that j > j0 =⇒ kuφ( j) − uk∞ < ε. Also choose j1 ∈ J such that j > j1 =⇒ φ( j) > i1 . Finally, pick j2 > j0 , j1 . Then, for j > j2 , u ≤ kuφ( j) − uk∞ + uφ( j) ≤ ε + v. Letting ε → 0, we conclude that u ≤ v for any supersolution v ≥ u, which proves item 1 of the lemma. Since each ui was obtained as a global minimizer 8

Chapter 1. Defining stability of a truncated energy, uφ( j) , j > j1 minimizes EΩ among all functions v ∈ C 2 (Ω) such that u ≤ v ≤ v. Passing to the limit, so does u. Exercise 1.1.3 • Prove that g defined by (1.7) is a continuous map and that the family of functions (g x ) x∈Ω defined by g x (u) := g(x, u) for u ∈ R is uniformly Lipschitz continuous on any compact interval [a, b]. • Prove that E b defined by (1.8) is bounded below. Exercise 1.1.4 • By looking at the variations of E b (t) = E b (u+tϕ), prove that a minimizer u of E b is a weak solution to (1.9). • Prove that u ∈ C 1,α (Ω). Exercise 1.1.5 Let u denote the minimal solution to (1.3) relative to u. Let ω denote a subdomain of Ω. We say that u2 ∈ C 2 (ω) is a local supersolution to (1.3) if −∆u2 ≥ f (u2 ) in ω, u2 ≥ u on ∂ ω and u2 ≥ u in ω. Prove that u2 ≥ u in ω for any local supersolution u2 .

1.2 1.2.1

Linearized stability Principal eigenvalue of the linearized operator

Take a second look at the definition of stability, Inequality (1.5). As observed in Remark 1.1.1, if Ω is bounded, Q u ≥ 0 on all of H01 (Ω), which we can rephrase as ˆ ˆ 2 0 2 inf |∇ϕ| d x − f (u)ϕ d x ≥ 0. (1.10) ϕ∈H01 (Ω),kϕk L 2 (Ω) =1

Ω

Ω

The left-hand side of the above inequality coincides with the principal eigenvalue λ1 (−∆ − f 0 (u); Ω) of the elliptic operator −∆ − f 0 (u) with Dirichlet boundary conditions (see Theorem A.6.1). By Theorem A.6.1, we also have λ1 (−∆ − f 0 (u); Ω) ≥ 0, if and only if there exists v ∈ C 2 (Ω) such that v > 0 in Ω and −∆v − f 0 (u)v ≥ 0 in Ω. (1.11) Recalling that ω 7→ λ1 (−∆ − f 0 (u); ω) is nonincreasing, we obtain the following equivalent definition of stability (which remains valid even if Ω is unbounded): 9

Stable solutions of elliptic PDEs Proposition 1.2.1 Let f ∈ C 1,α (R) and let Ω denote a domain of RN , N ≥ 1. A solution u ∈ C 2 (Ω) of (1.4) is stable if and only if either of the following conditions hold • λ1 (−∆ − f 0 (u); ω) ≥ 0 for every bounded subdomain ω ⊂⊂ Ω, or • there exists v ∈ C 2 (Ω) such that v > 0 in Ω and (1.11) holds. Remark 1.2.1 When Ω is bounded, the first condition simply reduces to λ1 (−∆ − f 0 (u); Ω) ≥ 0. Proof. Assume first that (1.5) holds and fix ω ⊂⊂ Ω. Then, Q u (ϕ) ≥ 0 for all ϕ ∈ H01 (ω), that is, λ1 (−∆ − f 0 (u); ω) ≥ 0. Assume next that λ1 (−∆ − f 0 (u); ω) ≥ 0 for every subdomain ω. Let (ωn ) denote an increasing sequence of bounded subdomains covering Ω and x 0 ∈ ∩ωn . Denote ϕ1,n > 0 the associated eigenfunction normalized by ϕ1,n (x 0 ) = 1. Using Harnack’s inequality (see Proposition A.3.2), (ϕ1,n ) is uniformly bounded on compact subsets of Ω. By standard elliptic regularity, a subsequence (ϕ1,kn ) converges uniformly on compact subsets to some function v ∈ C 2 (Ω) such that v ≥ 0 and −∆v − f 0 (u)v = λv ≥ 0 in Ω, where λ = limn→∞ λ1 (−∆ − f 0 (u); ωn ). By the strong maximum principle, either v ≡ 0, which is excluded by the normalization ϕ1,n (x 0 ) = 1, or v > 0 as requested in Definition 1.2.1. Assume at last that there exists v ∈ C 2 (Ω) such that v > 0 in Ω and (1.11) holds. Take ϕ ∈ Cc1 (Ω) and multiply (1.11) by ϕ 2 /v:

ˆ

ϕ2 dx −∆v − f 0 (u)v v Ω ˆ ϕ2 0 2 − f (u)ϕ ≤ ∇v · ∇ dx v Ω ˆ ˆ ˆ 2 ϕ ϕ 2 |∇v| d x + 2 ∇v · ∇ϕ d x − f 0 (u)ϕ 2 d x ≤− 2 Ω v Ω ˆ Ω v ˆ 2 ≤ ∇ϕ d x − f 0 (u)ϕ 2 d x,

0≤

Ω

Ω

where we used Young’s inequality in the last inequality. Equation (1.5) follows. Exercise 1.2.1 Let f ∈ C 1 (R) and let Ω denote a smoothly bounded domain of RN , N ≥ 1. Assume that there exists u, u ∈ C 2 (Ω), a sub- and a supersolution 10

Chapter 1. Defining stability to (1.3), such that u < u. Let u0 = u and for n ∈ N∗ , let un ∈ C 2 (Ω) denote the unique solution to −∆un + Λun = f (un−1 ) + Λun−1 in Ω, (1.12) un = 0 on ∂ Ω, where Λ = k f 0 k L ∞ (a,b) , a = minΩ u, b = maxΩ u. • Prove that un is well defined and that u ≤ un ≤ un+1 ≤ u for all n ∈ N. • Prove that u : Ω → R defined by u(x) := limn→+∞ un (x) for all x ∈ Ω is a classical solution to (1.3). • Prove that u is the minimal solution to (1.3) relative to u. • Let u" = u − "ϕ1 , where " > 0 and ϕ1 > 0 is an eigenfunction associated to λ1 = λ1 (−∆ − f 0 (u); Ω). Assume by contradiction that λ1 < 0. Prove that for small " > 0, u" is a supersolution to (1.3) such that u" > u, and obtain a contradiction. Conclude that u is stable.

1.2.2

New examples of stable solutions

Using Definition 1.2.1, we can produce new examples of stable solutions. Example 1.2.1 Monotone solutions are stable. Definition 1.2.1 A solution u ∈ C 2 (Ω) of (1.4) is monotone if up to a rotation of space ∂u >0 in Ω. ∂ xN Differentiate (1.4) with respect to x N : if u is monotone then v = ∂ u/∂ x N is positive and solves −∆v − f 0 (u)v = 0 in Ω. So, u is stable, by Proposition 1.2.1. Example 1.2.2 ([97]) Positive solutions on coercive epigraphs are stable. Definition 1.2.2 A domain Ω is a coercive epigraph if, up to a rotation of space, there exists g ∈ C(RN −1 , R) such that Ω = {x = (x 0 , x N ) ∈ RN −1 × R : x N > g(x 0 )} and lim|x 0 |→+∞ g(x 0 ) = +∞. 11

Stable solutions of elliptic PDEs xN

Ω

xλ Tλ x

Σλ

Figure 1.2: The moving-plane device.

Lemma 1.2.1 ([92]) Let Ω denote a coercive epigraph. If u > 0 solves (1.3), then u is monotone, hence stable. Proof. The argument relies on the celebrated moving-plane device: given λ > 0, consider a hyperplane Tλ = {(x 0 , x N ) ∈ RN −1 × R : x N = λ} and Σλ = {(x 0 , x N ) ∈ Ω : x N < λ}, the region below Tλ . Reflect Σλ through the hyperplane Tλ : for x ∈ Σλ , the reflection of x is given by xˆλ = (x 0 , 2λ − x N ). Observe that since Ω is an epigraph, x ∈ Σλ ⇒ xˆλ ∈ Ω and we may define ˆλ (x) = u(ˆ ˆλ − u. We shall prove u x λ ) for x ∈ Σλ , see Figure 1.2. Also let wλ = u that given any λ > 0, wλ > 0

in Σλ .

(1.13)

We establish (1.13) first for small λ > 0. Let Vλ (x) =

f (ˆ uλ ) − f (u) ˆλ − u u

ˆλ (x), whenever u(x) 6= u

(x),

and Vλ (x) = f 0 (u(x)) otherwise. Then, −∆wλ − Vλ (x)wλ = 0 12

in Σλ

(1.14)

Chapter 1. Defining stability and wλ ≥ 0 on ∂ Σλ . The maximum principle for small domains (Proposition A.7.1) implies that wλ ≥ 0 in Σλ , provided λ > 0 is small. Since u > 0 in Ω and u = 0 on ∂ Ω, wλ > 0 on ∂ Ω ∩ Σλ . By the strong maximum principle applied to (1.14), wλ > 0 in Σλ , that is, (1.13) holds for all small values of λ > 0. Now let λ0 denote the supremum of all µ > 0 such that (1.13) holds for all 0 < λ ≤ µ. Assume by contradiction that λ0 < ∞. By continuity, we have wλ0 ≥ 0 in Σλ0 . By the strong maximum principle, the inequality is strict: wλ0 > 0 in Σλ0 . Fix δ > 0 small and a compact set K ⊂⊂ Σλ0 such that Σ \ K ≤ δ/2. Since K is compact, w ≥ η > 0 in K for some constant λ0 λ0 η > 0. Choosing ε > 0 sufficiently small, it follows that wλ0 +" > 0 in K, while Σ λ0 +" \ K ≤ δ. So, −∆wλ0 +" + Vλ0 +" wλ0 +" = 0 wλ0 +" ≥ 0

in Σλ0 +" \ K, on ∂ Σλ0 +" \ K .

By the maximum principle for small domains, we conclude that wλ0 +" ≥ 0 in Σλ0 +" \ K. Since we already had wλ0 +" > 0 in K, the strong maximum principle then implies that (1.13) holds for λ = λ0 + ", contradicting the definition of λ0 . So, (1.13) holds for all λ > 0. We want to prove that u is monotone in the x N direction, that is, that ∂ u/∂ x N (x 0 , λ) > 0 for all (x 0 , λ) ∈ Tλ ∩ Ω and all λ > 0. To see this, fix λ > 0 and observe that wλ is a positive solution to (1.14) in Σλ and that wλ = 0 on Tλ ∩ Ω. By the boundary point lemma (Lemma A.5.1), we deduce that 2

∂u ∂ xN

(x 0 , λ) = −

∂ wλ ∂ xN

(x 0 , λ) > 0,

for all (x 0 , λ) ∈ Tλ ∩ Ω.

Exercise 1.2.2 ([123]) Let f ∈ C 1 (R, R) and let B denote the unit ball in RN , N ≥ 2. Let u ∈ C 2 (B) denote a positive solution to (1.18) below. We want to prove that u is radially symmetric. • Check that it suffices to prove that u is an even function of the variable xN . • Check that it suffices to prove that u(x 0 , x N ) ≤ u(x 0 , −x N ) for every x = (x 0 , x N ) ∈ B such that x N > 0. • For λ ∈ (0, 1), define Tλ = {(x 0 , x N ) ∈ RN −1 × R : x N = λ}, Σλ = ˆλ (x) = u(ˆ {(x 0 , x N ) ∈ B : λ < x N < 1}, xˆλ = (x 0 , 2λ − x N ), u x λ ), and 13

Stable solutions of elliptic PDEs ˆλ (x) − u(x), for x = (x 0 , x N ) ∈ Σλ . Prove that wλ is well wλ (x) = u defined and 0 < wλ in x ∈ Σλ , (1.15) for all λ ∈ (0, 1). Conclude that u is radially symmetric. • Prove that

∂u ∂r

< 0 in B \ {0}, where r = |x|.

N Remark 1.2.2 Lemma 1.2.1 remains valid if Ω = R+ and either N = 2 or N ≥ 3, f (0) ≥ 0, and f globally Lipschitz, or N ≥ 3, f (0) ≥ 0, and u bounded. See [16] and [70]. Unfortunately, positivity does not imply stability for a general domain Ω, as the following examples show.

Example 1.2.3 ([97]) Let I = (0, 1), p > 1 and v ∈ C 2 ([0, 1]), v > 0 be a solution to −v 00 = v p in (0, 1), (1.16) v(0) = v(1) = 0. Multiplying the above equation by v and integrating by parts and recalling Remark 1.1.1, we easily see that Q v (v) < 0, that is, v is unstable. So positive solutions defined on a bounded domain need not be stable. Since Lemma 1.2.1 relied on the existence of a privileged direction in which the solution u was monotone, one may hope that positivity implies stability for a larger class of unbounded domains, say for cylinders. Unfortunately, this is still not the case, for example, take N ≥ 2 and Ω = I ×RN −1 ⊂ RN . If v is a solution to (1.16), then u(x 1 , . . . , x N ) = v(x 1 ) is a positive solution to (1.3) with f (u) = u p . However, u is not stable. Take R > 0, the N − 1 diPN −1 2 0 N −1 x j < R} and mensional ball ωR = {x = (x 1 , . . . , x N −1 ) ∈ R : j=1 ϕR an eigenfunction associated to the principal eigenvalue λ1 (−∆; ωR ) of the Laplace operator on ωR . Normalize ϕR by kϕR k L 2 (ωR ) = 1 and set ψR = uϕR . Then, ˆ ˆ ˆ 2 2 2 Q u (ψR ) = ∇u∇(uϕR ) d x + u ∇ϕR d x − p u p+1 ϕR2 d x ˆΩ ˆ Ω ˆ Ω 2 = −∆u uϕR2 d x + u2 ∇ϕR d x − p u p+1 ϕR2 d x Ω Ω Ω ˆ ˆ 2 = (1 − p) u p+1 ϕR2 d x + u2 ∇ϕR d x Ω Ω ˆ 1 ˆ 1 p+1 = (1 − p) v d t + λ1 (−∆; ωR ) v 2 d t. 0

0

14

Chapter 1. Defining stability By a direct scaling argument, λ1 (−∆; ωR ) = λ1 (−∆; ω1 ) R12 . So, choosing R > 0 large, we deduce that Q u (ψR ) < 0. By Remark 1.1.1, u is not stable. ´ 1 2 Exercise 1.2.3 Let E (v) = 21 0 dd xv d x and let F denote the set of all func´1 tions v ∈ H01 (0, 1) such that 0 |v| p+1 d x = 1. Prove that min F E is achieved. What PDE does the minimizer solve? Deduce that there exists a positive solution to (1.16).

1.3 1.3.1

Elementary properties of stable solutions Uniqueness

If f is nonincreasing and Ω bounded, the maximum principle implies that (1.3) has a unique solution. Exercise 1.3.1 Prove it. Otherwise, the equation can have many solutions. But perhaps uniqueness holds in the class of stable solutions. This is indeed the case when the nonlinearity is convex as the following proposition shows. Proposition 1.3.1 ([32]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Let λ1 = λ1 (−∆; Ω) > 0 denote the principal eigenvalue of the Dirichlet Laplacian on Ω. Assume that f ∈ C 1 (R) is convex. Let u1 , u2 ∈ C 2 (Ω) denote two stable solutions of (1.3). Then, either u1 = u2 or f (u) = λ1 u on the ranges of u1 and u2 . Proof. Let u1 , u2 denote two stable solutions of (1.3). Then, w = u2 − u1 solves −∆w = f (u2 ) − f (u1 )

in Ω.

Multiply the above equality by w + , the positive part of w, and integrate by parts: ˆ ˆ ∇w + 2 d x = ( f (u ) − f (u ))w d x. 2 1 + Ω

Ω

Since u2 is stable, we also have ˆ ˆ ∇w + 2 d x ≥ f 0 (u2 )(w + )2 d x, Ω

Ω

15

Stable solutions of elliptic PDEs hence

ˆ Ω

f (u2 ) − f (u1 ) − f 0 (u2 )w + w + d x ≥ 0.

By convexity, the integrand in the above inequality is nonpositive. If f is strictly convex, we readily deduce that w+ = 0, that is, u2 ≤ u1 . The reverse inequality is obtained by exchanging the roles of u1 and u2 . Otherwise, f is linear on the union of all intervals with end points u1 (x), u2 (x), x ∈ Ω, hence on the whole ranges of u1 and u2 . So, u1 , u2 are stable solutions of a linear problem, which is possible only if they belong to the eigenspace associated to the principal eigenvalue λ1 . Exercise 1.3.2 Prove Proposition 1.3.1 for concave f . Assume that there exists a minimal solution to (1.3), denoted u. It follows from the above uniqueness result that if f is strictly convex, then u is the unique stable solution to (1.3). By minimality, u lies below all other solutions of the equation. Can all solutions of the equation be ordered in such a way? When f is strictly convex, the answer is always negative. Indeed, if u1 , u2 denote two solutions both different from u, then necessarily they must cross as the following proposition shows. Proposition 1.3.2 ([111, 112, 139]) Let Ω denote a smoothly bounded domain of RN . Assume that f ∈ C 1 (R) is strictly convex. Let u1 , u2 ∈ C 2 (Ω) denote two distinct solutions of (1.3) such that u1 ≤ u2

in Ω.

Then, u1 is the unique stable solution to (1.3). Proof. Let w = u2 − u1 . Then, w ≥ 0 in Ω and using convexity, −∆w = f (u2 ) − f (u1 ) ≥ f 0 (u1 )w

in Ω.

By the strong maximum principle (Corollary A.5.1), either w ≡ 0, which we have excluded or w > 0 in Ω, hence u1 is stable (by Proposition 1.2.1). By Proposition 1.3.1, we deduce that u1 is the unique stable solution to (1.3).

1.3.2

Nonuniqueness

The convexity assumption cannot be dropped in Proposition 1.3.1. A counterexample is given by the Allen-Cahn nonlinearity f (u) = u − u3 : 16

Chapter 1. Defining stability

Proposition 1.3.3 Let Ω denote a smoothly bounded domain of RN and λ1 = λ1 (−∆; Ω) > 0 denote the principal eigenvalue of the Dirichlet Laplacian on Ω. Let f (u) = u − u3 and λ ≥ 0. Consider the equation

−∆u = λ f (u) u=0

in Ω, on ∂ Ω.

(1.17)

• If λ ≤ λ1 , then u = 0 is the unique solution to (1.17) and it is stable. • If λ > λ1 , then there exists at least two stable nontrivial solutions of (1.17).

2 Figure 1.3: W (u) = 41 u2 − 1 is a double-well potential, indicating that at least two stable solutions may exist.

Proof. We begin with the case λ ≤ λ1 . Let X = H01 (Ω) ∩ L ∞ (Ω). Then, any (classical) solution to (1.17) is a critical point of the energy E : X → R defined for u ∈ X by EΩ (u) =

1 2

ˆ |∇u| d x + 2

Ω

λ 4

ˆ 2 u2 − 1 d x. Ω

We are going to prove that EΩ is strictly convex and bounded below by λ/4, which implies that u = 0 is the only critical point of EΩ . Since u = 0 is a global 17

Stable solutions of elliptic PDEs minimizer (or using a direct computation), we easily obtain that it is stable. We rewrite EΩ as EΩ = E1 + E2 , where E1 (u) =

1 2

ˆ Ω

ˆ |∇u| d x − λ 2

2

u dx

and

Ω

E2 (u) =

λ 4

ˆ u4 + 1 d x. Ω

Clearly, E1 ≥ 0 if λ ≤ λ1 and E2 ≥ λ/4 is strictly convex. So, it suffices to prove that E1 is convex. Now, since λ ≤ λ1 , E1 is a positive semidefinite quadratic form. Let B1 denote its associated symmetric bilinear form. Then, given t ∈ [0, 1], u, v ∈ X , E1 (tu + (1 − t)v) = t 2 E1 (u) + (1 − t)2 E1 (v) + 2t(1 − t)B1 (u, v) ≤ t 2 E1 (u) + (1 − t)2 E1 (v) + 2t(1 − t)E1 (u)1/2 E1 (v)1/2 2 ≤ tE1 (u)1/2 + (1 − t)E1 (v)1/2 ≤ tE1 (u) + (1 − t)E1 (v), where we used the Cauchy-Schwarz inequality in the first inequality and the convexity of x 7→ x 2 in the last. We turn next to the case λ > λ1 . u = 0 and u = 1 are ordered sub- and supersolutions of the equation, so by Lemma 1.1.1, there exists u, the minimal solution (relative to u = 0) of (1.17). Since u is minimal, u is stable. Since λ > λ1 , 0 is unstable and so u 6≡ 0. By the strong maximum principle, 0 < u < 1. By a direct computation, −u is another stable solution to the problem.

1.3.3

Symmetry

When an equation exhibits a certain symmetry, it is natural to ask whether its solutions are also symmetric. For example, one can ask whether any solution u of (1.3) in Ω = B, the unit ball of RN , is radially symmetric. This is indeed the case if u is stable, as the following proposition shows. Proposition 1.3.4 ([4]) Let f ∈ C 1 (R). Also let B denote the unit ball in RN , N ≥ 2 and u ∈ C 2 (B) denote a stable solution to

−∆u = f (u) u=0

in B, on ∂ B.

(1.18)

Then u is radially symmetric. Moreover, r 7→ u(r) is either constant, increasing or decreasing in (0, 1). 18

Chapter 1. Defining stability Remark 1.3.1 A celebrated result of Gidas, Ni, and Nirenberg [123] states that every positive solution to (1.18) is radially symmetric and decreasing. See Exercise 1.2.2. However, unstable sign-changing solutions of (1.18) can be nonradial. This is the case for example if f (u) = λ2 u, where λ2 is the second eigenvalue of the Dirichlet Laplace operator (see paragraph V.5.5 in [62]) or if f (u) = |u| p−1 u, p > 1 (see [2]). Proof. We first show that u is radial. It suffices to prove that any tangential derivative v = x i ∂ j u − x j ∂i u, i, j = 1, . . . , N is identically zero. Integration by parts implies on the one hand that ˆ ˆ v dx = x i un j − x j uni dσ = 0, (1.19) ∂B

B

since ni = x i is the i-th component of the normal unit vector to ∂ B pointing outwards. On the other hand, a direct calculation shows that v solves the linearized equation −∆v = f 0 (u)v in B, (1.20) v=0 on ∂ B, where the boundary condition follows from the facts that u is constant on ∂ B and that v is a tangential derivative. Multiplying (1.20) by v and integrating by parts, it follows that ˆ ˆ 2 f 0 (u)v 2 d x = 0. |∇v| d x − B

B

Since u is stable, it follows that if v 6≡ 0, v minimizes the Rayleigh quotient (1.10). Hence, the linearized operator −∆ − f 0 (u) has principal eigenvalue λ1 = 0 and v is an eigenfunction, so it must be everywhere positive or everywhere negative, contradicting (1.19). So, v ≡ 0 and u is radial. We prove next that u(r) is constant or monotone. If u is not constant, v = du/d r is nontrivial. Assume by contradiction that v(r0 ) = 0 for some r0 ∈ (0, 1]. By Remark 1.1.1, u is stable in B r0 and we may use v as a test function in (1.5). We obtain ˆ ˆ 2 |∇v| d x − f 0 (u)v 2 d x ≥ 0. (1.21) B r0

B r0

Differentiating (1.18) with respect to r we also have   −∆v + N − 1 v = f 0 (u)v in B r0 , r2  v=0 on ∂ B r0 . 19

Stable solutions of elliptic PDEs Multiplying the above equation by v and integrating by parts, we get ˆ ˆ ˆ N −1 2 2 0 2 |∇v| d x − f (u)v d x = − v d x < 0, r2 B r0 B r0 B r0 which contradicts (1.21). Exercise 1.3.3 Prove that Proposition 1.3.4 remains valid if assumption (1.5) is replaced by λ2 (−∆ − f 0 (u); Ω) > 0. What if λ2 (−∆ − f 0 (u); Ω) = 0?

1.4

Dynamical stability

At the beginning of this chapter, we introduced stability as the ability of a system to return to equilibrium after a small perturbation. This intuitive description can be stated mathematically as follows. Definition 1.4.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1. A solution u ∈ C 2 (Ω) of (1.3) is asymptotically stable if there exists ε > 0 such that for all u0 ∈ C 2 (Ω) with ku − u0 k L ∞ (Ω) < ε, the solution v ∈ C 2 (Ω × [0, T ]) of  ∂v in Ω × (0, T ),  ∂ t − ∆v = f (v) (1.22) v=0 on ∂ Ω × (0, T ),  v(x, 0) = u0 (x) for x ∈ Ω is defined for all times T > 0 and lim kv(t) − uk L ∞ (Ω) = 0.

t→+∞

(1.23)

Unfortunately, the notion of stability, as introduced in (1.5), is not equivalent to that of dynamical stability. Before looking at counter-examples, let us first state a positive result. Proposition 1.4.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let u ∈ C 2 (Ω) denote a solution to (1.3) such that the linearized operator is positive, that is, λ1 (−∆ − f 0 (u); Ω) > 0. Then, u is asymptotically stable. 20

Chapter 1. Defining stability Proof. We follow [193] and [28]. We start out by constructing a subsolution un and a supersolution un of (1.3) such that un 0, v is well defined and unique by the Lax-Milgram lemma, and v ∈ C 2 (Ω) by standard elliptic regularity. Let un = u + 1n (v + 1). Then, 1 1 −∆un − f (un ) = −∆u + (−∆v) − f u + (v + 1) n n 1 0 1 0 = f (u) + f (u)v + f (u) + 1 − f u + (v + 1) n n 1 0 = f (u) + f (u)v + f 0 (u) + 1 n 1 0 v+1 − f (u) + f (u)(v + 1) + o n n 1 ≥ + o(1/n) ≥ 0, n where the last inequality holds for n sufficiently large. Similar inequalities prove that un = u − 1n (v + 1) is a subsolution to the problem. We are left with proving that u is asymptotically stable. To do so, we remark that if " > 0 is chosen small enough, then any initial datum u0 ∈ C 2 (Ω) such that ku − u0 k L ∞ (Ω) < ε lies in the interval (un0 , un0 ) for some fixed n0 ∈ N. By the method of sub- and supersolutions, the solution v of (1.22) is well defined for all times and un0 ≤ v ≤ un0 . In particular, for all t > 0 kv(t) − uk L ∞ (Ω) ≤

1 n0

(kvk L ∞ (Ω) + 1).

(1.24)

Fix δ ∈ (0, λ1 ) small. Since f is C 1 , the inequality f (z) − f (u) − f 0 (u)(z − u) ≤ δ |z − u| holds for small enough |z − u|. Choosing n0 sufficiently large (which is allowed if " > 0 is small), we conclude that for all time t > 0, f (v) − f (u) − f 0 (u)(v − u) ≤ δ |v − u| . 21

Stable solutions of elliptic PDEs So, w := v − u satisfies ∂w ∂t

− ∆w − ( f 0 (u) + δ)w = f (v) − f (u) − ( f 0 (u) + δ)(v − u) ≤ 0.

The principal eigenvalue µ1 of the operator −∆ − ( f 0 (u) + δ) is given by µ1 = λ1 − δ > 0. By comparison, we deduce that w(t) ≤ C e−µ1 t . Similar arguments lead to −w(t) ≤ C e−µ1 t , so that u is asymptotically stable. Conversely, if a solution u of (1.3) is asymptotically stable, then necessarily it must be stable, that is, (1.5) holds. This is the object of the following proposition. Proposition 1.4.2 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let u ∈ C 2 (Ω) denote a solution to (1.3) such that λ1 (−∆ − f 0 (u); Ω) < 0. Then, u is not asymptotically stable. Proof. Take " > 0 and consider the function u = u − "ϕ1 , where ϕ1 > 0 is an eigenfunction associated to λ1 = λ1 (−∆ − f 0 (u); Ω). Then, −∆u − f (u) = −∆u − " −∆ϕ1 − f u − "ϕ1 = f (u) − " f 0 (u) + λ1 ϕ1 − f u − "ϕ1 = f (u) − " f 0 (u) + λ1 ϕ1 − f (u) − " f 0 (u)ϕ1 + o("ϕ1 ) = −"λ1 ϕ1 + o("ϕ1 ) ≥ 0, if " > 0 is sufficiently small. So, u is a supersolution to (1.3). It follows that the solution v of (1.22) with initial datum u0 = u is a nonincreasing function of time. So, for all times t > 0 for which v is well defined, u − v(t) ≥ u − u0 = "ϕ1 , contradicting lim t→∞ ku − v(t)k L ∞ (Ω) = 0.

There is still one scenario that we haven’t considered: the case λ1 (−∆ − f (u)) = 0. As we are about to see, examples show that no conclusion can be drawn in full generality. 0

22

Chapter 1. Defining stability Proposition 1.4.3 Let Ω denote a smoothly bounded domain of RN , N ≥ 3, let λ1 = λ1 (−∆; Ω) > 0 and let ϕ1 > 0 denote a corresponding eigenvector. Take " > 0, 1 < p < (N + 2)/(N − 2) and consider the equation  ∂v p−1 v in Ω × (0, T ),  ∂ t − ∆v = λ1 v + |v| (1.25) v=0 on ∂ Ω × (0, T ),  v(x, 0) = ±"ϕ1 (x) for x ∈ Ω. Then, u = 0 is a solution to (1.3) with f (u) = λ1 u + |u| p−1 u such that λ1 (−∆ − f 0 (0); Ω) = 0. Yet, u = 0 is not asymptotically stable. Proof. Let EΩ : H01 (Ω) → R denote the usual energy functional defined by ˆ ˆ ˆ λ1 1 1 2 2 |∇u| d x − u dx − |u| p+1 d x. EΩ (u) = 2 Ω 2 Ω p+1 Ω Note that EΩ (0) = 0, while EΩ (±"ϕ1 ) < 0. Let v denote the solution to (1.25). d dt

EΩ (v) = DEΩ (v).

∂v ∂t

=−

ˆ Ω

∂v ∂t

2 d x ≤ 0.

In particular, EΩ (v) ≤ EΩ (±"ϕ1 ) < 0. So, even if v were defined for all times t > 0, v could not converge to u = 0, which has zero energy. The reverse conclusion can also occur. Proposition 1.4.4 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let λ1 = λ1 (−∆; Ω) > 0. Take " > 0, 1 < p and consider the equation  ∂v p−1 v in Ω × (0, T ),  ∂ t − ∆v = λ1 v − |v| (1.26) v=0 on ∂ Ω × (0, T ),  v(x, 0) = u0 (x) for x ∈ Ω. Then, u = 0 is a solution to (1.3) with f (u) = λ1 u − |u| p−1 u such that λ1 (−∆ − f 0 (0); Ω) = 0 and u = 0 is asymptotically stable. Proof. Let ϕ1 > 0 denote a eigenvector associated to λ1 . One may easily check that un = − 1n ϕ1 and un = 1n ϕ1 are respectively a sub- and a supersolution to (1.26) provided u0 is small enough. We can then argue as in the proof of Proposition 1.4.1. 23

Stable solutions of elliptic PDEs

1.5

Stability outside a compact set

In this section, following [94], we generalize the notion of stability by requiring that (1.5) holds only for test functions supported away from a given compact set. Definition 1.5.1 Let f ∈ C 1 (R), let Ω denote an open set of RN , N ≥ 1, and let K ⊂⊂ Ω denote a compact subset of Ω. A solution u ∈ C 2 (Ω) of (1.4) is stable outside the compact set K if ˆ ˆ 2 (1.27) Q u (ϕ) := |∇ϕ| d x − f 0 (u)ϕ 2 d x ≥ 0, ∀ ϕ ∈ Cc1 (Ω \ K). Ω

Ω

Definition 1.5.1 encompasses the following fundamental class of solutions. Example 1.5.1 Solutions of finite Morse index are stable outside a compact set. Definition 1.5.2 Let f ∈ C 1 (R), let Ω denote a domain of RN , N ≥ 1. A solution u ∈ C 2 (Ω) of (1.4) has Morse index k ≥ 1 if k is the maximal dimension of a subspace X k of Cc1 (Ω) such that Q u (ϕ) < 0

∀ϕ ∈ X k \ {0}.

We then write k = ind (u). Remark 1.5.1 If u is stable, we write 0 = ind (u). Let us check that every solution of finite Morse index is stable outside a compact set. Assume ind (u) = k ∈ N∗ and take k independent vectors ϕ1 , . . . , ϕk ∈ Cc1 (Ω) spanning the associated space X k . Each ϕi has compact support so that K = ∪ki=1 supp ϕi is a compact subset of Ω. Then, Q u (ϕ) ≥ 0 for all ϕ ∈ Cc1 (Ω\K) for otherwise there would exist a function ϕ with supp ϕ 6⊂ K and Q u (ϕ) < 0. In particular, ϕ, ϕ1 , . . . , ϕk would be linearly independent, hence ind (u) ≥ k + 1, a contradiction. ¯ If we assume that Ω is a bounded domain, then every solution u ∈ C 2 (Ω) of (1.3) has finite Morse index and is thus stable outside a compact set. Proposition 1.5.1 Let f ∈ C 1 (R) and let Ω denote a smoothly bounded domain ¯ of (1.4) has finite Morse index. Furof RN , N ≥ 1. Every solution u ∈ C 2 (Ω) thermore, k = ind (u) if and only if the linearized operator −∆ − f 0 (u) (with Dirichlet boundary conditions) has exactly k strictly negative eigenvalues. 24

Chapter 1. Defining stability Remark 1.5.2 In the above statement, eigenvalues are repeated according to their geometric multiplicity. Proof. We just need to prove that the Morse index k of u is equal to the number ˜k of negative eigenvalues of the operator. Step 1. ˜k ≥ k. If k = 0, we refer the reader to Definition 1.2.1. If k = 1, then for any ψ ∈ X k \ {0}, Q u (ψ) < 0. In particular, λ1 = λ1 (−∆ − f 0 (u); Ω) < 0, since λ1 is given by (1.10). If k ≥ 2, the k-th eigenvalue λk = λk (−∆ − f 0 (u); Ω) of the linearized operator is given by ¨´ λk = inf

2 Ω |∇ϕ|

´ d x − Ω f 0 (u)ϕ 2 d x ´ 2 : ϕ ∈ H01 (Ω) \ {0} s.t. ϕ d x Ω ˆ ϕϕ j d x = 0 ∀ j = 1, . . . , k − 1 , (1.28) Ω

where the functions ϕ j , j = 1, . . . , k − 1 are linearly independent eigenvectors associated to the eigenvalues λ j < λk (repeated according to their geometric multiplicity). Consider´ the linear map Λ : X k → Rk−1 defined for ϕ ∈ X k by ´ the kernel of Λϕ = ( Ω ϕϕ1 d x, . . . , Ω ϕϕk−1 d x). Since X k is k-dimensional, ´ Λ is nontrivial and there exists ϕ ∈ X k \ {0} such that Ω ϕϕ j d x = 0 for all j = 1, . . . , k − 1. We deduce that λk ≤ Q u (ϕ) < 0. Step 2. ˜k ≤ k. Using the notation of Step 1, Q u (ϕ j ) < 0 for all j = 1, . . . , ˜k. Since ϕ j ∈ H01 (Ω), there exists ψ j ∈ Cc1 (Ω) such that Q u (ψ j ) < 0 for j = 1, . . . , ˜k. Furthermore, the functions (ψ j ) can be chosen linearly independant, for otherwise (ϕ j ) would be linearly dependent. So, ind (u) ≥ ˜k. Solutions of finite Morse index have the following additional stability property. Proposition 1.5.1 Let f ∈ C 1 (R) and let Ω denote an open set of RN , N ≥ 2. Let u be a solution to (1.4) with finite Morse index. Then, for every x 0 ∈ Ω, there exists r0 > 0 such that u is stable in B(x 0 , r0 ). Proof. We may always assume that B(0, 1) ⊂ Ω and it suffices to prove that u is stable near the origin. Assume first that ind u = 1. Either u is stable in B(0, 1/n) for some n ≥ 2 and we are done. Or, for all n ≥ 2, there exists a direction ϕn ∈ Cc1 (B(0, 1/n)) such that Q u (ϕn ) < 0. Since ind u = 1, this 25

Stable solutions of elliptic PDEs implies that u is stable in B(0, 1) \ B(0, 1/n). This being true for all n ≥ 2, we deduce that u is stable in B(0, 1) \ {0}. In fact, since N ≥ 2, u is stable in B(0, 1). Take an arbitrary test function ϕ ∈ Cc1 (B(0, 1)) and let ϕn = ϕζn , where  0 if |x| < 1/n2 ,   ln |x| ζn (x) = 2 − if 1/n2 ≤ |x| < 1/n,  ln 1/n  1

if |x| ≥ 1/n,

if N = 2, and ζn (x) = ζ(nx), with ζ ∈ C 1 (RN ) such that ζ ≡ 0 in B1 and ζ ≡ 1 in RN \ B2 , if N ≥ 3. Then, ϕn ∈ Cc0,1 (B(0, 1) \ {0}) and so Q u (ϕn ) ≥ 0. In addition, one easily verifies that ϕn → ϕ pointwise and in H 1 (B(0, 1)), and |ϕn | ≤ |ϕ|. Hence, Q u (ϕ) ≥ 0. So, every solution of index ind u = 1 is stable in a neighborhood of 0. Now take a solution u of index k. Working exactly as above, we deduce that u has index k − 1 in some ball B(0, r1 ). Working inductively on k, we deduce that u is stable in some ball B(0, rk ). Exercise 1.5.1 When working on unbounded domains, the notion of stability outside a compact is still insufficient to encompass all solutions. Check that the function u constructed in Example 1.2.3 is unstable outside every compact set of Ω = I × RN −1 .

1.6

Resolving an ambiguity

Consider again (1.4) in the special case Ω = R, that is, −

d 2u d x2

= f (u)

in R.

(1.29)

Assume that u = 0 is a stable solution to the above equation. Then, we must have f 0 (0) ≤ 0. To see this, simply apply stability with test function ϕ" (x) = ϕ(" x), where ϕ ∈ Cc1 (RN ) 6≡ 0, and let " → 0. Assume further that f 0 (0) < 0. By Proposition 1.4.1, u = 0 must be asymptotically stable for the corresponding parabolic equation, at least on any bounded open interval of R (or even just for the ODE du = f (u)). dt Now, we could have also thought of (1.29) as an ODE, where x is now interpreted as the time variable. For this interpretation, our notion of stability is opposite to the notion of asymptotic stability: 26

Chapter 1. Defining stability Proposition 1.6.1 ([214]) Let f ∈ C 1 (R) such that f (0) = 0 and f 0 (0) < 0. In particular, u = 0 is a stable solution to (1.29). However, u is asymptotically unstable for the ode d 2u − 2 = f (u). (1.30) dt Proof. Assume to the contrary that 0 is an asymptotically stable equilibrium of (1.30). Then, for " > 0 sufficiently small, the solution u(t) to (1.30) with u0 (0) = 0 and u(0) = ", is defined for all times t > 0, and lim u(t) = 0,

t→+∞

and so

lim inf |u0 (t)| = 0. t→+∞

Multiplying (1.30) by u0 and integrating between 0 and t, we obtain the following law of conservation of energy: 1 2 where F (t) =

´t 0

u0 (t)2 + F (u(t)) = F ("),

f (s) ds. Passing to the liminf as t → +∞, we deduce that F (") = F (0).

This being true for all " small, we deduce that f ≡ 0 in a neighborhood of 0, contradicting f 0 (0) < 0.

27

Chapter 2 The Gelfand problem In Chapter 1, we gained familiarity with the notion of stability for semilinear elliptic equations of the form (1.3). It is time to look at a concrete example. This chapter is devoted to the study of the following problem: −∆u = λeu in B, (2.1) u=0 on ∂ B, where λ > 0 is a parameter and B is the unit ball of RN , N ≥ 1. Equation (2.1) bears many names: Barenblatt, Bratu, Emden, Fowler, Frank-Kamenetskii, Gelfand, and Liouville are some of the famous scientists to whom the equation has been attributed. For short, we call (2.1) the Gelfand problem.1 It arises as a (crude) model in a number of interesting physical contexts,2 one of which we outline next.

2.1

Motivation

In dimension N = 1, 2, 3, Equation (2.1) can be derived from the thermal self-ignition model. The full model describes the reaction process in a combustible material during what is referred to as the ignition period. A solution u of (2.1) represents a dimensionless temperature inside a cylindrical vessel (which walls are ideally conducting), when the system has reached an intermediate-asymptotic steady state. The underlying space variable x ∈ B One may argue that our choice is not historically sound. See [26], [131]. We warn the reader that the considerations presented in this book have little impact on the understanding of the underlying physical phenomenon. Instead, we shall use physics (well, truly heuristics) to gain intuition on the mathematical analysis of (2.1). 1

2

29

Stable solutions of elliptic PDEs should be thought of as dimensionless, that is, the vessel’s size has been normalized. We refer the interested reader to [13] and Chapters VI-VII of [110] for the detailed derivation of the model. Now, take a second look at Equation (2.1). On the left-hand side, there is a diffusion operator, −∆, accounting for the diffusion of heat from the hot reactants to the cold boundary. On the right-hand side, we have a reaction term, eu . The exponential nonlinearity has to do with the so-called Arrhenius law. More precisely, it is an approximation of the nonlinear term given in this u (empirical) law, which truly takes the form f (u) = e 1+"u . This term models the production of heat induced by the chemical reaction. The diffusion operator and the reaction nonlinearity compete. In one kind of reaction, the produced heat does not have time to be carried away through the walls of the vessel: either the combustible rarifies and the reaction dies out, or there is so much combustible that a thermal explosion occurs. Either way, no steady-state, that is, no solution to (2.1), should be expected. In another kind of reaction, on the contrary, an equilibrium between the produced and the diffused heat quickly occurs, so that existence of solutions of (2.1) should hold. The balance between diffusion and reaction is quantified by the parameter λ > 0. This parameter is sometimes referred to as the Frank-Kamenetskii constant. According to our previous discussion, we should expect no solution to (2.1) if λ is large, while solutions should exist for small λ. This is indeed the case, as the following proposition demonstrates. Proposition 2.1.1 Assume N ≥ 1. Then, every solution u ∈ C 2 (B) of (2.1) is radial. Furthermore, there exists λ∗ = λ∗ (N ) > 0 such that • For 0 < λ < λ∗ , there exists the minimal solution uλ ∈ C 2 (B) of (2.1). In particular, uλ is stable. • For λ > λ∗ , there exists no solution u ∈ C 2 (B) of (2.1). We shall prove Proposition 2.1.1 for N = 1 in Proposition 2.2.1, N = 2 in Proposition 2.3.1, and N ≥ 3 in Proposition 2.4.1. See also Proposition 3.3.1 for a more general and streamlined argument.

2.2

Dimension N = 1

For N = 1, (2.1) reads −u00 = λeu ,

u(−1) = u(1) = 0.

We completely characterize solutions to the equation as follows. 30

(2.2)

Chapter 2. The Gelfand problem

Proposition 2.2.1 There exists λ∗ > 0 such that • For 0 < λ < λ∗ , there exists exactly two solutions u ∈ C 2 ([−1, 1]) of (2.2). One of them, denoted uλ , is minimal, hence stable. The other one, denoted Uλ , has Morse index 1. Both solutions are positive, even, strictly decreasing on (0, 1], and uniquely determined by their value at x = 0. In addition, the curve ¨ (0, λ∗ ) → C 2 ([−1, 1]) × C 2 ([−1, 1]), λ 7→ (uλ , Uλ ) is smooth and for all x ∈ (−1, 1), lim (uλ (x), Uλ (x)) = (0, +∞),

(2.3)

lim (uλ (x), Uλ (x)) = (u∗ (x), u∗ (x)),

(2.4)

λ→0+

λ→λ∗

where u∗ ∈ C 2 ([−1, 1]) is the unique solution to (2.2) for λ = λ∗ . • For λ > λ∗ , there exists no solution u ∈ C 2 ([−1, 1]) to (2.1). Plotting λ on the x-axis and the norm kuk∞ = u(0) on the y-axis for each solution u ∈ {uλ , Uλ }, we obtain the following bifurcation diagram summarizing Proposition 2.2.1. kuλkL∞(B)

λ∗

λ

Figure 2.1: Bifurcation diagram for the Gelfand problem in dimension N = 1.

31

Stable solutions of elliptic PDEs Proof. Step 1. Every solution is positive, even, radially decreasing and characterized by its value at 0. Let u denote a solution to (2.2). Since −u00 > 0 in (−1, 1), u cannot achieve an interior point of minimum. In particular, u > 0 in (−1, 1). Also, u is even and u0 (r) < 0 for r ∈ (0, 1). Indeed, assume by contradiction that u achieves its maximum at some x 0 ∈ (−1, 1) \ {0}. Without loss of generality, x 0 ∈ (0, 1). Since v(t) = u(x 0 + t) and v˜(t) = u(x 0 − t) satisfy the same initial value problem with initial conditions v(0) = u(x 0 ) and v 0 (0) = 0, they must coincide, that is, v is even. In particular, u(2x 0 − 1) = u(1) = 0, contradicting u > 0 in (−1, 1). So, u achieves its unique point of maximum at 0. It follows that u is even and that u0 (r) < 0 for r ∈ (0, 1). Now multiply (2.2) by u0 and integrate between 0 and r ∈ (0, 1): −

u02 2

= λ (eu − eu0 ) ,

(2.5)

where u0 = u(0), which we rewrite as −u0 p

2λ (eu0 − eu )

= 1.

Integrating once more between 0 and 1, it follows that every solution to (2.2) satisfies ˆ u0 dt = 1. (2.6) p 0 2λ(eu0 − e t ) Conversely, every time there exists u0 > 0 such that (2.6) holds, the even function u defined for r ∈ (0, 1) by ˆ u0 dt =r p u(r) 2λ(eu0 − e t ) solves (2.2). This completes Step 1. Step 2. For some λ∗ > 0, there are exactly two solutions for λ ∈ (0, λ∗ ), one for λ = λ∗ , and none for λ > λ∗ . In addition, (2.3) and (2.4) hold. According to Step 1, there exists a solution such that u(0) = u0 > 0 if and only if (2.6) holds. Using standard calculus, one easily sees that the realvalued function I : R∗+ → R defined for u0 > 0 by ˆ u0 dt I(u0 ) = p u e 0 − et 0 32

Chapter 2. The Gelfand problem takes values in some bounded interval (0, M ∗ ], achieves its maximum M ∗ at a unique point u∗0 , and satisfies limu0 →0+ I(u0 ) = limu0 →+∞ I(u0 ) = 0. Step 2 follows. Step 3. Any solution u is stable outside the compact set K = {0}. To see this, recalling that u is even, it suffices to prove that for every ϕ ∈ Cc1 (0, 1).

Q u (ϕ) ≥ 0

Since u0 < 0 in (0, 1), every test function ϕ ∈ Cc1 (0, 1) can be written in the form ϕ = u0 ψ, where ψ ∈ Cc1 (0, 1). So,

ˆ

1

Q u (ϕ) =

d(u0 ψ)

2

! − λeu (u0 ψ)2

dr

0

dr

ˆ 1 (u00 )2 ψ2 + 2u0 ψu00 ψ0 + (u0 )2 (ψ0 )2 − λeu (u0 )2 ψ2 d r = 0 ˆ 1 = u00 (u0 ψ2 )0 + (u0 )2 (ψ0 )2 − λeu (u0 )2 ψ2 d r. (2.7) 0

Now, differentiating (2.2), we obtain −u000 = λeu u0 . Multiplying by u0 ψ2 and integrating, it follows that

ˆ

ˆ

1

u (u ψ ) d r = λ 00

0

0

1

eu (u0 )2 ψ2 d r.

2 0

0

Using this in (2.7), we finally obtain

ˆ Q u (ϕ) =

1

(u0 )2 (ψ0 )2 d r ≥ 0. 0

Step 4. Uλ has Morse index 1. First observe that by Proposition 1.3.2, Uλ must have nonzero Morse index. Take a direction ψ ∈ Cc1 (−1, 1) such that Q Uλ (ψ) < 0. By Step 3 and by density, Q Uλ (ϕ) ≥ 0, for every ϕ ∈ Cc1 (−1, 1) such that ϕ(0) = 0. In particular, ψ(0) 6= 0 and we may as well assume that ψ(0) = 1. Now take any ϕ ∈ ˜ = ϕ − ϕ(0)ψ vanishes at 0 and so Q Uλ (ϕ) ˜ ≥ 0. It follows Cc1 (−1, 1). Then, ϕ that the Morse index of Uλ is at most 1. 33

Stable solutions of elliptic PDEs

2.3

Dimension N = 2

In dimension N = 2, the solutions to the Gelfand problem (2.1) can be explicitly computed. Proposition 2.3.1 ([147]) Let N = 2 and λ∗ = 2. Then, • For 0 < λ < λ∗ , there exists exactly two solutions, uλ , Uλ ∈ C 2 (B) to (2.1). uλ is minimal, hence stable, while Uλ is unstable. Both solutions are radial and explicitly given by uλ (r) = ln where b± =

8b−

Uλ (r) = ln

, (1 + λb− r 2 )2

p 4−λ± 16−8λ , λ2

8b+ (1 + λb+ r 2 )2

(2.8)

r ∈ [0, 1].

• For λ = λ∗ , there exists a unique solution given by u(r) = ln

4 (1 + r 2 )2

,

for r ∈ [0, 1].

(2.9)

• For λ > λ∗ , there exists no solution u ∈ C 2 (B) to (2.1). Plotting λ on the x-axis and the norm kuk∞ = u(0) on the y-axis for each solution u ∈ {uλ , Uλ }, we obtain a bifurcation diagram (Figure 2.2) summarizing Proposition 2.3.1. Remark 2.3.1 The bifurcation diagrams in dimension N = 1 and N = 2 are very similar. We point out one important difference. For N = 1, as λ → 0, the unstable solution Uλ blows up at every point x ∈ (−1, 1), by (2.4). For N = 2, Uλ blows up only at the origin. In fact, Uλ (r) → 4 ln 1r for r ∈ (0, 1), as follows from (2.8). Proof. By the maximum principle, every solution is positive. By the Gidas-NiNirenberg symmetry result (see Exercise 1.2.2), every solution is radial. One can verify by direct inspection that any function of the form (2.8) is a solution. Conversely, since there is at most one solution to the ODE −u00 − N −1 u0 = λeu r with the prescribed initial value u0 (0) = 0 and u(0) = a, every solution must be of the form (2.8) for some b ∈ R. Since solutions must also satisfy the boundary condition u(1) = 0, we must have 8b = (1 + λb)2 . This equation is quadratic in b and has respectively 2, 1, or 0 solutions if λ < λ∗ = 2, λ = λ∗ , and λ > λ∗ , respectively. 34

Chapter 2. The Gelfand problem kuλkL∞(B)

λ∗

λ

Figure 2.2: Bifurcation diagram for the Gelfand problem in dimension N = 2.

2.4

Dimension N ≥ 3

The structure of the solution set to (2.1) in dimension N ≥ 3 is radically different. In this section, we establish two bifurcation diagrams (Figures 2.3 and 2.5), and calculate the Morse index of solutions. Proposition 2.4.1 ([132]) Let 3 ≤ N ≤ 9. There exists λ∗ > 2(N − 2) such that • For 0 < λ < λ∗ , λ 6= 2(N − 2), there exists finitely many solutions u ∈ C 2 (B) to (2.1). • Given any k ∈ N, there exists ε > 0 such that for |λ − 2(N − 2)| < ε, there exists at least k solutions. • For λ = 2(N − 2), there exists infinitely many solutions. • For λ = λ∗ , there exists a unique solution. • For λ > λ∗ , there exists no solution. Proposition 2.4.1 is summarized in Figure 2.3. Proof. By the maximum principle, every solution is positive. By the Gidas-NiNirenberg symmetry result (see Exercise 1.2.2), every solution is radial and 35

Stable solutions of elliptic PDEs

!u!L∞ (B)

!u!L∞ (B)

!u!L∞ (B)

λ∗

λ

1≤N ≤2

2(N − 2)

λ∗

λ

3≤N ≤9

λ∗ N ≥ 10

Figure 2.3: Bifurcation diagram in dimension 3 ≤ N ≤ 9.

radially decreasing. So, every solution to (2.1) must satisfy the initial value problem ( 0 r −(N −1) r N −1 u0 + λeu = 0 (2.10) u(0) = a u0 (0) = 0, for some a > 0. Equation (2.10) has a unique maximal solution, obtained using the Picard fixed point theorem applied to the equivalent integral equation ˆ r ˆ s N −1 t u(r) = a − λ eu(t) d t ds. s 0 0 We claim that u is defined for all r ≥ 0. To see this, we apply the Emden transformation r 2(N − 2) t u(r) = w(t) − 2t + a with r= e, (2.11) λe a (2.10) is equivalent to the autonomous ODE w 00 + (N − 2)w 0 + 2(N − 2) (e w − 1) = 0, lim w(t) − 2t = lim e−t w 0 (t) − 2 = 0.

t→−∞

t→−∞

36

(2.12)

λ

Chapter 2. The Gelfand problem Let v(t) = w(t) − 2t. Then w solves (2.12) if and only if v solves the integral equation ˆ s ˆ t −(N −2)s N σ+v(σ) v(t) = −2(N − 2) e e dσ ds. (2.13) −∞

−∞

Applying the Picard fixed point theorem to (2.13), we deduce that there exists a unique solution to (2.12) defined on a maximal interval (−∞, T ). In fact, T = +∞. To see this, consider the Lyapunov function 1 L (w) = (w 0 )2 + 2(N − 2)(e w − w). 2 Then, using (2.12), dL (w) dt

= w 00 w 0 + 2(N − 2)(e w − 1)w 0 = −(N − 2)(w 0 )2 ≤ 0.

Hence, L (w) is bounded from above, and so w, w 0 remain bounded as t → T − . It follows that T = +∞. Now rewrite (2.12) as a system d w w0 = 0 −(N − 2)(w 0 + 2(e w − 1)) dt w and observe that (0, 0) is the unique stationary point. To determine its nature, linearize the system at (0, 0): d z z 0 1 = . 0 −2(N − 2) −(N − 2) z0 dt z The associated eigenvalues are given by p 1 N − 2 ± i (N − 2)(10 − N ) µ± = − 2 and so (0, 0) is a spiral attractor. Now let w denote the solution to (2.12). By definition, the orbit O = {(w, w 0 ) = (w(t), w 0 (t)) : t ∈ R} is asymptotic to the line w 0 = 2 at t = −∞ in the (w, w 0 )-phase plane. Consider the following four regions of this plane Ω1 = {w 0 > 0, w 0 + 2(e w − 1) > 0}, Ω2 = {w 0 < 0, w 0 + 2(e w − 1) > 0}, Ω3 = {w 0 < 0, w 0 + 2(e w − 1) < 0}, Ω4 = {w 0 < 0, w 0 + 2(e w − 1) < 0}. 37

Stable solutions of elliptic PDEs

We prove next that the orbit O is contained in the half-plane {(w, w 0 ) : w 0 < 2}, and that, starting in Ω1 , O then enters successively Ωi , i ∈ Z/4Z, spiraling toward the unique stationary point (0, 0). Observe that since u0 (r) < 0 for r > 0, O lies in {(w, w 0 ) : w 0 < 2}. To see that O starts in Ω1 , we calculate lim r −2 w 0 (t) + 2(e w(t) − 1) = lim+ r −2 ru0 (r) + 2eu(r)+2t−a t→−∞ r→0 λ 00 u(r) = lim+ u (0) + 2e r→0 2(N − 2) a λe = u00 (0) + N −2 1 1 a = λe − + > 0. N N −2 So, O starts in Ω1 . Since w 0 > 0 in Ω1 , (2.12) can be rewritten in this region as N −2 0 dw 0 =− (w + 2(e w − 1)). (2.14) dw w0 Since 0 < w 0 < 2 in Ω1 , (2.14) implies that the orbit cannot escape to infinity in Ω1 , that is, O ∩ Ω1 ∩ {(w, w 0 ) : w > M } is empty for large enough M . From (2.14), we also deduce that O goes right and downwards in Ω1 . So, either O remains in Ω1 , or O leaves Ω1 at some time t 1 . If the former case occurs, (w(t), w 0 (t)) remains in Ω1 and converges to the unique stationary point (0, 0) as t → +∞. This is excluded since (0, 0) is a spiral attractor. So, O leaves Ω1 at some time t 1 where w 0 (t 1 ) = 0 and e w(t 1 ) − 1 ≥ 0. In fact, e w(t 1 ) − 1 > 0. Otherwise, it would follow by uniqueness for the second order ODE (2.12) with initial datum w(t 1 ) = w 0 (t 1 ) = 0 that w ≡ 0. Hence, O must enter Ω2 at t 1 . A similar analysis shows that thereon, O enters successively Ωi , i ∈ Z/4Z and remains in a bounded region of the phase-plane. In particular, w must converge to the unique stationary point (0, 0). We summarize the previous results in Figure 2.4. Going back to (2.1), the boundary condition u(1) = 0 translates to w(τ) − 2τ + a = 0, where τ satisfies

r

2(N − 2) λe a 38

eτ = 1.

Chapter 2. The Gelfand problem

Figure 2.4: Phase portrait of O in the (w, w 0 ) plane.

This is equivalent to asking that w(τ) = ln

λ 2(N − 2)

.

From Figure 2.4, we find infinitely many such values of τ for λ = 2(N − 2), finitely many for λ in a bounded punctured interval (0, λ∗ ) \ {2(N − 2)}, more than k solutions for λ close enough to 2(N − 2), one solution for λ = λ∗ , and none for λ > λ∗ . Exercise 2.4.1 Let N ≥ 10 and λ∗ = 2(N − 2). Prove that (2.1) has a unique solution u ∈ C 2 (B) (which is stable) for 0 < λ < λ∗ , and no solution u ∈ C 2 (B) for λ > λ∗ . Prove that uλ converges pointwise to ln |x|1 2 , as λ → λ∗ .

2.4.1

Stability analysis

In Proposition 2.4.1, we have proved that in dimension 3 ≤ N ≤ 9, given any τ ∈ R, (λτ , uτ ) defined by

with r = e t−τ and

uτ (r) = w(t) − 2t − (w(τ) − 2τ) = w(ln r + τ) − w(τ) − 2 ln r,

(2.15)

λτ = 2(N − 2)e w(τ)

(2.16)

39

Stable solutions of elliptic PDEs

!u!L∞ (B)

!u!L∞ (B)

L∞ (B)

λ∗

λ

2(N − 2)

1≤N ≤2

λ∗

3≤N ≤9

λ

λ∗

λ

N ≥ 10

Figure 2.5: Bifurcation diagram in dimension N ≥ 10.

satisfies (2.1). Conversely, every solution to (2.1) is expressed in the form (2.15) and (2.16). Thus, we can parametrize the solution set S with τ ∈ R. That is, S = {(λτ , uτ ) : τ ∈ R}. From Figure 2.4, we infer that the orbit O crosses the w-axis infinitely many times. Let (τk ) denote the corresponding crossing times, that is, τ1 < τ2 < · · · < τk < . . . and w 0 (τk ) = 0. (2.17) Then,

w(τ2 ) < · · · < w(τ2k ) < w(τ2k−1 ) < · · · < w(τ1 ),

and w(t) achieves a local maximum and minimum at t = τ2k−1 and t = τ2k , respectively. Tk = (λτk , uτk ) is called a turning point. Proposition 2.4.2 ([165]) Let 3 ≤ N ≤ 9 and k ≥ 1. The Morse index of solutions to (2.1) belonging to the arc (Tk , Tk+1 ] is at most equal to k. Proof. Consider the eigenvalue problem −∆ϕ − λτ euτ ϕ = µϕ ϕ=0 40

in B, on ∂ B.

(2.18)

Chapter 2. The Gelfand problem We claim that all eigenfunctions must be radial if µ ≤ 0. Lemma 2.4.1 ([144]) Assume that µ ≤ 0. Then, any solution ϕ ∈ C 2 (B) to (2.18) is radial. Proof. For k ∈ N, let µk = k(k + N − 2) be the k-th eigenvalue of the LaplaceBeltrami operator −∆S N −1 on the sphere (see Theorem C.4.1). Also let {ϕk,l : k ≥ 0, l = 1, . . . , mk } be an orthonormal system in L 2 (S N −1 ) formed with eigenfunctions associated to each µk . Take a solution ϕ of (2.18) and set ˆ ak,l (r) = ϕϕk,l dσ. S

N −1

For k ≥ 1, ak,l (0) = ak,l (1) = 0 and ak,l solves µk N −1 0 uτ 00 ak,l + ak,l + λτ e − 2 ak,l + µak,l = 0. r r

(2.19)

We need to prove that ak,l ≡ 0 for k ≥ 1. Suppose to the contrary that ak,l 6≡ 0. Let r0 be the first zero of ak,l . Then, we may assume without loss of generality that ak,l > 0 in (0, r0 ). Recall that u = uτ is radial and radially decreasing. Then, v := −u0 > 0 in (0, r0 ], v(0) = 0, and differentiating (2.1) with respect to r, N −1 0 N −1 00 uτ v = 0, in (0, r0 ). (2.20) v + v + λτ e − r r2 Multiplying (2.19) by r N −1 v, integrating and using (2.20), we deduce that ˆ r0 N − 1 − µk 0 N −1 ak,l (r0 )v(r0 )r0 + ak,l (r)v(r)r N −1 d r = 2 r 0 ˆ r0 ak,l (r)v(r)r N −1 d r. −µ 0 0 Since ak,l (r0 ) < 0, ak,l , v > 0 in (0, r0 ), and N − 1 ≤ µk , the left-hand side is negative. Thus, µ must be positive, a contradiction.

Thanks to the previous lemma, (2.18) reduces to   ϕ 00 + N − 1 ϕ 0 + λ euτ ϕ + µϕ = 0 for r ∈ (0, 1) τ r  ϕ 0 (0) = 0, ϕ(1) = 0. 41

(2.21)

Stable solutions of elliptic PDEs At a turning point Tk , µ = 0 must be one of the eigenvalues of (2.21). To du see this, simply observe that v = dττ is an eigenfunction associated to µ = 0, as follows from differentiating (2.1) with respect to τ and using (2.16) and (2.17). So, 0 is the l-th eigenvalue of the linearized equation (2.21), for some l ∈ N. We want to prove that l = k. For this, we first show that the l-th eigenvalue µτ of (2.21) changes sign from + to −, as τ increases across τk . Note that dλτ (τk ) = 0, dτ by (2.16) and (2.17). Since λτ and uτ are smooth functions of τ, and since µτ is a simple eigenvalue, it follows from the implicit function theorem that µτ is du a smooth function of τ. So, letting v = dττ and differentiating (2.1) in the τ variable at u = uτ , we get   −∆v = λ euτ v + dλτ euτ in B, τ (2.22) dτ  v=0 on ∂ B. At τ = τk , (2.22) simplifies to ¨ −∆v = λτk euτk v

in B,

v=0

(2.23)

on ∂ B.

That is, v = duτ /dτ τ=τ is an eigenfunction associated to the l-th eigenvalue k 0 at τ = τk . Differentiating again (2.22) with respect to τ, we get for z = d 2 uτ , dτ2 τ=τk

  −∆z − λτ e k 

uτk

z = λτ k e

uτk

d 2 λτ v + dτ2 2

e uτ k

in B,

τ=τk

z=0

on ∂ B.

Multiplying by v and integrating, we deduce that ˆ ˆ 2 d λ τ uτk 3 λτ k e v d x + euτk v d x = 0. 2 dτ B τ=τk B Integrating (2.23), ˆ ˆ uτk λτ k e v d x = − B

(2.24)

∂v

∂B

∂n

42

dσ = −|S

N −1

|v 0 (1).

(2.25)

(2.26)

Chapter 2. The Gelfand problem Furthermore, by (2.15),

v 0 (1) = w 00 (τk )

and by (2.16),

d 2 λτ dτ2

τ=τk

So, (2.25) becomes

= λτk w 00 (τk ).

ˆ λτ k

euτk v 3 d x = |S

N −1

|w 00 (τk )2 .

(2.27)

B

For any τ ∈ R, let ϕτ denote an eigenfunction associated to the l-th eigen value, normalized by ϕτ (0) = duτ /dτ τ=τ , that is, k

−∆ϕτ − λτ euτ ϕτ = µτ ϕτ ϕτ = 0

and ϕτ k

duτ = dτ

in B, on ∂ B,

(2.28)

.

τ=τk dϕ

Differentiate (2.28) with respect to τ. We get for ψ = dττ ,   −∆ψ − λ euτ ψ − µ ψ = λ euτ duτ ϕ + dλτ euτ ϕ + dµτ ϕ τ τ τ τ τ τ dτ dτ dτ  ψ=0

in B, on ∂ B.

At τ = τk , the equation reduces to −∆ψ − λτk e

uτk

ψ = λτ k e

uτk

dµτ v + dτ 2

v.

τ=τk

Multiplying by v and integrating, we deduce that ˆ ˆ dµτ uτ k 3 λτ k e v d x + v 2 d x = 0. dτ τ=τk B B Using (2.27), we obtain dµτ dτ

τ=τk

ˆ v 2 d x = −|S B

43

N −1

|w 00 (τk )2 .

(2.29)

Stable solutions of elliptic PDEs

By (2.12) and (2.17), w 00 (t) never vanishes at t = τk . Hence, the l-th eigenvalue to (2.18) decreases from + to − as τ increases through τk . Equivalently, the Morse index increases by one, as τ increases through τk . We claim that on the arc (Tk Tk+1 ), zero is not an eigenvalue of (2.18) and hence the Morse index never changes there. If this were not the case, there would exist a time τ 6= τk and an eigenfunction ϕ 6≡ 0, which is radial by Lemma 2.4.1, solving −∆ϕ = λτ euτ ϕ in B, (2.30) ϕ=0 on ∂ B. Integrating the above equation, we obtain on the one hand ˆ λτ euτ ϕ d x = −|S N −1 |ϕ 0 (1).

(2.31)

B

Multiplying (2.22) by ϕ and integrating, we have on the other hand ˆ dλτ euτ ϕ d x = 0. dτ B

(2.32)

dλ

Since τ 6= τk , it follows from (2.16) that dττ 6= 0. Collecting (2.31) and (2.32), we deduce that ϕ 0 (1) = 0. By uniqueness for the ode (2.30) with initial value ϕ(1) = ϕ 0 (1) = 0, this forces ϕ ≡ 0, a contradiction. We have just proved that the Morse index remains constant on each arc (Tk Tk+1 ) and increases by one through each turning point Tk . Finally, observe that any solution on the lowest part of the bifurcation diagram (Figure 2.3), that is, any solution of the form (λ, u) = (λτ , uτ ), τ ∈ (−∞, τ1 ], is minimal, hence stable. Proposition 2.4.2 follows. Exercise 2.4.2 Let N = 2. Prove that the unstable solution Uλ to (2.1) has Morse index ind (Uλ ) = 1.

2.5

Summary

Let us review the main features of the bifurcation diagrams in Figure 2.6. The curve on in each of these diagrams is the graph G = {(λ, kuk L ∞ (B) ) : u is a solution to (2.1) with parameter λ}. All classical solutions of the equation are represented. 44

Chapter 2. The Gelfand problem kukL∞ (B)

kukL∞ (B)

kukL∞ (B)

λ∗ 1≤N ≤2

λ

2(N − 2)

λ∗

λ

3≤N ≤9

λ∗

λ

N ≥ 10

Figure 2.6: Bifurcation diagrams for the Gelfand problem.

1. Nonexistence. In any dimension N ≥ 1, there exists no solution to the equation for λ larger than a certain value λ∗ = λ∗ (N ), called the extremal parameter. For 1 ≤ N ≤ 9, λ∗ > 2(N −2) (for N = 1, λ∗ ' 0.88, for N = 2, λ∗ = 2 and for N = 3, λ∗ ' 3.32). For N ≥ 10, λ∗ = 2(N − 2). 2. Multiplicity. Starting from right to left, we see that • For N ≥ 10, there exists a unique solution for each λ ∈ (0, λ∗ ). • For 3 ≤ N ≤ 9, the problem has – a unique solution for λ sufficiently small, – finitely many solutions for λ 6= 2(N − 2), – more than any given number of solutions for λ sufficiently close to 2(N − 2), – infinitely many solutions for λ = 2(N − 2), – two solutions for λ close to λ∗ , and – a unique solution for λ = λ∗ . • For 1 ≤ N ≤ 2, the problem has exactly two solutions for every λ ∈ (0, λ∗ ) and a unique one for λ = λ∗ . 3. The stable branch. For each λ ∈ (0, λ∗ ), the minimal solution must be the one with the smallest L ∞ norm. Furthermore, it is stable. By 45

Stable solutions of elliptic PDEs Proposition 1.3.1, there exists at most one stable solution to the equation for each λ. So, the lowest piece of the diagrams is the branch of stable solutions. 4. Boundedness of the stable branch. In dimensions 1 ≤ N ≤ 9, all stable solutions are uniformly bounded, while this fails in dimensions N ≥ 10. 5. Turning points. In dimensions 1 ≤ N ≤ 9, the solution curve turns ∗ around at λ = λ∗ , where we have λ1 (−∆ − λ∗ eu , B) = 0. All solutions lying above the stable branch are unstable, so their Morse index is at least equal to one. In dimensions 3 ≤ N ≤ 9, the solution curve exhibits infinitely many turning points, accumulating toward λs = 2(N − 2), us = −2 ln |x|. At each of these points, the Morse index of solutions increases by one unit. 6. Boundedness of the branch of index k. It follows from the above discussion, that all solutions of Morse index at most k are uniformly bounded by a constant depending on k and N only. 7. Singular solutions. In dimensions N ≥ 2, the curve has a vertical asymptote at λ = λs = 2(N − 2). Solutions accumulate toward us (x) = −2 ln |x|, λs = 2(N − 2). Note that us is a solution to (2.1) (in the sense of distributions) only for N ≥ 3. Note also that by Hardy’s inequality (Proposition C.1.1), us is stable if N ≥ 10, while us is not even of finite Morse index if 1 ≤ N ≤ 9. Other singular solutions of the equation exist (see [186]) but they are not represented here.

46

Chapter 3 Extremal solutions In Chapter 2, we obtained a complete picture of the set of classical solutions to the Gelfand problem (2.1) (see Figure 2.6). Such a result was obtained for a very specific equation and one may wonder what happens, say, if the domain is no longer a ball or if the nonlinearity is not the exponential function. In this chapter, we consider the equation

−∆u = λ f (u) u=0

in Ω, on ∂ Ω,

(3.1)

where this time, Ω ⊂ RN , N ≥ 1, denotes any smoothly bounded domain. Throughout the chapter, the nonlinearity f ∈ C 1 (R) will be required to satisfy the following sign assumption: f (t) ≥ 0

for t ≥ 0.

(3.2)

See, for example, [197, 199] for more general nonlinearities. Singular solutions play an important role in the study of (3.1), even if one is interested solely in classical solutions. Recall that us (x) = −2 ln |x|

(3.3)

is a singular solution to the Gelfand problem (2.1) in the unit ball B ⊂ RN , N ≥ 3, for λ = λs := 2(N − 2). Also recall that, when N ≥ 3, us is the limit of the whole curve of regular solutions to the equation; the curve is said to bifurcate from infinity. Our first task, in the setting of (3.1), is to properly define weak solutions. 47

Stable solutions of elliptic PDEs

3.1 3.1.1

Weak solutions Defining weak solutions

In this section, we identify the relevant notion of weak solutions to (1.3). In particular, we need to decide whether the equation is satisfied up to the singular set of the solution or only away from it. To answer this question, consider again the function us given by (3.3). In dimension N ≥ 3, us solves (2.1) in D 0 (B) and is therefore a natural candidate. Now, for N = 2, Equation (2.1) with λ = λs = 0, simplifies to ∆u = 0, that is, we request that u be harmonic in B. For sure, us should be excluded from the solution set. Since us is harmonic away from the origin (but −∆us = 4πδ0 in D 0 (R2 )), we shall request that the equation is satisfied up to the singular set. For the equation to hold, in the sense of distributions, we must also ask that weak solutions be (locally) integrable. This leads us to the following definition. Definition 3.1.1 Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain, and let dΩ denote the distance to the boundary of Ω, that is, dΩ (x) = dist(x, ∂ Ω)

for all x ∈ RN .

(3.4)

Let f ∈ C(R). We say that u is an L 1 −weak solution (or simply a weak solution) to (1.3) if u ∈ L 1 (Ω), f (u)dΩ (x) ∈ L 1 (Ω), and ˆ ˆ f (u)ϕ d x, (3.5) u −∆ϕ d x = Ω

Ω

for all ϕ ∈ C02 (Ω), where ¦ © C02 (Ω) = ϕ ∈ C 2 (Ω) : ϕ ∂ Ω ≡ 0 .

(3.6)

2 Remark 3.1.1 Note that (3.5) makes sense for ϕ ∈ C0 (Ω). Indeed, by the meanvalue theorem, ϕ(x) ≤ C dΩ (x). Note also that the boundary condition in (1.3) is encoded in the chosen space C02 (Ω) of test functions.

Exercise 3.1.1 Prove that if u ∈ C 2 (Ω) is a classical solution to (1.3), then u is an L 1 -weak solution to (1.3). Prove that if u ∈ H01 (Ω), f (u)dΩ (x) ∈ L 1 (Ω), and ˆ ˆ Ω

∇u∇ϕ d x = 48

Ω

f (u)ϕ d x

Chapter 3. Extremal solutions for all ϕ ∈ Cc∞ (Ω), then u is a weak solution to (1.3). Conversely, prove that if u is a weak solution, f ∈ C 0,α (R) for some α ∈ (0, 1) and f (u) ∈ L p (Ω) for some p > N /2, then in fact u ∈ C 2 (Ω) and u solves the equation in the classical sense. Exercise 3.1.1 shows that L 1 -weak solutions are a natural extension of the standard notions of variational and classical solutions. It is however not clear at this stage why Definition 3.1.1 encompasses all the reasonable singularities one might encounter when dealing with equations of the form (1.3). The model singular solution given by (3.3) happens to belong to L 1 (B), but perhaps this is not general and weaker nonlinearities f might produce nonintegrable singularities. We prove next that singular solutions that do not belong to L 1 (Ω) are rather unlikely (at least under the assumption f ≥ 0). That is, they cannot be calculated by any reasonable approximation argument. Definition 3.1.2 Let f ∈ C 1 (R), f ≥ 0. Consider an arbitrary nondecreasing sequence ( f n ) of bounded continuous functions such that f n % f pointwise. Let un ∈ C 2 (Ω) denote a solution to (1.3) with nonlinearity f n . If for any choice of such ( f n ) and (un ), there holds lim

n→+∞

un (x) dΩ (x)

= +∞

uniformly in x ∈ Ω,

where dΩ is given by (3.4), then we say that there is complete blow-up in (1.3). Remark 3.1.2 A standard way of approximating f is to use the truncation f n of f at level n, defined for t ∈ R by f (t) if f (t) ≤ n, f n (t) = (3.7) n if f (t) > n. Proposition 3.1.1 ([27]) Let N ≥ 1 and let Ω ⊂ RN denote a smoothly bounded domain of RN . Assume that f ∈ C 1 (R) satisfies f ≥ 0. If there exists no weak solution to (1.3), then there is complete blow-up in (1.3). Proof. Take a nondecreasing sequence of bounded functions ( f n ) converging pointwise to f . Let ζ0 denote the unique solution to −∆ζ0 = 1 in Ω, (3.8) ζ0 = 0 on ∂ Ω. For each n ∈ N, since f n is bounded above by a constant Mn and below by 0, the functions u = 0 and u = Mn ζ0 form a sub- and a supersolution to the 49

Stable solutions of elliptic PDEs approximated problem (3.9) below. Furthermore, u < u in Ω. By the method of sub- and supersolutions (Lemma 1.1.1), there exists a minimal nonnegative solution un ∈ C 2 (Ω) of −∆un = f n (un ) in Ω, (3.9) un = 0 on ∂ Ω.

ˆ

We claim that lim

n→+∞

Ω

f n (un )dΩ (x) d x = +∞.

(3.10)

´If not, since (un ) is a nondecreasing sequence (each un being minimal), Ω f n (un )dΩ (x) d x ≤ C. Multiplying (3.9) with the solution ζ0 of (3.8), we deduce that kun k L 1 (Ω) ≤ C. By monotone convergence, (un ) converges in L 1 (Ω) to a weak solution to (1.3), a contradiction. We have just proved (3.10). We may now apply the boundary point lemma (Proposition A.4.2) to conclude that for some constant c = c(Ω) > 0, ˆ un ≥ c f n (un )dΩ (x) d x dΩ . Ω

So complete blow-up occurs for the sequence of minimal solutions (un ). Since ˜n ≥ un for any other solution u ˜n of (3.9), the result follows. u 2 Exercise 3.1.2 Let Ω = (−1, 1), λ > 0, and let u ∈ L loc (Ω \ {0}). Assume that u solves −x 2 u00 = u2 + λ, in D 0 (Ω \ {0}). (3.11)

• Prove that u ∈ C ∞ (Ω \ {0}) and that |u(x)| ≥ λ| ln x| − C,

near x = 0.

• Let ζn ∈ Cc∞ (Ω \ {0}) denote a cutoff function such that 0 ≤ ζn ≤ 1 in Ω, ζn (x) = 0 for |x| ≤ 1/n, and ζn (x) = 1 for |x| ≥ 2/n. Multiplying (3.11) by ζ4n /x 2 , prove that ˆ 3/n 2 u d x ≤ C n. 2 2/n x 2 • Deduce that there is no u ∈ L loc (Ω \ {0}), solving (3.11).

• Given n ∈ N∗ , prove that there exists λn > 0 and un solution to ¨ −(x 2 + 1/n)u00n = u2n + λn in Ω, un (−1) = un (1) = 0. 50

(3.12)

Chapter 3. Extremal solutions • Let λ∗n denote the supremum of all λn such that (3.12) has a (classical) solution. Prove that λ∗n → 0, as n → +∞.

3.2

Stable weak solutions

The notion of stability for weak solutions is naturally defined as follows. Definition 3.2.1 Let N ≥ 1, let Ω denote an open set of RN and let f ∈ C 1 (R). Let u ∈ L l1oc (Ω) satisfy f (u) ∈ L l1oc (Ω) and −∆u = f (u),

in D 0 (Ω).

1 We say that u is stable if f 0 (u) ∈ Lloc (Ω) and (1.5) holds.

Remark 3.2.1 By Hardy’s inequality (Proposition C.1.1), the function us (x) = −2 ln |x| is a stable weak solution to the Gelfand problem (2.1) if and only if N ≥ 10.

3.2.1

Uniqueness of stable weak solutions

Recall that for (strictly) convex f , there exists at most one classical stable solution to (1.3), see Proposition 1.3.1. For weak solutions, the situation is more delicate. A first partial answer is provided by the following proposition. Proposition 3.2.1 ([32]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Let λ1 = λ1 (−∆; Ω) > 0 denote the principal eigenvalue of the Dirichlet Laplacian on Ω. Assume that f ∈ C 1 (R) is convex. Let u1 , u2 denote two stable weak solutions of (1.3). In addition, assume that u1 , u2 ∈ H01 (Ω). Then, either u1 = u2 almost everywhere (a.e.) or f (u) = λ1 u on the essential ranges of u1 and u2 . In the latter case, u1 and u2 belong to the eigenspace associated to λ1 . In particular, they are colinear. Proof. Simply repeat the proof of Proposition 1.3.1. 1 When solutions do not belong to the energy space H0 (Ω), uniqueness fails in general, as the following example demonstrates. 51

Stable solutions of elliptic PDEs Example 3.2.1 ([32]) Let N ≥ 3 and let Ω = B denote the unit ball in RN . Given any p in the range p N +2 N −1 N 0 and consider

−∆u = λs (1 + u) p u=0

in B, on ∂ B.

(3.14)

There exists at least two stable solutions of (3.14). One of them belongs to − 2 H01 (B), while the other does not and is given by us (x) = |x| p−1 − 1. Proof. Thanks to Hardy’s inequality (Proposition C.1.1), a direct computation shows that us is a stable weak solution that does not belong to H01 (B), for p in the range (3.13). By the method of sub- and supersolutions, there exists a stable solution u, satisfying 0 ≤ u ≤ us in Ω, obtained as the monotone limit of the sequence (un ) given by u0 = 0 and for n ≥ 1, −∆un = λs (1 + un−1 ) p in B, (3.15) un = 0 on ∂ B. Since us 6∈ H01 (B), it suffices to prove that u ∈ H01 (B) to deduce that u 6= us . To do so, we observe by an obvious inductive argument, that each un is smooth. So, we need only prove that there exists a constant M such that for all n ∈ N, k∇un k L 2 (B) ≤ M .

(3.16)

Multiply (3.15) by un and integrate. Then, ˆ ˆ ˆ 2 p ∇u d x = λ (1 + un−1 ) un d x ≤ λs (1 + un ) p un d x. (3.17) n s B

B

B

Since us is stable, we also have ˆ ˆ ˆ 2 p−1 2 ∇u d x ≥ λ p (1 + u ) u d x ≥ λ p (1 + u ) p−1 u2 d x. n s s s n n n B

B

It follows that

B

ˆ

ˆ

p B

(1 + un ) p−1 u2n

(1 + un ) p un d x,

dx ≤ B

52

Chapter 3. Extremal solutions and so,

ˆ (p − 1)

ˆ

(1 + un ) un d x ≤ p (1 + un ) p−1 un d x B B ˆ ˆ p−1 ≤p (1 + un ) un d x + p (1 + un ) p−1 un d x p

p+1

[ p−1
≤p

p+1

[ p−1 ≥un ]

ˆ p+1

[ p−1
(1 + un ) p−1 un d x + C(p, N ) ≤

It follows that

p−1 2

ˆ p+1

[ p−1
(1 + un ) p un d x + C.

ˆ (1 + un ) p un d x ≤ C. B

Recalling (3.17), we deduce (3.16).

3.2.2

Approximation of stable weak solutions

As we have seen in Chapter 2, when N ≥ 10, the singular solution us given by (3.3) is the limit of the branch of stable solutions for the Gelfand problem (2.1). In other words, us can be approximated by a curve of smooth stable solutions to the same equation (with a slightly smaller value of the parameter λ). This approximation procedure turns out to hold in a more general setting, as the following two results demonstrate. Theorem 3.2.1 ([29]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f ≥ 0, and f is convex. Assume that u ∈ L 1 (Ω) is a stable weak solution to (1.3). Then, for every " > 0, there exists a stable solution to −∆u" = (1 − ") f (u" ) in Ω, (3.18) u" = 0 on ∂ Ω, which is bounded, hence regular. Combining this with the uniqueness result Proposition 3.2.1, we obtain the following corollary. 53

Stable solutions of elliptic PDEs Corollary 3.2.1 Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f ≥ 0, and f is convex. Assume that u ∈ H01 (Ω) is a weak solution to (1.3). Then, u = lim"→0 u" , where u" is the classical stable solution to (3.18) and where convergence holds in H01 (Ω). To prove Theorem 3.2.1, we establish first the following lemma, known as the concave truncation method, or Kato’s inequality. Lemma 3.2.1 ([133], [29]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Let dΩ denote the distance to the boundary of Ω, as defined in (3.4). Given f ∈ L 1 (Ω, dΩ (x)d x), let u ∈ L 1 (Ω) denote the unique weak solution to −∆u = f in Ω, u=0 on ∂ Ω, given by Lemma A.9.1. Let Φ ∈ C 1 (R) denote a concave function such that Φ0 is bounded and Φ(0) ≥ 0. Then, Φ(u) ∈ L 1 (Ω) and −∆Φ(u) ≥ Φ0 (u) f in the sense that ˆ ˆ − Φ(u)∆ϕ d x ≥ Φ0 (u) f ϕ d x, Ω

Ω

in Ω,

for all ϕ ∈ C02 (Ω), ϕ ≥ 0.

Remark 3.2.2 The lemma remains valid for Φ(u) = −u+ , that is, ˆ ˆ + u ∆ϕ d x ≥ − f ϕ d x, ∀ ϕ ∈ C02 (Ω), ϕ ≥ 0. Ω

(3.19)

[u≥0]

Proof. Take f n ∈ Cc1 (Ω) such that f n → f in L 1 (Ω, dΩ (x)d x). Let un ∈ C 2 (Ω) denote the solution to −∆un = f n in Ω, un = 0 on ∂ Ω. Then, −∆Φ(un ) = −Φ0 (un )∆un − Φ00 (un )|∇un |2 ≥ −Φ0 (un )∆un = Φ0 (un ) f n . 54

Chapter 3. Extremal solutions Therefore, given ϕ ∈ C02 (Ω), ϕ ≥ 0,

ˆ −

ˆ

Ω

Φ(un )∆ϕ d x ≥

Ω

Φ0 (un ) f n ϕ d x.

By Lemma A.9.1, un → u in L 1 (Ω). Since Φ0 is bounded, |Φ(t)| ≤ C|t| + Φ(0), for all t ∈ R. So, we also have Φ(un ) → Φ(u) ∈ L 1 (Ω). The lemma follows letting n → +∞. To see that Remark 3.2.2 holds, take a sequence of smooth concave functions in R such that Φn (t) = −t if t ≥ 0 and |Φn (t)| ≤ 1/n if t < 0. In particular, 0 ≥ Φ0n ≥ −1 in R. So, we may apply Lemma 3.2.1: given ϕ ∈ C02 (Ω), ϕ ≥ 0,

ˆ −

ˆ

Ω

Φn (u)∆ϕ d x ≥

Ω

Φ0n (u) f ϕ d x.

Letting n → +∞, we deduce (3.19).

Proof of Theorem 3.2.1. Thanks to Lemma 3.2.1, we may construct a bounded supersolution U ≥ 0 to (3.18) as follows. If f (t 0 ) = 0 for some t 0 ≥ 0, then t 0 is a supersolution to (3.18). By the method of sub- and supersolutions, we deduce that there exists a bounded stable solution to (3.18). If f (t) > 0 for t ≥ 0, we seek a supersolution to the form U = Φ(u), where Φ satisfies the assumptions of Lemma 3.2.1. Then, U is a supersolution to (3.18) as soon as Φ0 (u) f (u) ≥ (1 − ") f (Φ(u)). So, we set Φ to be a solution to the differential equation Φ0 (t) f (t) = (1 − ") f (Φ(t)),

for all t > 0.

with initial value Φ(0) = 0. In other words, for t > 0, Φ(t) is the unique real number such that ˆ Φ(t) ˆ t ds ds = (1 − ") . (3.20) f (s) 0 f (s) 0 55

Stable solutions of elliptic PDEs Note that 0 ≤ Φ(t) ≤ t, that Φ is C 2 , that Φ0 ≥ 0, and that f (Φ(t)) 0 00 Φ (t) = (1 − ") f (t) 1−" 0 = ( f (Φ(t))Φ0 (t) f (t) − f (Φ(t)) f 0 (t)) 2 f (t) (1 − ") f (Φ(t)) ((1 − ") f 0 (Φ(t)) − f 0 (t)) = f (t)2 (1 − ") f (Φ(t)) 0 ≤ ( f (Φ(t)) − f 0 (t)). f (t)2 Since Φ(t) ≤ t and f is convex, we deduce that Φ is concave. In particular, 0 ≤ Φ0 (t) ≤ Φ0 (0) for t ≥ 0, that is, Φ0 is bounded. So, we may apply Lemma 3.2.1 and deduce that U is a supersolution to (3.18). We split the rest of the proof in two cases: Case 1. Assume that ˆ +∞ ds < +∞. f (s) 0 Then, it follows from (3.20) that Φ is bounded and so must be U. By the method of sub- and supersolutions, we deduce that there exists a classical stable solution to (3.18). Case 2. Assume that ˆ +∞ ds = +∞. f (s) 0 If f is nonincreasing, then f is bounded and U = u provides a bounded supersolution to (3.18). Otherwise, since f is convex, there exists t 0 ≥ 0 such that f 0 (t) > 0 for t ≥ t 0 . So, the function h defined for t ≥ 0 by ˆ t ds h(t) = 0 f (s) is concave in [t 0 , +∞) and for t 0 ≤ U ≤ u, h(u) ≤ h(U) + h0 (U)(u − U) = h(U) +

u−U f (U)

.

Apply the above inequality with U = Φ(u). Then, h(U) = (1 − ")h(u) and " f (U) ≤

u−U h(u)

≤

56

u h(u)

≤ Cu,

Chapter 3. Extremal solutions whenever u ≥ U ≥ t 0 . Clearly, we also have f (U) ≤ C for 0 ≤ U ≤ t 0 and so f (U) ≤ C(1 + u),

(3.21)

for all 0 ≤ U ≤ u. By the definition of Φ and Lemma 3.2.1, U(x) = Φ(u(x)) is a weak supersolution to −∆u1 = (1 − ") f (u1 ) in Ω, u1 = 0 on ∂ Ω. By the method of sub- and supersolutions, there exists a stable weak solution u1 of the above equation, such that 0 ≤ u1 ≤ U. In particular, we have 0 ≤ f (u1 ) ≤ f (U). By (3.21) and Corollary A.9.1, we deduce1 that u1 ∈ L p (Ω) for all 1 ≤ p < N /(N − 1). (3.22) By the same construction, we find a solution to ¨ −∆u2 = (1 − ")2 f (u2 ) u2 = 0

in Ω, on ∂ Ω.

such that 0 ≤ u2 ≤ u1 and f (u2 ) ≤ C(1 + u1 ). In particular, f (u2 ) ∈ L p (Ω) for any p in the range (3.22). This implies that u2 ∈ L q (Ω) for all q < NN−3 . By iteration, the solution uk to the equation ¨ −∆uk = (1 − ")k f (uk ) in Ω, uk = 0 on ∂ Ω, is bounded provided k ≥ [(N + 1)/2] + 1. Since " ∈ (0, 1), this completes the proof. Proof of Corollary 3.2.1. By Theorem 3.2.1, the stable solution u" of (3.18) is classical. u" is also minimal, hence 0 ≤ u" ≤ u"0 ≤ u for " 0 ≤ ". By monotone convergence, (u" ) converges in L 1 (Ω) to a stable weak solution v ∈ L 1 (Ω) such that 0 ≤ v ≤ u. Also, since f is convex, the function g(t) = f (t) + Λt, with Λ = f 0 (0)− , is nondecreasing in R+ . Multiplying (3.18) by u" and integrating, we obtain ˆ ˆ ˆ ∇u 2 d x = (1 − ") f (u" )u" d x ≤ g(u" )u" d x " Ω Ω Ω ˆ ˆ ˆ 2 ≤ g(u)u d x = |∇u| d x + Λ u2 d x < +∞. (3.23) Ω

Ω

Ω

In fact, we could even take p < N /(N − 2), since f (U) is L 1 (Ω) and not merely L (Ω, dΩ d x) as in Corollary A.9.1. 1

1

57

Stable solutions of elliptic PDEs It follows that (u" ) is bounded in H01 (Ω), hence v ∈ H01 (Ω). By Proposition 3.2.1, u = v. Returning to (3.23), we deduce from the monotone convergence theorem and the compactness of (u" ) in L 2 (Ω) that ˆ ˆ ∇u 2 d x = (1 − ") f (u" )u" d x " Ω Ω ˆ ˆ 2 = (1 − ") g(u" )u" d x − Λ u" d x Ω ˆ ˆ ˆΩ → g(u)u d x − Λ u2 d x = f (u)u d x Ω Ω Ω ˆ = |∇u|2 d x, Ω

as " → 0+ and so u" → u in H01 (Ω). Exercise 3.2.1 Assume that there exists a classical solution to  in Ω × (0, +∞),  vt − ∆v = f (v) v=0 on ∂ Ω × (0, +∞),  v=0 in Ω × {0},

(3.24)

where f ∈ C 2 (R) is such that f , f 0 , f 00 > 0. • Prove that there exists a bounded global solution v = v" to 

 vt − ∆v = (1 − ") f (v) v=0  v=0

in Ω × (0, +∞), on ∂ Ω × (0, +∞), in Ω × {0}.

• Prove that u" = lim t→+∞ v" (t) solves (3.18).

3.3

The stable branch

In this section, we construct and analyze the branch to stable solutions of (3.1). 58

Chapter 3. Extremal solutions Proposition 3.3.1 ([29, 63, 136, 137]) Assume that N ≥ 1. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f ≥ 0. Then, there exists λ∗ = λ∗ (Ω, N , f ) ∈ (0, +∞] such that • For 0 < λ < λ∗ , there exists the minimal solution uλ ∈ C 2 (Ω) of (3.1). In particular, uλ is stable. If in addition f is convex, then uλ is the unique classical stable solution to (3.1). • For λ > λ∗ , there exists no classical solution u ∈ C 2 (Ω) of (3.1). In addition, if f is convex or if Ω is a ball, then there exists no weak solution to (3.1) either. Remark 3.3.1 λ∗ is called the extremal parameter of (3.1). Remark 3.3.2 As we have seen in Chapter 1, the minimal solution to a semilinear elliptic problem is defined relatively to a subsolution. Here, since f ≥ 0, u = 0 is a subsolution to (3.1) and by the maximum principle (Proposition A.2.2), every solution to (3.1) is positive. So, uλ is minimal among all solutions of the problem. Proof. Let ζ0 ∈ C 2 (Ω) denote the solution to (3.8). For λ > 0 sufficiently small, we have 1 ≥ λ f (ζ0 ). So, u = ζ0 is a supersolution to (3.1), while u = 0 is a subsolution. In addition, u > u. So, we may apply the method of sub- and supersolutions (Lemma 1.1.1) and obtain the minimal solution uλ to (3.1) for λ > 0 small. Define λ∗ = sup{λ > 0 : (3.1) has a classical solution u ∈ C 2 (Ω)}. We claim that there exists solutions to (3.1) for all λ ∈ (0, λ∗ ). Fix such a λ. By definition of λ∗ , there exists µ ∈ (λ, λ∗ ) and a function uµ ∈ C 2 (Ω) solving ¨

−∆uµ = µ f (uµ ) uµ = 0

in Ω, on ∂ Ω.

In particular, since µ > λ, u = uµ is a supersolution to (3.1), while u = 0 is a subsolution, and u ≥ u. By the method of sub- and supersolutions, there exists the minimal solution uλ ∈ C 2 (Ω) to (3.1). By Proposition 1.3.1, if f is convex, then uλ is the unique stable solution to (3.1). It remains to prove that there exists no weak solution to (3.1) for λ > λ∗ , whenever f is convex or Ω is a ball. Assume by contradiction that there exists a weak solution u ∈ L 1 (Ω) of (3.1) for some λ > λ∗ . By the method of sub- and supersolutions, we may always assume that u is the minimal solution to (3.1). 59

Stable solutions of elliptic PDEs If f is convex, Theorem 3.2.1 implies that there exists a classical solution to the problem for all µ ∈ [0, λ), which contradicts the definition of λ∗ . Assume now that Ω is a ball B. Assume for simplicity that B is the unit ball centered at the origin. Since u is minimal, u must be radial. Indeed, given any rotation of space R, v(x) = u(Rx) is a weak solution to (3.1). Since u is minimal, u(x) ≤ v(x) = u(Rx) for x ∈ B. Applying the inequality at y = R−1 x, we deduce that u(x) = u(Rx) for all x ∈ B and so u is radial. In particular, u must be regular away from the origin and ∂ u/∂ r < 0 in B \ {0}. Given " > 0, set V (x) = V (r) := u(r + "), for r = |x| ≤ 1 − ". Then, V is a bounded function and for r ∈ (0, 1 − "), we have N −1 0 1 1 N −1 0 00 00 V = −V − V − (N − 1) − V 0 ≥ λ f (V ). −V − r r +" r r +" Take a test function ϕ ∈ C02 (B1−" ), ϕ ≥ 0 and δ > 0. Since V is bounded, it follows that ˆ ˆ ˆ ∂V ∂ϕ +ϕ dσ V (−∆ϕ) d x = (−∆V )ϕ d x − −V ∂r ∂r B1−" \Bδ B1−" \Bδ ∂ Bδ ˆ f (V )ϕ d x − kV k∞ k∇ϕk∞ |∂ Bδ |. ≥λ B1−" \Bδ

Letting δ → 0, we deduce that ˆ ˆ V (−∆ϕ) d x ≥ λ

f (V )ϕ d x. B1−"

B1−"

It follows that the function U defined by U(x) = V ((1 − ")x) for x ∈ B is a bounded weak supersolution to ¨ −∆u = (1 − ")2 λ f (u) in B, u=0 on ∂ B. By the method of sub- and supersolutions, we deduce that there exists a classical solution to the aformentioned equation, contradicting the definition of λ∗ . Exercise 3.3.1 Consider the semilinear boundary value problem  ∆u = 0 in Ω   ∂u (3.25) = λ f (u) on Γ1  ∂ν u=0 on Γ2 60

Chapter 3. Extremal solutions where Γ1 , Γ2 is a partition of ∂ Ω into surfaces separated by a smooth interface. We say that u is a weak solution to (3.25) if u ∈ W 1,1 (Ω), f (u) ∈ L 1 (Γ1 ), and ˆ ˆ ∂ ϕ 2 ¯ ≡ 0. u(−∆ϕ) d x = λ f (u)ϕ dσ ∀ϕ ∈ C (Ω) s.t. ϕ ≡ 0 and Γ2 ∂ ν Γ1 Ω Γ1 Assume that f ∈ C 1 (R), f convex, and f ≥ 0. Prove that there exists λ∗ ∈ (0, ∞] such that • (3.25) has a smooth solution for 0 ≤ λ < λ∗ , and • (3.25) has no solution for λ > λ∗ (even in the weak sense). Moreover, prove that for 0 ≤ λ < λ∗ , there exists a minimal solution uλ which is bounded, positive, and stable, in the sense that ˆ ˆ 2 |∇ϕ| d x − λ f 0 (uλ )ϕ 2 dσ ≥ 0. inf ϕ∈C 1 (Ω),ϕ=0 on Γ2

Ω

Γ1

Proposition 3.3.1 gives rise to a number of natural questions, which we address now.

3.3.1

When is λ∗ finite?

Proposition 3.3.1 Assume N ≥ 1. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f ≥ 0, f (0) 6= 0. • If inf t∈R+ • If inf t∈R+

f (t) t

> 0, then λ∗ < +∞.

k f k L ∞ (0,t) t

= 0, then λ∗ = +∞.

Remark 3.3.3 If f (0) = 0, then 0 is a trivial solution to (3.1) for all λ > 0 and so λ∗ = +∞. Remark 3.3.4 Under the additional assumption that f is nondecreasing, the f (t) proposition is sharp: λ∗ < +∞ if and only if inf t∈R+ t > 0. Proof. Assume first that a := inf t∈R+ f (t)/t > 0. Let ϕ1 > 0 denote an eigenvector associated to the principal eigenvalue λ1 = λ1 (−∆; Ω). Assume that u ∈ C 2 (Ω) is a solution to (3.1). Multiply (3.1) by ϕ1 and integrate by parts: ˆ ˆ ˆ ˆ λ f (u)ϕ1 d x = −∆u ϕ1 d x = −∆ϕ1 u d x = λ1 uϕ1 d x. Ω

Ω

Ω

61

Ω

Stable solutions of elliptic PDEs Since f (t) ≥ at for t ≥ 0,

ˆ (aλ − λ1 )

Ω

uϕ1 d x ≤ 0.

By the strong maximum principle (Proposition A.2.2) applied to (3.1), u > 0 and we deduce that aλ ≤ λ1 . In particular, no classical solution to (3.1) exists for large values of λ. kf k ∞ Assume now that inf t∈R+ Lt (0,t) = 0. Let ζ0 denote the solution to (3.8). Also let M = kζ0 k L ∞ (Ω) . Fix λ > 0 and take t 0 > 0 such that ζ

k f k L ∞ (0,t 0 ) t0

≤

1 . λM

Set u = t 0 M0 . Then, −∆u =

t0 M

≥ λk f k L ∞ (0,t 0 ) ≥ λ f (u).

Using the method of sub- and supersolutions, we obtain a solution to (3.1) for any λ > 0.

3.3.2

What happens at λ = λ∗ ?

We now remain with the case where λ∗ is finite. In particular, we must assume f > 0 (otherwise, if t 0 ≥ 0 is such that f (t 0 ) = 0, then u = t 0 is a supersolution to (3.1) and so the equation is solvable for all λ > 0). Results vary according to the behavior of f at infinity. We describe next the case of a superlinear nonlinearity. The superlinear case Proposition 3.3.2 ([63, 136, 137, 149]) Assume N ≥ 1. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f > 0. In addition, assume that f is nonndecreasing and that f is superlinear in the following sense: lim

t→+∞

f (t) t

= +∞.

(3.26)

Then, the family (uλ )0<λ<λ∗ of minimal solutions of (3.1) converges to a weak solution u∗ ∈ L 1 (Ω) of (3.1) for λ = λ∗ . If in addition f is convex, then u∗ is the unique weak solution to (3.1) for λ = λ∗ . Remark 3.3.5 Since f > 0 and f satisfies (3.26), infR+ f (t)/t > 0. By Proposition 3.3.1, we are in the situation where λ∗ < +∞. 62

Chapter 3. Extremal solutions Definition 3.3.1 The solution u∗ is called the extremal solution to (3.1). Before proving Proposition 3.3.2, we give a convenient criterion for identifying the extremal solution. Corollary 3.3.1 ([32]) Assume N ≥ 1. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R), f > 0, and that f is nondecreasing and convex. If u ∈ H01 (Ω) is an unbounded stable solution to (3.1) associated to a parameter λ > 0, then necessarily u = u∗ and λ = λ∗ . Proof of Corollary 3.3.1. Assume by contradiction that λ < λ∗ . By Proposition 3.2.1, u must coincide with the minimal classical solution uλ , a contradiction. So, λ = λ∗ and by minimality of uλ0 , λ0 < λ∗ , we must have uλ0 ≤ u, hence u∗ ≤ u. Also, ˆ ˆ ˆ ˆ ∗ ∇u 0 2 d x = λ0 f (uλ0 )uλ0 d x ≤ λ f (u)u d x = |∇u|2 d x < +∞. λ Ω

Ω

∗

So, u ∈

H01 (Ω)

Ω

Ω

and by Proposition 3.2.1, u = u . ∗

Proof of Proposition 3.3.2. We claim that there exists a constant C > 0 independent of λ, such that kuλ k L 1 (Ω) ≤ C

for all λ ∈ (0, λ∗ ).

(3.27)

Fix λ ∈ (0, λ ) and multiply (3.1) by ϕ1 > 0, an eigenvector associated to λ1 = λ1 (−∆; Ω). ˆ ˆ f (uλ )ϕ1 d x. (3.28) λ 1 uλ ϕ 1 d x = λ ∗

Ω

Ω

f is superlinear, so for all " > 0, there exists C" > 0 such that for all t ≥ 0, f (t) ≥ 1ε t − Cε . Hence, ˆ λ C≥ − λ1 uλ ϕ1 d x. ε Ω Choosing " = 2λλ , we obtain that (uλ dΩ (x)) is bounded in L 1 (Ω). By (3.28), 1 so is ( f (uλ )dΩ ). Test again (3.1), this time with ζ0 solving (3.8). ˆ ˆ ˆ uλ · 1 d x = (−∆uλ )ζ0 d x = λ f (uλ )ζ0 d x Ω

Ω

Ω

and (3.27) follows. Since uλ is minimal, the sequence (uλ ) is nondecreasing in λ. So, uλ % u∗ ∈ L 1 (Ω). Since f is nondecreasing, f (uλ ) % f (u∗ ) in L 1 (Ω, dΩ (x) d x). Passing to the limit in (3.5) as we may, we deduce that u∗ is a weak solution to (3.1) for λ = λ∗ . At last, u∗ is unique when f is convex, thanks to Corollary 3.3.2 below. 63

Stable solutions of elliptic PDEs Theorem 3.3.1 ([149]) Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain and let f ∈ C 1 (R) denote a nondecreasing convex function such that f > 0. Let λ∗ denote the associated extremal parameter. Assume that there exists v ∈ L 1 (Ω) such that v ≥ 0 a.e., f (v)dΩ ∈ L 1 (Ω), where dΩ is given by (3.4), and ˆ ˆ ∗ − v∆ϕ d x ≥ λ f (v)ϕ d x for all ϕ ∈ C02 (Ω) such that ϕ ≥ 0. Ω

Ω

Then, v = u∗ is the extremal solution to (3.1). The following corollary is immediate. Corollary 3.3.2 ([149]) Under the assumptions of Theorem 3.3.1, there is at most one weak solution to (3.1) for λ = λ∗ . We establish two intermediate lemmata in order to prove Theorem 3.3.1. Lemma 3.3.1 ([149]) Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain and let f ∈ C 1 (R), f > 0, denote a nondecreasing convex function. Also fix ε > 0. Assume that there exists a weak solution w ∈ L 1 (Ω) to −∆w = f (w) + ε in Ω, (3.29) w=0 on ∂ Ω. Then, there exists a classical solution u ∈ C 2 (Ω) to −∆u = (1 + α) f (u) in Ω, u=0 on ∂ Ω,

(3.30)

for some α > 0. Proof. Step 1. There exists a classical solution v ∈ C 2 (Ω) of   −∆v = f (v) + ε in Ω, 2  v=0 on ∂ Ω. Exercise 3.3.2 Prove Step 1, using the concave truncation technique of Theorem 3.2.1. Step 2. Now consider the function ζ0 ∈ C 2 (Ω) solving (3.8). Applying the mean-value theorem to v on the one hand and the boundary point lemma 64

Chapter 3. Extremal solutions (Lemma A.4.1) to ζ0 on the other hand, it follows that there exists α > 0 such that 2αv ≤ εζ0 . Set ε u = v + αv − ζ0 . 2 Clearly, 0 < u ≤ v in Ω. Furthermore, since f is nondecreasing, u satisfies   −∆u = f (v) + α f (v) + αε ≥ (1 + α) f (u) in Ω, 2  u=0 on ∂ Ω. In particular, u is a bounded supersolution to (3.30), while u = 0 is a subsolution and u < u. The lemma follows. Taking advantage of this lemma, we establish that if (1.3) has a stable singular weak solution, then no strict supersolution to (1.3) exists. Lemma 3.3.2 ([149]) Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain and let f ∈ C 1 (R), f > 0, denote a nondecreasing convex function. Let λ∗ denote the associated extremal parameter. Assume v ∈ L 1 (Ω), v ≥ 0 a.e., verifies ˆ f (v)dΩ d x < +∞, (3.31) Ω

where dΩ is given by (3.4). Assume further that for all ϕ ∈ C02 (Ω) such that ϕ ≥ 0, ˆ ˆ ∗ − v∆ϕ d x ≥ λ f (v)ϕ d x. (3.32) Ω

Then, in fact

Ω

ˆ

ˆ −

Ω

v∆ϕ d x = λ

∗ Ω

f (v)ϕ d x,

(3.33)

for all ϕ ∈ C02 (Ω) such that ϕ ≥ 0. Proof. We argue by contradiction and assume that there exists a nonnegative measure µ 6≡ 0, such that dΩ is µ-integrable and ˆ ˆ ˆ ∗ − v∆ϕ d x = λ f (v)ϕ d x + ϕ dµ, (3.34) Ω

Ω

for all ϕ ∈ C02 (Ω). Consider ζ1 ∈ L 1 (Ω), solving −∆ζ1 = µ in Ω, ζ1 = 0 on ∂ Ω. 65

Ω

(3.35)

Stable solutions of elliptic PDEs Such a solution exists and is unique by Corollary A.9.1. Since µ 6≡ 0, it follows from the boundary point lemma (Corollary A.9.2) on the one hand, and the mean value theorem on the other hand, that εζ0 ≤ ζ1 , for some ε > 0, and where ζ0 is the solution to (3.8). Set u = v + εζ0 − ζ1 . Clearly, 0 < u ≤ v. In addition, ˆ ˆ ˆ ∗ − u∆ϕ d x = (λ f (v) + ε)ϕ d x ≥ (λ∗ f (u) + ε)ϕ d x Ω

Ω

Ω

for all ϕ ∈ C02 (Ω), ϕ ≥ 0. By the method of sub- and supersolutions, there exists a weak solution 0 ≤ w ≤ u to −∆w = λ∗ f (w) + ε in Ω, w=0 on ∂ Ω. Lemma 3.3.1 now contradicts the definition of λ∗ .

Proof of Theorem 3.3.1. Let v denote a weak supersolution to the equation, as defined in Theorem 3.3.1. By the method of sub- and supersolutions, there ˜ such that 0 ≤ u ˜ ≤ v. Since u ˜ is minimal, it exists a minimal weak solution u ˜ = u∗ is the extremal solution. Assume now by contradiction that follows that u v 6= u∗ . Step 1. There exists A ⊂ u∗ (Ω), |A| 6= 0, such that f 00 (s) > 0

for all s ∈ A.

If not, we have f (u∗ ) = f (0) + f 0 (0)u∗ a.e. in Ω, and everything happens for u∗ as though f was linear. This contradicts Proposition 3.3.3. Therefore, there exists η > 0 and 0 < K1 < K2 ≤ ku∗ k L ∞ (Ω) such that f 00 (s) ≥ η

for all s ∈ [K1 , K2 ].

Step 2. We show the existence of a weak strict supersolution to the equation. First note that by Lemma 3.3.2, the function v satisfies ˆ ˆ ∗ − v∆ϕ d x = λ f (v)ϕ d x, Ω

Ω

for all ϕ ∈ C02 (Ω), such that ϕ ≥ 0. Take ψ = λ∗ ( f (v) − f (u∗ )) ≥ 0 and consider w ∈ L 1 (Ω) the solution to −∆w = ψ in Ω, w=0 on ∂ Ω. 66

Chapter 3. Extremal solutions By assumption, ψdΩ ∈ L 1 (Ω). In addition, ψ 6= 0, otherwise f (v) = f (u∗ ) a.e. in Ω, and, by Lemma A.9.1, v = u∗ a.e. in Ω. Using the boundary point lemma (Lemma A.9.2), we deduce that w ≥ c dΩ , for some constant c > 0. Since ˆ − (v − u∗ − w)∆ϕ d x = 0, Ω

for all ϕ ∈ C02 (Ω) such that ϕ ≥ 0, we deduce that v − u∗ = w ≥ cdΩ . Step 3. Now set u =

ˆ −

Ω

v+u∗ . 2

u∆ϕ d x =

λ∗ 2

Then,

ˆ

ˆ ( f (v) + f (u ))ϕ d x = ∗

Ω

Ω

( f (u) + h)ϕ d x,

for all ϕ ∈ C02 (Ω) such that ϕ ≥ 0 and where h is given by ˆ ˆ t v + u∗ 1 v 1 ∗ = h = ( f (v) + f (u )) − f d t ∗ f 00 (s) ds. u +t 2 2 2 u∗ 2 Clearly, h dΩ ∈ L 1 (Ω) and h ≥ 0 in Ω. In addition, by Steps 1 and 2, we have h 6≡ 0. It follows that u is a strict supersolution to the equation, contradicting Lemma 3.3.2. Exercise 3.3.3 Consider the following semilinear problem involving the fractional Laplacian (−∆)s u = λ f (u) in B1 , (3.36) u=0 on ∂ B1 . Here, B1 denotes the unit-ball RN , N ≥ 2, and s ∈ (0, 1). The operator (−∆)s in ∞ is defined as follows. Let ϕk k=1 denote an orthonormal basis of L 2 (B1 ) consisting of eigenfunctions of −∆ in B1 with homogeneous Dirichlet boundary conditions, associated to the eigenvalues {λk }∞ . The operator (−∆)s is dek=1 fined for any u in the Hilbert space H = {u ∈ L 2 (B1 ) : kuk2H =

∞ X

λsk |uk |2 < +∞},

k=1

by (−∆) u = s

∞ X k=1

67

λsk uk ϕk ,

Stable solutions of elliptic PDEs where u=

∞ X

ˆ uk ϕk ,

and uk =

uϕk d x. B1

k=1

• Let ψ ∈ Cc∞ (B1 ), and let ϕ := (−∆)−s ψ denote the unique solution in H to (−∆)s ϕ = ψ. Prove that there exists a constant C > 0 such that |(−∆)−s ψ| ≤ Cϕ1 . We assume that the nonlinearity f is smooth, nondecreasing, positive, and ´ superlinear. A measurable function u in B1 such that B1 |u|ϕ1 d x < +∞ and ´ B1 f (u)ϕ1 d x < +∞, is a weak solution to (3.36) if

ˆ

ˆ uψ d x = λ B1

f (u)(−∆)−s ψ d x, B1

for all ψ ∈ Cc∞ (B1 ).

In addition, we say that u is stable if for all ψ ∈ Cc∞ (B1 ) we have ˆ ˆ s 2 f 0 (u)ψ2 d x. |(−∆) 2 ψ| d x ≥ B1

B1

Let s ∈ (0, 1). Prove that there exists λ∗ > 0 such that • For 0 < λ < λ∗ , there exists a minimal solution uλ ∈ H ∩ L ∞ (B1 ) of (3.36). In addition, uλ is stable and increasing with λ. • For λ = λ∗ , the function u∗ = limλ%λ∗ uλ is a weak solution to (3.36). • For λ > λ∗ , (3.36) has no solution u ∈ H ∩ L ∞ (B1 ).

The affine case When the nonlinearity is not superlinear, the extremal solution need not exist, as the following simple example shows. Proposition 3.3.3 ([152]) Let N ≥ 1, let Ω denote a smoothly bounded domain of RN , and let λ1 = λ1 (−∆; Ω). If f (t) = at + b when t ≥ 0, with a, b > 0, then the extremal parameter associated to (3.1) verifies (i) λ∗ = λ1 /a, and (ii) (3.1) has no solution for λ = λ∗ . 68

Chapter 3. Extremal solutions

Proof. If λ ∈ (0, λ1 /a) then the problem −∆u − λau = λb u=0

in Ω, on ∂ Ω,

(3.37)

has a unique solution u ∈ H01 (Ω). By elliptic regularity, u ∈ C 2 (Ω). By the maximum principle, u > 0 in Ω. We claim that (3.37) has no solution for λ∗ = λ1 /a. If u were such a solution, multiply (3.37) by ϕ1 > 0, an eigenfunction ´ associated to λ1 = λ1 (−∆, Ω), to get Ω ϕ1 d x = 0, contradicting ϕ1 > 0. For a thorough investigation of asymptotically linear nonlinearities, we refer the reader to [152].

3.3.3

Is the stable branch a (smooth) curve?

Figure 3.1: A possible piece of the solution curve. Stable solutions are represented by the solid line. Let (uλ )λ∈(0,λ∗ ) denote the branch of minimal solutions constructed in Proposition 3.3.1. The minimality property of uλ implies that the mapping λ 7→ uλ is nondecreasing. We claim that this mapping is also left-continuous. Take, for example, 0 < µ ≤ λ < λ∗ . Using elliptic regularity, uµ → v, as µ → λ− where v ∈ C 2 (Ω) is a solution to (3.1). Since uµ ≤ uλ , v ≤ uλ . Since uλ is the minimal solution to (3.1), we also have uλ ≤ v. Hence, uλ = v and the claim follows. 69

Stable solutions of elliptic PDEs But jump discontinuities can still occur. This is precisely what happens when the branch of solutions is S-shaped (see Figure 3.1). In this section, we establish that if f is convex, then in fact the stable branch is a C 1 curve. For examples where the bifurcation diagram is S-shaped (so that the stable branch has a jump discontinuity), we refer the reader to [56]. Proposition 3.3.4 ([56]) Let N ≥ 1, let Ω denote a smoothly bounded domain of RN . Let f ∈ C 2 (R), f ≥ 0, denote a nondecreasing function and let (uλ )λ∈(0,λ∗ ) denote the branch of minimal solutions of (3.1). Given λ ∈ (0, λ∗ ), the following properties are equivalent: (i) λ1 (−∆ − f 0 (uλ ); Ω) > 0. (ii) The map µ 7→ uµ is C 1 from a neighborhood of λ to L ∞ (Ω). ´ (iii) Ω |uλ − uµ |2 dΩ (x) d x = o(|λ − µ|), as µ → λ. In addition, if f is convex (or if f is concave), these properties hold for all λ ∈ (0, λ∗ ). Proof. Step 1. If f is convex (or if f is concave), then (i) holds for all λ ∈ (0, λ∗ ). Let λ, µ ∈ (0, λ∗ ), let uλ , uµ denote the corresponding minimal solutions to (3.1) and let ϕ1 > 0 denote an eigenfunction associated to λ1 := λ1 (−∆ − f 0 (uλ ); Ω). Assume by contradiction that λ1 = 0. Then, −∆ϕ1 = λ f 0 (uλ )ϕ1 ,

in Ω.

(3.38)

Multiply (3.38) by uλ , (1.3) by ϕ1 , integrate and subtract these expressions. Then, ˆ (λ f (uλ ) − λuλ f 0 (uλ ))ϕ1 d x = 0. (3.39) Ω

Similarly,

ˆ Ω

(µ f (uµ ) − λuµ f 0 (uλ ))ϕ1 d x = 0.

(3.40)

Subtracting (3.39) from (3.40), we deduce that ˆ ˆ 0 λ ( f (uµ ) − f (uλ ) − (uµ − uλ ) f (uλ ))ϕ1 d x = (λ − µ) f (uµ )ϕ1 d x. (3.41) Ω

Ω

If f is convex, the left-hand side of (3.41) is nonnegative and we get a contradiction by choosing µ > λ. If f is concave, then the left-hand side of (3.41) is nonpositive and we get a contradiction by choosing µ < λ. 70

Chapter 3. Extremal solutions Step 2. Property (ii) implies (iii). This is immediate. Step 3. Property (iii) implies (i). Assume by contradiction that λ1 := λ1 (−∆− f 0 (uλ ); Ω) = 0. Fix λ ∈ (λ, λ∗ ) and set M = kuλ k L ∞ (Ω) . Since f is C 2 , there exists C > 0 such that | f (t) − f (s) − (t − s) f 0 (s)| ≤ C|t − s|2 ,

whenever 0 ≤ s, t ≤ M .

So, we deduce from (3.41) that for λ < µ < λ, ˆ ˆ |λ − µ| f (uµ )ϕ1 d x ≤ Cλ |uλ − uµ |2 ϕ1 d x, Ω

(3.42)

Ω

where ϕ1 > 0 is an eigenfunction associated to λ1 . Since ϕ1 ≤ C dΩ , we deduce ´ from (iii) and (3.42) that |λ − µ| Ω f (uµ )ϕ1´ d x = o(|λ − µ|). Recalling that uµ (x) → uλ (x) as µ → λ− , we deduce that Ω f (uλ )ϕ1 d x = 0. So, f ≡ 0 on the range of uλ and λ1 (−∆ − f 0 (uλ ); Ω) = λ1 (−∆; Ω) > 0, a contradiction. Step 4. Property (i) implies (ii). We first show that kuµ − uλ k L ∞ (Ω) → 0,

as µ → λ.

(3.43)

Assume that f vanishes on the range of uλ . Then, uλ ≡ 0, and the same must be true of uµ , for all µ ≥ 0. So, we may assume that f does not vanish identically on the ranges of the functions uµ , µ ∈ (0, λ∗ ). By the boundary point lemma (Proposition A.8.1 and Lemma A.5.1), given µ > ν, there exists " > 0 such that uµ ≥ uν + "dΩ . Set u = lim uµ and u = lim uµ . µ↑λ

µ↓λ

It is clear that u ≤ uλ and that u is a solution to (3.1). So, u = uλ . We claim that u = uλ . Indeed, since (i) holds, there exists the unique solution ζ1 ∈ C02 (Ω) to in Ω, −∆ζ1 − λ f 0 (uλ )ζ1 = 1 ζ1 = 0 on ∂ Ω. We set v = uλ + δζ1 for δ > 0, so that −∆v − (λ + θ ) f (v) = (δ − θ f (v)) − λ f (v) − f (uλ ) − (v − uλ ) f 0 (uλ ) . Since f (v) ≤ sup[0,kuλ k∞ +δkζ1 k∞ ] f and f (v) − f (uλ ) − (v − uλ ) f 0 (uλ ) = o(δ), we deduce that for δ > 0 sufficiently small, there exists θ = θ (δ) > 0 such that −∆v − (λ + θ ) f (v) ≥ 0 in Ω. In particular, uλ+θ ≤ v and so u ≤ v. Letting 71

Stable solutions of elliptic PDEs δ ↓ 0, we obtain u ≤ uλ , thus u = uλ . So, uµ (x) → uλ (x) as µ → λ, for all x ∈ Ω. Since uµ is nondecreasing in µ and uµ ∈ C(Ω) for all µ < λ∗ , the convergence is uniform and (3.43) holds. It then follows easily from (3.43) that λ1 (−∆ − µ f 0 (uµ ); Ω) → λ1 (−∆ − λ f 0 (uλ ); Ω), as µ → λ. In particular, we deduce from (i) that there exists δ, η > 0 such that λ1 (−∆ − µ f 0 (uµ ); Ω) > η,

(3.44)

for |µ − λ| < δ. Hence, (i) holds with λ replaced by µ such that |µ − λ| < δ and so we deduce from (3.43) that the mapping µ 7→ uµ is continuous from (λ − δ, λ + δ) to L ∞ (Ω).

(3.45)

We show next that there exists a constant C > 0 such that kuµ − uν k∞ ≤ C|µ − ν|,

(3.46)

for |µ − λ|, |ν − λ| < δ. Indeed, it follows from (3.44) that ˆ ˆ 2 2 ∇(uµ − uν ) d x − µ f 0 (uµ )(uµ − uν )2 d x ηkuµ − uν k L 2 (Ω) ≤ Ω Ω ˆ = (uµ − uν )[−∆(uµ − uν ) − µ f 0 (uµ )(uµ − uν )] d x Ω ˆ = µ (uµ − uν )[ f (uµ ) − f (uν ) − f 0 (uµ )(uµ − uν )] d x+ Ω ˆ (µ − ν) f (uν )(uµ − uν ) d x. Ω

Since µ| f (uµ ) − f (uν ) − f (uµ )(uµ − uν )| ≤ "(|µ − ν|)|uµ − uν |, with "(t) → 0 as t → 0, we obtain 0

ηkuµ − uν k2L 2 (Ω) ≤ "(|µ − ν|)kuµ − uν k L 2 (Ω) + C|µ − ν|kuµ − uν k L 2 (Ω) , so that kuµ − uν k L 2 (Ω) ≤ C|µ − ν|. Since −∆(uµ − uν ) = µ( f (uµ ) − f (uν )) + (µ − ν) f (uν ) and |µ( f (uµ ) − f (uν )) + (µ − ν) f (uν )| ≤ C|uµ − uν | + C|µ − ν|, (3.46) follows from the L 2 estimate and a direct bootstrap argument. Suppose now that |µ − λ| < δ. By (3.44), there exists the unique solution wµ ∈ C02 (Ω) of ¨ −∆wµ − µ f 0 (uµ )wµ = f (uµ ) in Ω, wµ = 0 72

on ∂ Ω.

Chapter 3. Extremal solutions By (3.44), (wµ ) is bounded in H01 (Ω) and so by standard elliptic regularity (wµ ) is bounded in C 1 (Ω). Using (3.46), we deduce that wµ is continuous d uµ , from (λ − δ, λ + δ) to L ∞ (Ω). Property (ii) follows if we show that wµ = dµ that is, uσ − uµ − (σ − µ)wµ ψ := → 0 in L ∞ (Ω), as σ → µ. σ−µ We have −∆ψ−µ f (uµ )ψ = (uσ −uµ ) f (uµ )+σ 0

0

uσ − uµ f (uσ ) − f (uµ ) − (uσ − uµ ) f 0 (uµ ) σ−µ

uσ − uµ

and it follows from (3.46) that the right-hand side in the above equality converges to 0 in L ∞ (Ω) as σ → µ. Using (3.44), we conclude that kψk∞ → 0 as σ → µ. Exercise 3.3.4 Assume that f ∈ C 2 (R), f ≥ 0, is a convex nondecreasing function. Prove that the map λ ∈ (0, λ∗ ) → uλ ∈ L ∞ (Ω) is convex.

3.3.4

Is the extremal solution bounded?

As follows from Chapter 2, when Ω is the unit ball and f (u) = exp(u), the extremal solution is bounded (hence smooth) if and only if N ≤ 9. Deciding whether u∗ is bounded or not in the general setting of (3.1) is a delicate question, which we shall address in the two following chapters.

73

,

Chapter 4 Regularity theory of stable solutions Recall that there exists singular solutions to PDEs of the form (1.3). Take, for example, us (x) = −2 ln |x| , (4.1) solving (1.3) in the unit ball B ⊂ RN , N ≥ 3, for the nonlinearity f (u) = λeu , with λ = λs := 2(N − 2). Also recall that, when N ≥ 10, us is the monotone limit of a curve of (regular) stable solutions uλ , λ < λs . So, us is stable. In addition, even if our interest rested solely in studying the nicest possible solutions—say the smooth, radial, and stable ones—no a priori bound holds true in full generality, as examplified by the aforementioned curve uλ . This chapter is devoted to the regularity theory of stable solutions. In other words, we shall try to answer the following questions: when is any stable solution to (1.3) regular? When can we obtain a priori estimates on families of (regular) stable solutions?

4.1

The radial case

Example 3.2.1 shows that no regularity result can hold in full generality for stable solutions that do not belong to the energy space H01 (Ω). This being said, we have the following sharp result, when Ω is the unit ball. Theorem 4.1.1 ([38]) Let N ≥ 2 and let Ω = B denote the unit ball of RN . Assume that f is a locally Lipschitz function and let u denote a stable radial weak solution to (1.3), such that u ∈ H01 (B). 75

Stable solutions of elliptic PDEs Then, u is either constant, radially increasing, or radially decreasing in B \ { 0} and there exists a constant C = C(N ) > 0 such that kuk L ∞ (B) ≤ C kuk L 1 (B) + k f (u)dB k L 1 (B) , if 1 ≤ N ≤ 9, (4.2) Remark 4.1.1 Recall that for N ≥ 3, the function us (x) = −2 ln |x| solves −∆u = λs eu in B, u=0 on ∂ B, with λs = 2(N − 2). By Hardy’s inequality, us is stable if and only if N ≥ 10. In particular, (4.2) fails for N ≥ 10. Still, sharp estimates of the (possibly) singular behavior of solutions are available (see [38] and [217]). Proof of Theorem 4.1.1. We establish two auxiliary lemmata. Lemma 4.1.1 ([38]) Let N ≥ 1 and let u ∈ H 1 (B) denote a radial solution to (1.4) in Ω = B \ {0}. Then, for every ϕ ∈ H 1 (B) ∩ L ∞ (B) with compact support in B \ {0}, there holds ˆ 2 2 2 Q u (ru r ϕ) = u r ∇(rϕ) − (N − 1)ϕ d x, (4.3) B

where, as usual,

ˆ ˆ 2 f 0 (u)ϕ 2 d x. ∇ϕ d x − Q u (ϕ) = B

B

Proof of Lemma 4.1.1. Take ϕ ∈ H 1 (B) ∩ L ∞ (B) with compact support in 2 B \ {0}. Also let c be any function in H loc (B \ {0}) ∩ L ∞ (B \ {0}). Note that φ = rϕc ∈ H 1 (B) ∩ L ∞ (B) has compact support in B \ {0}. Use φ as a test function in the definition of Q u : ˆ 2 Q u (rϕc) = r 2 ϕ 2 |∇c|2 + c 2 ∇(rϕ) + c∇c · ∇(r 2 ϕ 2 ) − f 0 (u)r 2 ϕ 2 c 2 d x ˆB 2 = r 2 ϕ 2 |∇c|2 + c 2 ∇(rϕ) − r 2 ϕ 2 ∇ · (c∇c) − f 0 (u)r 2 ϕ 2 c 2 d x ˆB 2 2 2 2 0 2 = c ∇(rϕ) − r ϕ (c∆c + f (u)c ) d x. B

Differentiate (1.3) with respect to r to get −∆u r +

N −1 r2

u r = f 0 (u)u r 76

in B \ {0}.

(4.4)

Chapter 4. Regularity theory of stable solutions Using local elliptic regularity and the fact that u is radial, we have that u r ∈ H l2oc (B \ {0}) ∩ L ∞ (B \ {0}). So, we can take c := u r in the previous computations. Using (4.4), we deduce (4.3). Lemma 4.1.2 ([38]) Let N ≥ 2 and let B denote the unit ball in RN . Assume that f is a locally Lipschitz function and let u denote a radial weak solution to (1.3). Let α satisfy p 1 ≤ α < 1 + N − 1. (4.5) Then, ˆ B1/2

u2r r −2α

CN

2 2 dx ≤ kuk L 1 (B) + k f (u)dB k L 1 (B) , (4.6) (N − 1) − (α − 1)2

where CN is a constant depending on N only and where dB is the distance to the boundary of B, defined by (3.4). Proof of Lemma 4.1.2. By appromixation, since u is stable, Q u (φ) ≥ 0 for every φ ∈ H 1 (B) with compact support in B \ {0}. By Lemma 4.1.1, ˆ ˆ 2 2 2 (4.7) (N − 1) u r ϕ d x ≤ u2r ∇(rϕ) d x, B

B

for every ϕ ∈ H 1 (B)∩ L ∞ (B) with compact support in B\{0}. In fact, (4.7) also holds for ϕ ∈ H 1 (B) ∩ L ∞ (B) with compact support in B such that ∇(rϕ) ∈ L ∞ (B). To see this, take ζ ∈ C 1 (RN ) such that 0 ≤ ζ ≤ 1, ζ ≡ 0 in B and ζ ≡ 1 in RN \ B2 . Let ζδ (x) = ζ(x/δ) for δ > 0, x ∈ RN . Applying (4.7) with test function ζδ ϕ, we obtain ˆ ˆ 2 2 2 2 (N − 1) u r ϕ ζδ d x ≤ u2r ∇(rϕζδ ) d x. B

B

Now, ˆ

2 u2r ∇(rϕζδ ) d x = B ˆ 2 2 = u2r ∇(rϕ) ζ2δ + r 2 ϕ 2 ∇ζδ + ζδ ∇ζδ · ∇(r 2 ϕ 2 ) d x ≤ B ˆ ˆ 2 r2 r 2 2 2 ϕ + ζ ∇(rϕ) d x ≤ ≤ u r ∇(rϕ) ζδ d x + C u r ϕ δ δ2 δ B B2δ \Bδ ˆ ˆ 2 2 2 ≤ u r ∇(rϕ) ζδ d x + C u2r d x. B

77

B2δ \Bδ

Stable solutions of elliptic PDEs In the last inequality, we used that ϕ and ∇(rϕ) are bounded. Since u ∈ H01 (B), the last term tends to zero as δ → 0. By monotone convergence, we deduce that (4.7) holds for every ϕ ∈ H 1 (B) ∩ L ∞ (B) with compact support in B and such that ∇(rϕ) ∈ L ∞ (B). Let " ∈ (0, 1/2). For α ≥ 1 in the range (4.5), apply (4.7) with ϕ = ϕ" given by  −α −α if 0 ≤ r ≤ ",  " − (1/2) −α −α ϕ" (r) = r − (1/2) if " < r ≤ 1/2,  0 if 1/2 < r. Note that ϕ" and ∇(rϕ" ) are bounded. We obtain ˆ ˆ 2 −α −α 2 −α −α 2 u r (r − (1/2) ) d x + (N − 1)(" − (1/2) ) (N − 1) u2r d x ≤ B B \B ˆ 1/2 " ˆ " ≤ u2r ((1 − α)r −α − (1/2)−α )2 d x + (" −α − (1/2)−α )2 u2r d x. B1/2 \B"

B"

Since N ≥ 2, it follows that ˆ ˆ −α 2 2 −α u r (r − (1/2) ) d x ≤ (N − 1) B1/2 \B"

B1/2 \B"

u2r ((1 − α)r −α − (1/2)−α )2 d x.

Expanding squares, using N ≥ 2 and (4.5), we find the estimate ˆ ˆ CN 2 −2α ur r dx ≤ u2 r −α d x. (N − 1) − (α − 1)2 B1/2 \B" r B1/2 \B" Now, choose a positive constant Cα,N such that CN (N − 1) − (α − 1)

2

It follows that

r −α ≤

2

r −2α + Cα,N r N −1 ,

ˆ

for all r ∈ (0, 1).

ˆ B1/2 \B"

u2r r −2α

Next, we claim that ˆ ˆ 2 N −1 ur r d x = |∂ B| B1/2

1

0

1/2

d x ≤ Cα,N

B1/2 \B"

u2r r N −1 d x.

(4.8)

u2r r 2N −2 d r ≤ CN kuk2L 1 (B) + k f (u)dB k2L 1 (B) . (4.9) 78

Chapter 4. Regularity theory of stable solutions Assume this claim for the moment. Using (4.8) and (4.9) and letting " → 0, we obtain ˆ u2r r −2α d x ≤ Cα,N kuk2L 1 (B) + k f (u)dB k2L 1 (B) . B1/2

p In particular, taking α = (1 + N − 1)/2, we deduce ˆ p u2r r −(1+ N −1) d x ≤ CN kuk2L 1 (B) + k f (u)dB k2L 1 (B) .

(4.10)

B1/2

p

Finally, since r −α ≤ r −(1+ N −1) in B, the desired estimate follows after letting " → 0. It remains to establish (4.9). First, since u is radially decreasing, ˆ 1/2 ur N −1 d r ≤ CN kuk L 1 (B) . (4.11) u(1/2) ≤ CN 1/4

Let ρ ∈ (1/2, 3/4) be chosen such that −u r (ρ) = −

u(3/4) − u(1/2) 1/4

= 4u(1/2) − 4u(3/4) ≤ 4u(1/2).

For s ≤ 1/2, integrate (r N −1 u r ) r = − f (u)r N −1 with respect to r, from s to ρ: ˆ ρ N −1 N −1 − u r (s)s = −u r (ρ)ρ − f (u)r N −1 d r ≤ s ≤ CN u(1/2) + k f (u)dB k L 1 (B) . Collecting the aformentioned estimates, it follows that 0 ≤ −u r (s)s N −1 ≤ CN kuk2L 1 (B) + k f (u)dB k2L 1 (B) , for all s ≤ 1/2. Squaring this inequality and integrating it in s, from 0 to 1/2, we conclude that (4.9) holds. Proof of Theorem 4.1.1 Completed. Take α in the range (4.5). For s ∈ (0, 1/2], we have ˆ 1/2 ˆ 1/2 N −1 N −1 u(s) − u(1/2) = −u r d r = −u r r −α+ 2 r α− 2 d r ≤ s

s

ˆ ≤ CN B1

1/2 ˆ

u2r r −2α

dx

79

1/2

1/2

r s

2α+1−N

dr

. (4.12)

Stable solutions of elliptic PDEs Using Lemma 4.1.1, we deduce that for all s ∈ (0, 1/2], u(s) ≤ u(1/2)+ +p

CN (N − 1) − (α − 1)2

ˆ

1/2

1/2

r 2α+1−N d r s

kuk L 1 (B) + k f (u)dB k L 1 (B) . (4.13)

Assume N ≤ 9. The integral in the right-hand side of the above inequality is finite with s = 0 if we take 2α + 1 − N > −1. Such a choice of α in the range (4.5) is possible since N ≤ 9. Exercise 4.1.1 Under the assumptions of Theorem 4.1.1, prove that if N = 10, then any stable radial weak solution u of (1.3) such that u ∈ H01 (B) satisfies |u(r)| ≤ C kuk L 1 (B) + k f (u)dB k L 1 (B) | ln r|. Is the estimate sharp?

4.2

Back to the Gelfand problem

We will now turn to nonradial solutions. To this end, we begin by returning to the study of the extremal solution for the Gelfand problem, posed this time in an arbitrary bounded domain:

−∆u = λeu u=0

in Ω, on ∂ Ω.

(4.14)

Theorem 4.2.1 ([154], [63]) Let 1 ≤ N ≤ 9 and let Ω ⊂ RN denote a bounded domain with C 2,α boundary. Then, the extremal solution to (4.14) is classical. Furthermore, there exists a constant C = C(N , Ω) > 0 such that for all 0 ≤ λ ≤ λ∗ , kuλ k L ∞ (Ω) ≤ C, (4.15) where uλ denotes the unique classical stable solution to (4.14). Remark 4.2.1 It follows from Chapter 2 that the restriction N ≤ 9 is sharp since the extremal solution is given by u∗ (x) = −2 ln |x|, when Ω = B is the unit ball and N ≥ 10. 80

Chapter 4. Regularity theory of stable solutions Proof. By elliptic regularity, it suffices to establish (4.15). We may also restrict to the case 0 < λ < λ∗ since u∗ is obtained as the monotone limit of uλ as λ % λ∗ . Fix such λ and let u = uλ . For α ∈ (0, 2), introduce the test functions ψ = e2αu − 1 and ϕ = eαu − 1. We are going to multiply (4.14) by ψ on the one hand, and to use the stability inequality (1.5) with test function ϕ on the other hand. The former calculation reads as follows. ˆ ˆ ∇u∇ψ d x = λ eu ψ d x = ˆ Ω ˆΩ 2 2αu 2α |∇u| e d x = λ eu e2αu − 1 d x = Ω ˆ ˆΩ 2 |∇ (eαu − 1)|2 d x = λ e(2α+1)u − eu d x. α Ω Ω Using ϕ = eαu − 1 in (1.5), we obtain ˆ ˆ 2 αu |∇ (e − 1)| d x ≥ λ eu (eαu − 1)2 d x Ω Ω ˆ =λ e(2α+1)u + eu − 2e(α+1)u d x. Ω

Combining the two equations, it follows that ˆ ˆ 2 (2α+1)u u e − e dx ≥ e(2α+1)u + eu − 2e(α+1)u d x, α Ω Ω which simplifies to ˆ ˆ 1 2−α (α+1)u u 4e − (α + 2)e d x ≥ e(2α+1)u d x. α Ω α Ω In particular,

ˆ e

(α+1)u

Ω

dx ≥

2−α

ˆ

4

e(2α+1)u d x. Ω

Applying Hölder’s inequality, we obtain |Ω|

α 2α+1

ˆ e

(2α+1)u

α+1

2α+1

≥

dx

Ω

81

2−α 4

ˆ e(2α+1)u d x. Ω

Stable solutions of elliptic PDEs This implies that eu is bounded in L p (Ω) for any p = 2α + 1 with α ∈ (0, 2). Since N ≤ 9, there exists α ∈ (0, 2) such that p = 2α + 1 > N /2. Since u solves (4.14), (4.15) follows by elliptic regularity. Exercise 4.2.1 Given p > 1, consider the problem −∆u = λ(1 + u) p in Ω, u=0 on ∂ Ω. Prove that the extremal solution is bounded if p < pc (N ), where  +∞ if N ≤ 10,  p 2 pc (N ) = (N − 2) − 4N + 8 N − 1  if N ≥ 11. (N − 2)(N − 10)

4.3

(4.16)

Dimensions N = 1, 2, 3

We return to the question of regularity for (1.3), with general nonlinearity f : if u is a stable weak solution to (1.3), is u in fact a classical solution? In dimensions 1 ≤ N ≤ 3, a first partial answer is given by the following theorem. Theorem 4.3.1 ([166]) Let 1 ≤ N ≤ 3 and let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 1 (R) is a nondecreasing convex function such that f (0) > 0. In addition, assume that f is superlinear, that is, lim

t→+∞

f (t) t

= +∞.

(4.17)

Then, any stable weak solution u to (1.3) such that u ∈ H01 (Ω), is in fact classical and there exists a constant C = C(Ω, N , f ) such that kuk L ∞ (Ω) ≤ C.

(4.18)

Proof. By Corollary 3.2.1, we need only prove (4.18) for classical solutions of (1.3). Following Theorem 4.2.1 in spirit, we introduce ´ u 0 2 two related test ˜ functions ϕ = f (u) = f (u) − f (0) and ψ = g(u) = 0 f (t) d t. We multiply (1.3) by ψ on the one hand, and use the stability inequality (1.5) with test function ϕ on the other hand. The former calculation reads ˆ ˆ 2 0 2 |∇u| f (u) d x = f (u)g(u) d x, Ω

Ω

82

Chapter 4. Regularity theory of stable solutions while testing the stability inequality (1.5) with ϕ = f˜(u) leads to ˆ ˆ 0 2 ˜ f (u) f (u) d x ≤ |∇u|2 f 0 (u)2 d x. Ω

So,

Ω

ˆ

ˆ f (u) f˜(u)2 d x ≤

ˆ

0

Ω

Ω

f˜(u)g(u) d x + f (0)

g(u) d x.

(4.19)

Ω

Now we estimate the difference f˜(t)2 f 0 (t) − f˜(t)g(t) for t > 0. By definition of g, we have

ˆ

ˆ t ˜ f (s) ds − f (t) f 0 (s)2 ds 0 ˆ t = f˜(t) f 0 (s) f 0 (t) − f 0 (s) ds.

t

f˜(t)2 f 0 (t) − f˜(t)g(t) = f˜(t) f 0 (t)

0

0

0

So, denoting

ˆ

t

h(t) =

f 0 (s) f 0 (t) − f 0 (s) ds, 0

we obtain from (4.19) ˆ Ω

ˆ f˜(u)h(u) d x ≤ f (0)

We claim that lim

t→+∞

h(t) f 0 (t)

g(u) d x.

(4.20)

Ω

= +∞.

(4.21)

Indeed, let C > 0 and choose s0 > 1 such that f 0 (s0 ) > 2C. By assumption, f is superlinear, that is, (4.17) holds. So, for t sufficiently large, there holds f 0 (t) > 2 f 0 (s0 + 1). And, for such t,

ˆ h(t) =

ˆ

t

f (s) f (t) − f (s) ds ≥ 0

0

0

0

ˆ

s0 +1

f 0 (s) f 0 (t) − f 0 (s) ds

s0 s0 +1

f 0 (s) f 0 (t) − f 0 (s0 + 1) ds ≥ C f 0 (t).

≥ s0

Claim (4.21) is proved. In addition, ˆ ˆ t 0 2 g(t) = f (s) ds ≤ 0

t

f 0 (s) f 0 (t) ds = f 0 (t) f˜(t).

0

83

Stable solutions of elliptic PDEs By the above equation and (4.21), we deduce that lim t→+∞ f˜(t)h(t)/g(t) = +∞, which combined with (4.20) leads to ˆ f˜(u)h(u) d x ≤ C. Ω

Using (4.21) again, we deduce that ˆ f˜(u) f 0 (u) d x ≤ C.

(4.22)

Ω

By Kato’s inequality (Lemma 3.2.1), −∆ f˜(u) ≤ f 0 (u) f (u),

in Ω.

Using the maximum principle, it follows that 0 ≤ f˜(u) ≤ v, where v solves −∆v = f 0 (u) f (u) in Ω, v=0 on ∂ Ω. By (4.22) and elliptic regularity (see Exercise A.9.1), it follows that v is uniformly bounded in L p (Ω), for all 1 ≤ p < NN−2 . Hence, f (u) is uniformly bounded in L p (Ω). Applying elliptic regularity (Theorem B.3.1) again in (1.3), we deduce that u is uniformly bounded, provided N ≤ 3. Exercise 4.3.1 Under the assumptions of Theorem 4.3.1, prove that for any N ≥ 4, u is bounded in L p (Ω) for all p < NN−4 (p < +∞ if N = 4). Theorem 4.3.1 applies to solutions belonging to H01 (Ω). In fact, using the same proof, the theorem remains valid for extremal solutions. It is interesting to note that in any dimension, extremal solutions always belong to H01 (Ω), at least if the domain is convex. Theorem 4.3.2 ([167]) Let 1 ≤ N and let Ω ⊂ RN denote a smoothly bounded convex domain. Assume that f ∈ C 1 (R) is a nondecreasing function such that f (0) > 0. In addition, assume that f is superlinear, that is, (4.17) holds. Then, the extremal solution u∗ to (3.1) belongs to H01 (Ω). Proof. Take λ < λ∗ and uλ the associated minimal solution. Then, by Lemma 1.1.1, uλ minimizes the energy ˆ ˆ 1 2 |∇u| d x − λ F (u) d x, EΩ (u) = 2 Ω Ω 84

Chapter 4. Regularity theory of stable solutions among all functions lying between 0 and uλ . In particular, EΩ (u) ≤ EΩ (0) = 0.

(4.23)

Pohozaev’s identity (see (8.9)) asserts ˆ ˆ ˆ 1 2N 2 λ F (uλ ) d x − |∇uλ | d x = |∇uλ |2 x · n dσ. N −2 Ω N − 2 ∂Ω Ω

(4.24)

We may suppose that N > 2 (since for N = 2, 3, we already know that u∗ is regular by Theorem 4.3.1). Combining (4.23) and (4.24), we obtain ˆ ˆ 1 2 |∇uλ | d x ≤ |∇uλ |2 x · n dσ. 2 ∂Ω Ω Finally, using the boundary estimates of Theorem 4.5.3 below, we deduce that (uλ ) is bounded in H01 (Ω).

4.4

A geometric Poincaré formula

In the light of Theorem 4.3.1, two natural questions arise: are stable solutions classical for N = 4, . . . , 9 and functions f that are nondecreasing, convex, and superlinear? Are the assumptions on f needed for the result to hold? In order to go any further, we pause to establish the following geometric restatement of stability. Theorem 4.4.1 ([207, 208]) Let N ≥ 1 and let Ω ⊂ RN denote an open set. Let u ∈ C 2 (Ω) denote a stable solution to (1.4). Then, for any ϕ ∈ Cc1 (Ω), there holds ˆ ˆ ∇ |∇u| 2 + |B|2 |∇u|2 ϕ 2 d x ≤ |∇u|2 ∇ϕ 2 d x, (4.25) T [∇u6=0]

Ω

where ∇ T denotes the tangential gradient along a given level set of u and where |B|2 denotes the sum of the squares of the principal curvatures of such a level set. Remark 4.4.1 By the implicit function theorem, each level set M = [u = t] of u is an N − 1 dimensional submanifold on [∇u 6= 0]. In particular, the tangential gradient (defined as the orthogonal projection of the gradient on the tangent space to M ), as well as the principal curvatures of M are well defined. 85

Stable solutions of elliptic PDEs Remark 4.4.2 In the case where Ω is a ball and u is a radial monotone function, any given level set of u is a hypersphere. It follows that its principal curvatures are equal and constant, so |B|2 = (N − 1)/r 2 with r = |x|, x ∈ B. The gradient of u is also radial so that ∇ T |∇u| = 0. Gathering these facts, we see that (4.25) reduces to ˆ 2 N − 1 2 2 0 ≤ u r ∇ϕ − ϕ d x. r2 B In particular, observe that (4.3) is simply a restatement of the geometric Poincaré formula in the radial setting. Proof of Theorem 4.4.1. Differentiate (1.3) with respect to x i , i = 1, . . . , N . Then, ui = the linearized equation −∆ui = f 0 (u)ui ,

∂u ∂ xi

solves

in Ω.

Take a test function ϕ ∈ Cc1 (Ω), multiply the previous equation by ui ϕ 2 , integrate, and sum over i. Then,

ˆ f (u)|∇u| ϕ d x = 0

Ω

2

2

N ˆ X i=1

=

Ω

N ˆ X Ω

ˆ

i=1

=

(−∆ui )ui ϕ 2 d x |∇ui | ϕ d x +

|D u| d x + 2

Ω

ˆ 2

2

2

2

1 2

ˆ Ω

Ω

ui ∇ui · ∇ϕ d x

∇|∇u|2 · ∇ϕ 2 d x,

(4.26)

PN where |D2 u|2 = i, j=1 u2i j . Let ψ = |∇u|ϕ. Then, ψ ∈ H 1 (Ω), ψ has compact support, and ∇ψ = ϕ∇|∇u| + |∇u|∇ϕ. Hence, |∇ψ|2 = |∇|∇u||2 ϕ 2 + |∇u|2 |∇ϕ|2 + 2ϕ|∇u|∇ϕ · ∇|∇u| 1 = |∇|∇u||2 ϕ 2 + |∇u|2 |∇ϕ|2 + ∇ϕ 2 · ∇|∇u|2 . 2 Apply the stability inequality (1.5) with test function ψ. Then, ˆ ˆ 1 2 2 0 2 2 2 2 2 2 f (u)|∇u| ϕ d x ≤ |∇|∇u|| ϕ + |∇u| |∇ϕ| + ∇ϕ · ∇|∇u| d x. 2 Ω Ω 86

Chapter 4. Regularity theory of stable solutions Using (4.26), we obtain ˆ ˆ 2 2 2 2 |D u| − |∇|∇u|| ϕ d x ≤ |∇u|2 |∇ϕ|2 d x. Ω

Ω

Equation (4.25) then follows from the following geometric identity. 2 |D2 u|2 − |∇|∇u||2 = ∇ T |∇u| + |B|2 |∇u|2 on {x ∈ Ω : ∇u(x) 6= 0}. (4.27) We prove at last Identity (4.27). Take a point x ∈ Ω such that ∇u(x) 6= 0 and let t = u(x). Without loss of generality, we may assume that x = 0, that ∇u(0) = |∇u(0)|eN , and that the level set L t := { y : y ∈ Ω, ∇u( y) 6= 0, u( y) = t} takes the form L t = { y : y = ( y 0 , yN ) ∈ Ω, ∇u( y) 6= 0, yN = Φ( y 0 )}, for some C 2 function Φ, such that D2 Φ(0) is a diagonal matrix which eigenvalues λ1 , . . . , λN −1 are the principal curvatures of the level set L t . Let us compute |∇|∇u||2 at x = 0. N N N X X ∇u · ∇ui 2 X 2 = u2iN . |∇|∇u|| = (∂i |∇u|) = |∇u| i=1 i=1 i=1 2

(4.28)

Similarly, at x = 0, N −1 N −1 X X 2 ∇ |∇u| 2 = (∂ |∇u|) = u2iN . T i i=1

(4.29)

i=1

By definition of Φ, u(x 0 , Φ(x 0 )) = t,

for all x = (x 0 , Φ(x 0 )) ∈ L t .

Differentiating with respect to x i , i = 1, . . . , N − 1, it follows that ∂i u(x 0 , Φ(x 0 )) + ∂N u(x 0 , Φ(x 0 ))∂i Φ(x 0 ) = 0. Differentiating again with respect to x j , j = 1, . . . , N − 1, we get ∂i j u + ∂iN u∂ j φ + ∂ jN u∂i Φ + ∂N N u∂i Φ∂ j Φ + ∂N u∂i j Φ = 0 at x = (x 0 , Φ(x 0 )). Evaluated at x = 0, the previous expression simplifies: ∂i j u + |∇u|∂i j Φ = 0. 87

Stable solutions of elliptic PDEs So, at x = 0, 

−|∇u|λ1

  D2 u =  

..

u1N .. .

.

−|∇u|λN −1 uN −1,N uN −1,N uN N

···

u1N

   . 

Hence, at x = 0, |D u| = |B| |∇u| + 2 2

2

2

2

N −1 X

u2iN + u2N N .

(4.30)

i=1

Using (4.28) and (4.29), Identity (4.27) follows.

4.5

Dimension N = 4

In this section, we exploit the geometric Poincaré formula to obtain a priori bounds on positive classical stable solutions of (1.3) posed in a convex domain Ω ⊂ RN , N ≤ 4. Theorem 4.5.1 ([34]) Let f ∈ C ∞ (R) and Ω ⊂ RN denote a smoothly bounded and convex domain. Assume that f ≥ 0. Then, there exists a constant cΩ , depending on Ω only, such that given any stable solution u ∈ C 2 (Ω) to (1.3), we have kuk L ∞ (Ω) ≤ C(Ω, kuk L 1 (Ω) , k f k L ∞ (cΩ kuk L1 (Ω) ) ), (4.31) where C(·) depends only on the quantities within the parentheses. As a consequence, we obtain the following regularity result for extremal solutions. Corollary 4.5.1 ([34]) Let 1 ≤ N ≤ 4 and let Ω ⊂ RN be a smoothly bounded and convex domain of RN . Assume that f ∈ C 1 (R) is nondecreasing, f (0) > 0, and f is superlinear in the sense of (4.17). Then, the extremal solution to (3.1) is classical.

4.5.1

Interior estimates

Theorem 4.5.2 ([34]) Let f ∈ C ∞ (R) and let Ω ⊂ RN a smoothly bounded domain. Assume 2 ≤ N ≤ 4. Let u denote a classical stable solution to (1.3). Assume u>0 in Ω. 88

Chapter 4. Regularity theory of stable solutions Then, for every t > 0, kuk L ∞ (Ω) ≤ t +

C t

|Ω|

ˆ

4−N 2N

2

[u
|∇u|4 d x

,

(4.32)

where C is a universal constant, independent of f , Ω, or u. Proof. Let T = kuk L ∞ (Ω) and note that by Sard’s theorem (Theorem C.3.1), almost every s ∈ (0, T ) is a regular value of u, that is, ∇u(x) 6= 0 for x ∈ [u = s]. Now apply the geometric Poincaré formula (Theorem 4.4.1) with ϕ = ϕ(u(x)), where ϕ is Lipschitz in [0, T ] and ϕ(0) = 0. Using the coarea formula (C.23), the right-hand side of (4.25) becomes ˆ ˆ 2 2 |∇u| |∇ϕ| d x = |∇u|4 ϕ 0 (u)2 d x Ω Ω ˆ T ˆ =

[u=s]

0

|∇u|3 dσ ϕ 0 (s)2 ds.

Given δ > 0, the left-hand side of (4.25) is bounded below by

ˆ

∇ |∇u| 2 + |B|2 |∇u|2 ϕ(u)2 d x = T [|∇u|>δ] ˆ T ˆ 2 1 ∇ |∇u| + |B|2 |∇u|2 dσ ϕ(s)2 ds T |∇u| 0 [u=s]∩[|∇u|>δ] ˆ Tˆ 2 2 dσ ϕ(s)2 ds. = 4 ∇ T |∇u|1/2 + |B| |∇u|1/2 0

[u=s]∩[|∇u|>δ]

Letting δ & 0 and using the monotone convergence theorem, we deduce that

ˆ

ˆ

T

T

h1 (s)ϕ(s) ds ≤

h2 (s)ϕ 0 (s)2 ds,

2

0

ˆ

where h1 (s) =

(4.33)

0

[u=s]

1/2 2 1/2 2 4 ∇ T |∇u| + |B| |∇u| dσ

(4.34)

ˆ

and h2 (s) =

[u=s]

89

|∇u|3 dσ,

(4.35)

Stable solutions of elliptic PDEs are defined at every regular value s of u. The rest of the proof differs in every dimension N = 4, 3, 2. Case N = 4. Given a regular value s of u, apply the Sobolev inequality (C.13) with M = [u = s], p = 2, and v = |∇u|1/2 . Noting that the mean curvature H of [u = s] satisfies |H| ≤ |B|, we obtain ˆ [u=s]

|∇u|

N −1 N −3

N −3 N −1

≤ C(N )h1 (s).

dσ

(4.36)

For N = 4, (4.36) reduces to 1/3

h2 ≤ C h1

a.e. in (0, T ).

(4.37)

For every regular value s of u, we have h2 (s) > 0 and h1 (s) < ∞. This together with (4.37) gives h1 /h2 ∈ (0, +∞) a.e. in (0, T ). So, for any regular value s and any integer k ≥ 1, h1 (s) g k (s) = min k, h2 (s) is well defined and g k ∈ L ∞ (0, T ). In addition, g k (s) %

h1 (s)

for a.e. s ∈ (0, T ), as k → +∞.

h2 (s)

Since g k is bounded, the function 

if s ≤ t

s/t ˆ sp 1 ϕk (t) =  exp p g k (τ)dτ 2 t 

(4.38)

if t < s ≤ T

(4.39)

is well defined, Lipschitz in [0, T ], and satisfies ϕk (0) = 0. Since 2 1 1 h2 ϕk0 = h2 g k ϕk2 ≤ h1 ϕk2 2 2

in (t, T ),

(4.33) used with ϕ = ϕk leads to ˆ T ˆ ˆ 2 t 2 2 h1 ϕk ds ≤ 2 h2 ds = 2 |∇u|4 d x. t t t 0 [u
1 t2

ˆ [u
|∇u|4 d x = 90

1 t2

h2 (s) ds.

(4.40)

(4.41)

Chapter 4. Regularity theory of stable solutions 1/2

We need to establish that T − t ≤ C B t . By (4.38), we have ˆ T ˆ T h2 g k 1/4 T−t= ds. ds = sup h1 k≥1 t t Using (4.40) and the Cauchy-Schwarz inequality, we have that 1/4 ! ˆ T ˆ Tp 1 h2 g k h2 g k 1/4 h1 ϕ k ds ds = h1 ϕk h31 t t !1/2 ˆ TÈ h g 1 2 k ≤ (2B t )1/2 ds h31 ϕk2 t 1/2 ˆ T p 1 1/2 . g k 2 ds ≤ (2B t ) C ϕk t

(4.42)

(4.43)

(4.44)

In the last inequality, we used the crucial estimate (4.37). Finally, using the definition (4.39) of ϕk , we bound the integral in (4.44) as follows. ˆ T ˆ T ϕk0 p 1 p 1 g k 2 ds = gk 2 1 p ds ϕk ϕk p g k ϕk t t 2 p ˆ T 0 p ϕk 2 −2 ds = [ϕk (s)]s=t = 2 s=T 3 2 t ϕk p p 2 −2 2 ≤ ϕk (t) = . 2 2 This bound, together with (4.42), (4.43), and (4.44), yields the desired proof for N = 4. Case N = 2, 3. In this case, we use the simpler test function s/t if s ≤ t ϕ(t) = 1 if t < s ≤ T. ´ By (4.34), h1 (s) ≥ [u=s] |B|2 |∇u| dσ, hence, using the aforementioned test function and (4.33), we obtain ˆ Tˆ ˆ T 2 |B| |∇u| dσ ds ≤ h1 (s)ϕ 2 (s) ds t [u=s] 0 ˆ ˆ t 1 1 |∇u|4 d x = B t . (4.45) ≤ h2 (s) 2 ds = 2 t t [u
Stable solutions of elliptic PDEs Next, we use the following geometric inequality: |[u = s]|

N −2 N −1

ˆ ≤ C(N )

[u=s]

|H| dσ,

(4.46)

where H is the mean curvature of [u = s], C(N ) is a constant depending only on N , and s is a regular value of u. In dimension N = 2, this simply follows from the Gauss-Bonnet formula. For N ≥ 3, (4.46) follows from the Sobolev inequality (C.13). Indeed, taking v = 1 and n = N − 1 > 1 = p in (C.13), we deduce (4.46). In addition, the classical isoperimetric inequality (C.1) gives N

|[u > s]| ≤ C(N )|[u = s]| N −1 .

(4.47)

Now, using (4.46) and (4.47), there exists a constant C = C(N ) such that |[u > s]|

N −2 N

ˆ ≤C

1 ˆ

ˆ

[u=s]

|H| dσ ≤ C

[u=s]

|B| |∇u| dσ

1

dσ

2

2

[u=s]

2

|∇u|

.

In the above, we assumed that s is a regular value and we used the CauchySchwarz inequality as well as the inequality |H| ≤ |B|. From this, we deduce that

ˆ

T

T−t=

ds ≤ t

ˆ

T

ˆ

1/2

C ˆ

t T

≤C t

ˆ

[u=s]

[u=s]

|[u > s]|

|B|2 |∇u| dσ

1/2 ˆ

T

|[u > s]| t

2(2−N ) N

dσ

1/2 ds

|∇u| ˆ

[u=s]

|[u > s]| t

ˆ ≤

ˆ

T

|B|2 |∇u| dσ ds C B 1/2 t

2(2−N ) N

2(2−N ) N

ˆ [u=s]

[u=s]

dσ |∇u|

dσ |∇u| 1/2

ds

1/2 ds , (4.48)

where we used (4.45) in the last inequality. By the coarea formula, the mapping s → |[u > s]| is differentiable almost everywhere and −

d ds

ˆ |[u > s]| =

[u=s]

dσ |∇u| 92

for a.e. s ∈ (0, T ).

Chapter 4. Regularity theory of stable solutions In addition, for N ≤ 3, |[u > s]| variation satisfies |Ω|

4−N N

4−N N

is nonincreasing in s. Thus, its total

h is=t 4−N 4−N ≥ |[u > t]| N = |[u > s]| N s=T ˆ T 4−N d 2(2−N ) ≥ |[u > s]| N − |[u > s]| ds N ds t ˆ T ˆ 4−N dσ 2(2−N ) = |[u > s]| N ds. N t [u=s] |∇u|

From this and (4.48), we conclude that for N ≤ 3, T − t ≤ C(N )B 1/2 |Ω| t

4−N 2N

,

which is the desired inequality. Using the interior estimates of Theorem 4.5.2, we easily obtain the following proposition. Proposition 4.5.1 Let f ∈ C ∞ (R). Let Ω be a smoothly bounded domain of RN , with 2 ≤ N ≤ 4. Let u be a classical stable solution to (1.3). In addition, assume that u(x) ≥ c1 dist(x, ∂ Ω) for all x ∈ Ω (4.49) and kuk L ∞ (Ωρ ) ≤ c2 ,

(4.50)

where Ωρ = {x ∈ Ω : dist(x, ∂ Ω) < ρ}. Then, kuk L ∞ (Ω) ≤ C(Ω, ρ, c1 , c2 , k f k L ∞ (0,c2 ) ).

(4.51)

Proof. Taking ρ smaller if necessary, we may assume that Ωδ is smooth for ρ every δ ∈ (0, ρ). We apply Theorem 4.5.2 with t = c1 2 . So, by (4.49), [u < t] ⊂ Ωρ/2 . By (4.32), it suffices to estimate kukW 1,4 (Ωρ/2 ) . But u solves −∆u = f (u) in Ωρ and u = 0 on ∂ Ω (which is one part of ∂ Ωρ ). In addition, ∂ Ω ∪ Ωρ/2 has compact closure contained in ∂ Ω ∪ Ωρ , and both sets are smooth. By (4.50), kuk L ∞ (Ωρ ) ≤ c2 and thus k f (u)k L ∞ (Ωρ ) ≤ k f k L ∞ (0,c2 ) . By elliptic regularity, we deduce a bound on kukW 1,4 (Ωρ/2 ) depending on the quantities in the right-hand side of (4.51). 93

Stable solutions of elliptic PDEs

4.5.2

Boundary estimates

In order to apply Proposition 4.5.1, we need to estimate solutions near the boundary. Theorem 4.5.3 ([57, 83, 123]) Let f be a locally Lipschitz function and let Ω be a smoothly bounded and convex domain of RN , N ≥ 2. Let u denote any positive classical solution to (1.3). Then, there exists constants ρ, γ > 0, depending only on Ω, such that kuk L ∞ (Ωρ ) ≤

1 γ

kuk L 1 (Ω) ,

(4.52)

where Ωρ = {x ∈ Ω : dist(x, ∂ Ω) < ρ}. Proof. We follow [185]. Clearly, it suffices to prove that for every x ∈ Ωρ , there exists a set I x such that |I x | ≥ γ and u(x) ≤ u( y), for all y ∈ I x . To see this, we apply the moving planes device. For λ > 0, x 0 ∈ ∂ Ω, let n = n(x 0 ) denote the exterior unit normal to ∂ Ω at x 0 and Σλ := x ∈ Ω : 0 < −(x − x 0 ) · n < λ . Step 1. There exists a value λ0 > 0, depending on Ω only, such that ∂n u < 0 in Σλ0 . Indeed, since Ω is convex, there exists λ0 > 0 such that the reflection of Σλ through the hyperplane Tλ = {x : −(x − x 0 ) · n = λ} remains inside Ω, for evey λ ≤ λ0 . One may then apply the moving planes procedure, exactly as in the proof of Lemma 1.2.1. Step 2. There exists a neighborhood Θ of the direction n = n(x 0 ) in S N −1 , depending on Ω only, such that for all θ ∈ Θ, 3 1 ∂θ u < 0 in Σ := x ∈ Ω : λ0 < −(x − x 0 ) · n(x 0 ) < λ0 . 8 8 Indeed, apply Step 1 at every point x˜0 ∈ ∂ Ω in a neighborhood of x 0 . In particular, assuming for simplicity that all curvatures of ∂ Ω are positive at x 0 , we obtain a neighborhood Θ of n(x 0 ) in S N −1 , such that for every θ ∈ Θ, λ0 ∂θ u < 0 in x ∈ Ω : 0 < −(x − x 0 ) · θ < . 2 By taking a smaller neighborhood Θ, we may assume that 1 |(x − x 0 ) · (θ − n(x 0 ))| < λ0 , 8 94

for all x ∈ Σλ0 and θ ∈ Θ.

Chapter 4. Regularity theory of stable solutions Now, since −(x − x 0 ) · θ = −(x − x 0 ) · (θ − n(x 0 )) − (x − x 0 ) · n(x 0 ), we have for any x ∈ Σ, λ0

1 3 1 1 = λ0 + λ0 > −(x − x 0 ) · θ > λ0 − λ0 = 0 2 8 8 8 8

and so

∂θ u(x) < 0.

Step 3. Now take ρ = λ0 /8, where λ0 is given in Step 1. Fix a point x ∈ Ωρ = {x ∈ Ω : dist(x, ∂ Ω) < ρ} and let x 0 denote its projection on ∂ Ω. By Step 1, u(x) ≤ u(x 1 ), where x 1 = x 0 − ρn(x 0 ). By Step 2, u(x 1 ) ≤ u( y), for all y in the cone I x ⊂ Σ having vertex at x 1 , opening angle Θ, and height λ0 /4.

4.5.3

Proof of Theorem 4.5.1 and Corollary 4.5.1

Proof of Theorem 4.5.1. By the boundary point lemma (Proposition A.4.2), there exists a constant c = c(Ω) such that ˆ u≥c f (u)dΩ (x) d x dΩ , Ω

where dΩ (x) = dist(x, ∂ Ω). Let ζ0 be the solution to (3.8). Then, integrating (1.3) against ζ0 yields ˆ ˆ f (u)dΩ (x) d x. f (u)ζ0 (x) d x ≤ C kuk L 1 (Ω) = Ω

Ω

So, u ≥ c1 dΩ , for some c1 depending on Ω and kuk L 1 (Ω) only. In addition, by the boundary estimate of Theorem 4.5.3, kuk L ∞ (Ωρ ) ≤ c2 , for some constants ρ, c2 depending on Ω and kuk L 1 (Ω) only. Applying Proposition 4.5.1, Theorem 4.5.1 follows. Proof of Corollary 4.5.1. Since f > 0, all solutions of (3.1) are positive. In particular, up to modifying f for negative values of t, we may always assume that f (t) ≥ f (0)/2 > 0 for all t ∈ R. Recall that the extremal solution u∗ is the increasing L 1 -limit, as λ % λ∗ , of the minimal solutions uλ to (3.1). So, if f ∈ C ∞ , we may simply apply Theorem 4.5.1 with nonlinearity λ f , λ∗ /2 < λ < λ∗ and obtain estimates for kuλ k L ∞ (Ω) , which are uniform in λ. Letting λ % λ∗ , we conclude that u∗ ∈ L ∞ (Ω). 95

Stable solutions of elliptic PDEs If f is only C 1 , let ρk be a C ∞ mollifier with support in (0, 1/k), of the form ρk (t) = kρk (kt). We replace f by

ˆ

ˆ

s

f k (s) =

1

f (t)ρk (s − t) d t =

f (s − t/k)ρ(t) d t. 0

s−1/k

Given any k ∈ N, note that f k ≤ f k+1 ≤ f in R, f k ∈ C ∞ (R), and f k is nondecreasing. In addition, f k (0) > 0 and f k is superlinear in the sense of (4.17). Since f (u∗ ) ≥ f k (u∗ ), u∗ is a weak supersolution to

−∆u = λ f k (u) u=0

in Ω, on ∂ Ω,

(4.53)

for λ = λ∗ . By the method of sub- and supersolutions (Lemma 1.1.1), (4.53) is solvable for λ = λ∗ , and so λ∗k ≥ λ∗ , where λ∗k is the extremal parameter associated to (4.53). Hence uk , the minimal solution to (4.53) for λ = λ∗ −1/k is classical. By Theorem 4.5.1, (uk ) is uniformly bounded in L ∞ (Ω). Note that uk ≤ uk+1 ≤ u∗ . Thus, uk increases in L 1 (Ω) toward a solution to (3.1) (with λ = λ∗ ) smaller than or equal to u∗ , hence identical to u∗ . Since uk was uniformly bounded in L ∞ (Ω), u∗ ∈ L ∞ (Ω).

4.6

Regularity of solutions of bounded Morse index

Consider again the bifurcation diagrams for the Gelfand problem in the unit ball, Figure P.1. From the analysis in Chapter 2, we see that no regularity theory can be developped for solutions of arbitrarily large Morse index. On the one hand, a singular solution exists in any dimension N ≥ 3 (and its Morse index is infinite for 3 ≤ N ≤ 9). On the other hand, there exists smooth solutions (thus having finite Morse index) having arbitrarily large L ∞ norm. Still, one may ask whether solutions with bounded Morse index are bounded. This is indeed the case for the Gelfand problem and 1 ≤ N ≤ 9. Theorem 4.6.1 ([71]) Assume that 3 ≤ N ≤ 9 and let Ω ⊂ RN denote a bounded domain with C 2,α boundary. Fix M ∈ N, and let (λ, u) be a solution to the Gelfand problem (4.14), with Morse index at most M . Then, there exists a constant C = C(N , Ω, M ) > 0, such that kuk L ∞ (Ω) ≤ C. 96

(4.54)

Chapter 4. Regularity theory of stable solutions Remark 4.6.1 As follows from Chapter 2, in dimension 1 ≤ N ≤ 2 (look at Figure P.1), (4.54) cannot hold with a constant independant of λ. Remark 4.6.2 A similar result holds for the Lane-Emden nonlinearity f (u) = |u| p−1 u, see [97]. Proof. By standard elliptic regularity, it suffices to show that u ≤ C in Ω. We assume to the contrary that there exists a sequence of solutions (λn, un ), with index at most M , such that Mn := maxΩ un → +∞, as n → +∞. Let x n denote a corresponding point of maximum of un and dn = dist(x n , ∂ Ω). Passing to a subsequence if necessary, we may assume that there exists λ0 ∈ R+ , x 0 ∈ Ω, such that λn → λ0 and x n → x 0 , as n → +∞. Furthermore, λ0 > 0, by Theorems 4.2.1 and 8.3.4. We use a rescaling (also called a blow-up) argument. Let rn = e−Mn /2 and vn (x) = un (x n + rn x) − Mn , for x ∈ Ωn := r1 (Ω − x n ). Then, vn solves n

−∆vn = λn e vn vn = −Mn

in Ωn , on ∂ Ωn .

(4.55)

We distinguish two cases. Case 1. Assume that (dn /rn ) is unbounded. Taking a subsequence if necessary, we may assume that limn→+∞ dn /rn = +∞. This implies that Ωn → RN , as n → +∞. We claim that (vn ) is uniformly bounded on compact sets of RN . To see this, fix a ball BR and n so large that BR ⊂ Ωn . On the one hand, since vn ≤ 0 , the solution w n to −∆w n = λn e vn in BR , wn = 0 on ∂ BR , is uniformly bounded in BR . On the other hand, zn = vn − w n is harmonic in BR , and zn (0) = −w n (0) is bounded. By Harnack’s inequality (Proposition A.3.1), zn is uniformly bounded in BR/2 , and so must be vn . So, we may pass to the limit in (4.55) and find a solution v of Morse index at most M to −∆v = λ0 e v

in RN .

This is impossible, in virtue of Theorem 6.3.3. Case 2. Assume that (dn /rn ) is bounded. Taking a subsequence if necessary, we may assume that limn→+∞ dn /rn = c ≥ 0. Let x˜n ∈ ∂ Ω be such that dn = |x n − x˜n |. For any fixed n, take a coordinate chart y = ( y1 , . . . , yN ) at x˜n , mapping some neighborhood V of x˜n onto the cylinder B(0, 1) × (−1, 1), and such that Ω ∩ V is mapped onto B(0, 1) ∩ (0, 1), ∂ Ω ∩ V onto B(0, 1) × {0}, x˜n 97

Stable solutions of elliptic PDEs to 0, and x n to yn = (0, . . . , dn ). We may always assume that the local charts are uniformly bounded in C 2 , independently of n. Then, v = vn ( y) = un (x) solves −Lvn = λn e vn in B(0, 1) × (0, 1), vn = 0 on B(0, 1) × {0}, P P P ∂y ∂y where Lv = i, j ai j ∂ yi y j v + i bi ∂ yi v, ai j = k ∂ x i ∂ x j , and bi = ∆ yi . The local k

k

charts being uniformly bounded in C 2 independently of n, the same holds true of ai j in C 1 and bi in L ∞ . Also note that ai j (0) = δi j . Now set 1 Mn − vn ( yn + rn x) , w n (x) = Mn for x ∈ Ωn := B(0, 1/rn ) × (−dn /rn , (1 − dn )/rn ). Then, w n solves   −L 0 w = λn e−Mn wn in Ωn , n Mn  wn = 1 on B(0, 1/rn ) × {−dn /rn },

P P where L 0 w = i, j ai j ( yn + rn x)∂ x i x j w + rn i bi ( yn + rn x)∂ x i w. Note that 0 ≤ w n ≤ 1 in Ωn . By elliptic regularity, w n converges to a function w such that 0 ≤ w ≤ 1 and −∆w = 0 in [x N > −c], w=1 on [x N = −c]. By the strong maximum principle, w > 0 in [x N ≥ −c]. limn→+∞ w n (0) = 0, a contradiction.

98

But w(0) =

Chapter 5 Singular stable solutions In the previous chapter, we looked for optimal conditions under which stable solutions to (1.3) must be bounded. This chapter is concerned with the study of singular stable solutions. In the first section, we prove that in large dimensions, the extremal solution to the Gelfand problem (4.14) is singular for a large class of domains obtained as perturbations of the unit ball. It should be pointed out that these solutions are singular at only one point and that the construction of stable solutions having a bigger singular set is an open problem. In the second section, we prove that this result is optimal: in any dimension, one can find many (smoothly bounded convex) domains such that the extremal solution remains bounded. In the last section, in the context of the Lane-Emden nonlinearity, we prove that if a weak solution is stable, then the Hausdorff dimension of its singular set cannot be large.

5.1

The Gelfand problem in the perturbed ball

According to Theorem 4.2.1, singular stable solutions to the Gelfand problem (4.14) can only exist in dimension N ≥ 10. But until now, we know of only one domain where the extremal solution is singular: the ball. In this section, we show that singular stable solutions persist in any domain obtained as a C 2 -diffeomorphic perturbation of the ball B. Theorem 5.1.1 ([73]) Let N ≥ 11, let ψ : B → RN denote a C 2 map, t > 0 and define Ω t = {x + tψ(x) : x ∈ B}. 99

Stable solutions of elliptic PDEs Let u∗ (t) denote the extremal solution to −∆u = λeu u=0

in Ω t , on ∂ Ω t .

(5.1)

Then there exists t 0 = t 0 (N , ψ) > 0 such that if t < t 0 , u∗ (t) is singular. In 1 addition, there exists ξ(t) ∈ B such that ku∗ (t) − log |x−ξ(t)| 2 k L ∞ (Ω t ) → 0 as t → 0. The behavior of the singular solution at the origin is characterized as follows: Corollary 5.1.1 ([73]) Fix t < t 0 and let (λ∗ (t), u∗ (t), ξ(t)) denote the extremal solution to (5.1) given by Theorem 5.1.1. Then, u (t) = ln ∗

1 |x − ξ(t)|2

+ ln

λ∗ (0)

λ∗ (t)

+ ε(|x − ξ(t)|),

(5.2)

where lims→0 ε(s) = 0. Remark 5.1.1 It is not known whether Theorem 5.1.1 holds in dimension N = 10. Proof of Theorem 5.1.1. Recall that Ω t = { x + tψ(x) : x ∈ B }, where t is small and ψ : B → RN a C 2 map. We change variables to replace (5.1) with a problem in the unit ball. The map id + tψ is invertible for t small ˜ y). Define v by and we write the inverse of y = x + tψ(x) as x = y + t ψ( ˜ y)). u( y) = v( y + t ψ( Then, ∆ y u = ∆ x v + L t v, where L t is a second-order operator given by L t v = 2t

X i,k

vx i x k

˜k ∂ψ ∂ yi

+t

X

vx k

˜k ∂ 2ψ

i,k

100

∂ yi2

+ t2

X i, j,k

vx j x k

˜j ∂ ψ ˜k ∂ψ ∂ yi ∂ yi

.

Chapter 5. Singular stable solutions We look for a solution to (5.1) of the form v(x) = log

1 |x − ξ|2

+ φ,

λ = λs + µ,

where λs = 2(N − 2), where ξ ∈ B is close to the origin, µ ∈ R is small, and where φ is a small bounded perturbation. Then (5.1) is equivalent to λs λs µ φ −∆φ − φ = L φ + (e − 1 − φ) + eφ t   |x − ξ|2 |x − ξ|2 |x − ξ|2    1 in B, + L t log  |x − ξ|2     1 φ = − log on ∂ B. |x − ξ|2 (5.3) 

To solve the above equation, we shall use the elliptic regularity theory developed for the inverse-square potential in Section B.5. Observe that for N ≥ 11 and λs = 2(N − 2), we have 0 < λs < (N − 2)2 /4 and α− > 0, α− < 0, where 1 2 α− are the indicial roots defined in Section B.5.1. So, Theorem B.5.1 holds k with k1 = 1 and ν = 0. For such ν, k1 , Theorem B.5.1 states that in order to find a bounded solution to (5.3) one must guarantee that the right-hand side and the boundary data of the equation satisfy the N + 1 orthogonality conditions (B.88). These conditions need not hold a priori for (5.3). So, we first modify the right-hand side of (5.3) so that the orthogonality conditions automatically hold: let ε0 > 0 and η ∈ C ∞ (R) such that 0 ≤ η ≤ 1, η 6≡ 0 and supp (η) ⊂ [ 41 , 12 ]. For ξ ∈ B1/2 we construct functions V`,ξ as Vl,ξ (x) = η(|x − ξ|)W1,l,ξ (x)

` = 1, . . . , N ,

where W1,`,ξ is constructed in Section B.5. Also let ˜ f (x, t) = L t log

1

|x − ξ|2

and note that k f˜(x, t)|x − ξ|2 k0,α,−2,ξ ≤ C|t|. 101

(5.4)

Stable solutions of elliptic PDEs Instead of (5.3), we first consider λs λs 1 φ −∆φ − φ = L φ + (e − 1 − φ) + µ eφ + t 0  2 2 2  |x − ξ| |x − ξ| |x − ξ|    N X  + f˜(x, t) + µi Vi,ξ in B,   i=1     1 φ = − log on ∂ B, |x − ξ|2 (5.5) 

and prove the following lemma. Lemma 5.1.1 There exists ε0 > 0 such that if |ξ| < ε0 , |t| < ε0 , there exists 2,α unique φ ∈ C0,ξ (B) and µ0 , . . . , µN ∈ R solving (5.5). Proof. Let ε0 > 0 and consider the Banach space X of functions φ(x, ξ) defined for x ∈ B, ξ ∈ Bε0 , which are twice continuously differentiable with respect to x and continuous with respect to ξ for x 6= ξ for which the following norm is finite: kφkX = sup kφ(·, ξ)k2,α,0,ξ;B . ξ∈Bε0

Let BR = {φ ∈ X | kφkX ≤ R}. By Theorem B.5.1, given ψ ∈ BR , there exists a unique φ ∈ X solving   −∆φ −

λs |x − ξ|2

φ = g in B,

(5.6)

φ = h on ∂ B,

 with λs

1

ψ

ψ

(e − 1 − ψ) + µ0 e + f˜(x, t) + g = Lt ψ + |x − ξ|2 |x − ξ|2 and h = − log

1 |x − ξ|2

102

,

N X

µi Vi,ξ (5.7)

i=1

(5.8)

Chapter 5. Singular stable solutions if and only if the N + 1 orthogonality relations

ˆ

ˆ gWk,l,ξ d x = B

h ∂B

∂ Wk,l,ξ ∂n

dσ,

for (k, l) = (0, 1) and k = 1, l = 1 . . . N

(5.9) are satisfied. Equation (5.9) is a linear system in the unknown µ0 , µ1 , . . . , µN and it is uniquely solvable provided the associated matrix of coefficients is nonsingular. This is indeed the case when ξ = 0 and ψ = 0, since by definition of Wk,l , the matrix is diagonal with nonzero diagonal entries. By continuity, the matrix remains nonsingular for small ξ and R. So, we may define a nonlinear map F : BR → X by F (ψ) = φ, where φ(·, ξ) is the solution to (5.6) with g, h given by (5.7) and (5.8). Let us show that if t is small then one can choose R small so that F : BR → BR . Indeed, let ψ ∈ BR and φ = F (ψ). Then by Theorem B.5.1, we have kφk2,α,0,ξ;B ≤ C tkψk0,α,0,ξ;B + kλs (eψ − 1 − ψ) + |x − ξ|2 f˜(x, t)k0,α,0,ξ;B + ! N X + |µi | . (5.10) i=0

From (5.9), we infer that for t, R and |ξ| small, N X

|µi | ≤ C tkψk0,α,0,ξ;B + kλs (eψ − 1 − ψ) + |x − ξ|2 f˜(x, t)k0,α,0,ξ;B + |ξ|

i=0

and so kφk2,α,0,ξ;B ≤ ≤ C tkψk0,α,0,ξ;B + kλs (eψ − 1 − ψ) + |x − ξ|2 f˜(x, t)k0,α,0,ξ;B + |ξ| R ≤ C(|t|R + R2 + |t| + |ξ|) < , (5.11) 2 provided R is first taken small enough and then |t| and |ξ| < ε0 are chosen small. Next we show that F is a contraction on BR . Let ψ1 , ψ2 ∈ BR and φ` = (`) F (ψ` ), ` = 1, 2. Let µi , i = 0, . . . , N be the constants in (5.6) associated with ψ` . 103

Stable solutions of elliptic PDEs Let φ = φ1 − φ2 . Then φ satisfies  λs φ = L t (ψ1 − ψ2 )+ −∆φ −  2  |x − ξ|   ψ  ψ2  1 −1−ψ  e e − 1 − ψ 2 1   + λs −   |x − ξ|2 |x − ξ|2     ψ1  − e ψ2  (2) e  + µ0 |x − ξ|2    e ψ1  (1) (2)  + (µ0 − µ0 )   |x − ξ|2     N X   (1) (2)  + (µi − µi )Vi,ξ in B,    i=1  φ=0

on ∂ B. (5.12)

Using Theorem B.5.1 and working as previously, we deduce that kφk2,α,0,ξ +

N X

(1)

(2)

|µi − µi | ≤ Ckgk0,α,0,ξ ,

(5.13)

i=0

where g := L t (ψ1 −ψ2 )+λs

e ψ1 − 1 − ψ 1 |x − ξ|2

−

e ψ2 − 1 − ψ 2 |x − ξ|2

(2) e +µ0

ψ1

− e ψ2

|x − ξ|2

. (5.14)

(2)

Using (5.9), we have in particular that |µ0 | ≤ CR and it follows from (5.14) and (5.13) that kφ1 − φ2 k2,α,0,ξ ≤ CRkψ1 − ψ2 k2,α,0,ξ .

(5.15)

and so kF (ψ1 ) − F (ψ2 )kX ≤ CRkψ1 − ψ2 kX . So, F is a contraction if R is small enough, and the lemma follows.

Proof of Theorem 5.1.1 continued. Recall that we want to find a bounded solution to (5.3). Thanks to Lemma 5.1.1, it suffices to prove that the point ξ can be selected so that the parameters µ1 , . . . , µN in (5.5) all vanish. To do so, 104

Chapter 5. Singular stable solutions we define the map (ξ, t) 7→ φ(ξ, t) as the small solution to (5.5) constructed in Lemma 5.1.1 for t, ξ small. We need to show that for t small enough there is a choice of ξ such that µi = 0 for i = 1, . . . , N . Let bj,ξ = W1, j (x − ξ)η1 (|x − ξ|), V

j = 1, . . . , N ,

where η1 ∈ C ∞ (R) is a cutoff function such that 0 ≤ η1 ≤ 1, ( η1 (r) = 0 for r ≤ 81 , η1 (r) = 1 for r ≥

1 4

(5.16)

(5.17)

bj,ξ and and where W1, j is defined in Section B.5.1. Multiplication of (5.5) by V integration in B gives ˆ ˆ bj,ξ ∂V λs 1 bj,ξ − L t V bj,ξ − bj,ξ φ d x + −∆V log V dσ |x − ξ|2 |x − ξ|2 ∂ n B ∂B ˆ ∂φ bj,ξ dσ − V ∂B ∂ n ˆ ˆ λs eφ φ b b dx = (e − 1 − φ)Vj,ξ d x + µ0 V 2 2 j,ξ B |x − ξ| B |x − ξ| ˆ ˆ N X ˜ b bj,ξ d x. + f (x, t)Vj,ξ d x + µi Vi,ξ V B

i=1

B

When ξ = 0 the matrix A = A(ξ) defined by ˆ bj,ξ d x for i, j = 1 . . . N Ai, j (ξ) = Vi,ξ V B

is diagonal and invertible and by continuity it is still invertible for small ξ. Thus, we see that µi = 0 for i = 1, . . . , N if and only if H j (ξ, t) = 0,

∀ j = 1, . . . , N ,

(5.18)

where, given j = 1, . . . , N , ˆ ˆ eφ λs φ b b dx H j (ξ, t) = (e − 1 − φ)Vj,ξ d x + µ0 V 2 2 j,ξ B |x − ξ| B |x − ξ| ˆ ˆ ˆ bj,ξ ∂V 1 ∂φ ˜ b bj,ξ dσ + f (x, t)Vj,ξ d x − log dσ + V |x − ξ|2 ∂ n B ∂B ∂ B1 ∂ n ˆ λs bj,ξ − L t V bj,ξ − bj,ξ φ d x. − −∆V V |x − ξ|2 B 105

Stable solutions of elliptic PDEs If this holds, then µ1 (ξ, t) = · · · = µN (ξ, t) = 0 and φ(ξ, t) is the desired solution to (5.3) (with µ in (5.3) equal to µ0 (ξ, t)). Observe that ˆ

∂ ∂ ξk

log ∂B

1

bj,ξ ∂V

|x − ξ|2 ∂ n

ˆ

=2

dσ

∂B

ξ=0

xk

bj,0 ∂V

ˆ

∂n

dσ

bj,ξ ∂ ∂V dσ + log |x − ξ|2 ∂ ξk ∂ n ξ=0 ∂B ˆ bj,0 ∂V =2 xk dσ. (5.19) ∂n ∂B 1

We shall use the Brouwer fixed point theorem as follows. (H1 , . . . , H N ) and

ˆ B(ξ) = (B1 , . . . , BN ) with

B j (ξ) =

log ∂B +

1

∂ W1, j

|x − ξ|2 ∂ n

Define H =

(x − ξ) dσ.

−

x For j = 1, . . . , N we have W1, j (x) = (|x|−α j − |x|−α j )ϕ j ( |x| ), and hence ∂ W1, j ∂n

(x) = (α−j − α+j )ϕ j (x) =

α−j −α+j ´ ( S N −1 x 2j )1/2

x j , for x ∈ ∂ B. Using this and (5.19),

we deduce that B is differentiable and DB(0) is invertible. Equation (5.18) is then equivalent to ξ = G(ξ), where G(ξ) = DB(0)−1 (DB(0)ξ − H(ξ, t)) . To apply the Brouwer fixed point theorem, it suffices to prove that for t, ρ small, G is a continuous function of ξ and G : B ρ → B ρ . This is the object of the next two lemmata. Lemma 5.1.2 G is continuous for t, ξ small. Proof. Observe first that for t, ξ small such that kφk L ∞ (B) ≤ R we have kφk L ∞ (B) ≤ C(kλs (eφ − 1 − φ) + |x − ξ|2 f˜(x, t)k L ∞ (B) + |ξ|) ≤ C(Rkφk L ∞ (B) + |t| + |ξ|), 106

Chapter 5. Singular stable solutions and we deduce (taking R smaller if necessary) kφk L ∞ (B) ≤ C(|t| + |ξ|).

(5.20)

Similarly, |µi | ≤ C(|t| + |ξ|),

∀i = 0, . . . , N .

(5.21) (k)

Now, take a sequence ξk → ξ and let φk = φ(ξk , t), µi be the solutions and parameters associated to (5.5). By (5.20) and (5.5) and using elliptic estimates we see that φk is bounded in C 1,α on compact sets of B \ {ξ}. By passing to a subsequence we may assume that φk → φ uniformly on compact (k) sets of B \ {ξ} and by (5.21) that µi → µi . Then φ is a solution to (5.5) with kφk L ∞ (B) ≤ R and with parameters ξ and µi . This solution is unique by Lemma 5.1.1 and this shows that in fact, the complete sequence converges. Then all terms in the definition of H(ξ, t) converge. In fact ˆ ˆ eφ − 1 − φ e φk − 1 − φ k bj,ξ d x → λs bj,ξ d x as k → ∞, V V λs |x − ξ|2 |x − ξ|2 B B by dominated convergence, because e φk − 1 − φ C k b V . ≤ j,ξ |x − ξ|2 |x − ξ|2 Similarly, (k) µ0

ˆ

e φk

B

bj,ξ d x → µ0 V |x − ξ|2

ˆ

eφ B

|x − ξ|2

bj,ξ d x V

as k → ∞.

(5.22)

Lemma 5.1.3 If ρ > 0 and |t| are small enough then G : Bρ → Bρ . Proof. By (5.20) ˆ φ(ξ) e − 1 − φ(ξ) bj,ξ d x ≤ Ckφk2 ∞ ≤ C(|ξ| + |t|)2 . V λs L (B) B |x − ξ|2 Let σ > 0 to be fixed later. From (5.22) we have ˆ eφ b Vj,ξ d x ≤ σ 2 B |x − ξ| 107

Stable solutions of elliptic PDEs if t and ξ are small enough. Also, |DB(0)ξ − B(ξ)| ≤ C|ξ|2 , and

ˆ f˜(x, t)V bj,ξ d x ≤ C|t| B

for some constant C. Thus if |ξ| < ρ and ρ is small we have |G(ξ)| ≤ C ρ 2 + |t| + σρ . First, fix σ such that Cσ < 14 . We can then fix ρ > 0 so small that C(ρ 2 +σρ) < ρ . Then, for |t| small, |G(ξ)| ≤ ρ. 2 Proof of Theorem 5.1.1 completed. Thanks to the previous two lemmata, we deduce that for t sufficiently small, there exists a point ξ = ξ(t), a small parameter µ = µ(t), and a small bounded function φ = φ(x, t) solving (5.3). ˜ y) = φ(x), where x = y + t ψ( ˜ y) for y ∈ Ω t . Change variables and let φ( Then, λ(t) φ˜ 1 −∆ y φ˜ = e + ∆ ln in Ω t . y |x − ξ|2 |x − ξ|2 ˜ ˜ y) = φ( ˜ y) + ln | y−ξ| Letting ξ˜ = ξ + tψ(ξ) and ψ( , the above equation can be |x−ξ|2 rewritten as λ(0) λ(t) ψ˜ ˜= e − in Ω t , −∆ y ψ ˜2 ˜2 | y − ξ| | y − ξ| 2

λ(0) ˜ where we used the fact that ∆ y ln | y−1ξ| ˜ 2 = − | y−ξ| ˜ 2 . Then, u = u(t) = ψ +

ln | y−1ξ| ˜ 2 is a solution to (5.1) associated to λ = λ(t), and

1

u(t) − ln

2 ˜

| y − ξ(t)|

+ |λ(t) − 2(N − 2)| → 0,

as t → 0.

(5.23)

L ∞ (Ω t )

It remains to prove that u(t) is the extremal solution to (5.1). By Proposition (N −2)2 3.3.1, it suffices to show that u(t) is stable. Since N ≥ 11, 2(N − 2) < 4 and it follows from (5.23) that if t is chosen small enough, λ(t)e

u−ln

1 2 ∞ ˜ | y−ξ(t)| L (Ω t )

108

<

(N − 2)2 4

.

Chapter 5. Singular stable solutions So, for ϕ ∈ Cc1 (Ω t ), ˆ ˆ ˆ (N − 2)2 ϕ2 u 2 λ(t) e ϕ dy ≤ dy ≤ |∇ϕ|2 d y, 2 N N 4 | y − ξ(t)| Ωt R R in virtue of Hardy’s inequality (Proposition C.1.1). Hence, u(t) is stable, and so u(t) is the extremal solution to (5.1). ˜ is bounded, it Proof of Corollary 5.1.1. Using the above notation, since ψ ˜ that ψ ˜ is follows by Corollary B.5.1 and the fixed point characterization of ψ ˜ Define the sequence (ψn ) by continuous at y = ξ. 1 ˜ ˜ ˜ + ξ. ˜ ˜ ( y − ξ) + ξ , for y ∈ Ωn := n(Ω t − ξ) ψn ( y) = ψ n ˜ ξ). ˜ Also, ψn solves Clearly, (ψn ) converges pointwise to the constant ψ( −∆ y ψn =

λ(0) λ(t) ψ e n− 2 ˜ ˜2 | y − ξ| | y − ξ|

in Ωn .

(5.24)

˜ the right-hand side in the above equality remains bounded. Away from y = ξ, It follows by elliptic regularity that up to a subsequence, (ψn ) converges to ˜ ξ) ˜ in the topology of C ∞ (RN \ {ξ}). ˜ ψ( In particular, passing to the limit for y 6= ξ˜ in (5.24), we obtain 0=

λ(t) ψ( λ(0) ˜ ˜ e ξ) − , ˜2 ˜2 | y − ξ| | y − ξ|

˜ ξ) ˜ = ln λ(0) . Since the solution u(t) of (5.1) we constructed is given hence ψ( λ(t) 1 ˜ by u(t) = ln | y−ξ| ˜ 2 + ψ, we have just proved Corollary 5.1.1. Exercise 5.1.1 Given p > 1, consider the problem −∆u = λ(1 + u) p in Ω t , u=0 on ∂ Ω t .

(5.25)

• When t = 0, that is, when the domain is the unit ball, prove that the extremal solution is unbounded and given by u∗ = |x|−2/(p−1) − 1, if and only if N ≥ 11 and p ≥ pc (N ), where p (N − 2)2 − 4N + 8 N − 1 . pc (N ) = (N − 2)(N − 10) Compare to Exercise 4.2.1. 109

Stable solutions of elliptic PDEs • Let N ≥ 11 and p > pc (N ). Given t > 0 small, let u∗ (t) denote the extremal solution to (5.25). Show that there exists t 0 = t 0 (N , ψ, p) > 0 such that if t < t 0 , u∗ (t) is singular. In addition, prove that there exists ξ(t) ∈ B1 such that ku(x, t)−(|x −ξ(t)|−2/(p−1) −1)k L ∞ (Ω t ) → 0, as t → 0.

5.2

Flat domains

In the previous section, we proved that the extremal solution to the Gelfand problem (4.14) is singular in any dimension N ≥ 11 and for any domain close to the ball. In this section, we prove that the situation is completely different when the domain is chosen close to an infinite cylinder with cross-section in RN2 , N2 ≤ 9. Theorem 5.2.1 ([69]) Let N = N1 + N2 ≥ 10. Let Ω ⊂ RN denote a smoothly bounded domain. We assume furthermore that Ω is convex and ∂ Ω is uniformly convex, that is, its principal curvatures are bounded away from zero. Given " > 0, let u∗" be the extremal solution to −∆u = λeu in Ω" , (5.26) u=0 on ∂ Ω" , where, writing RN = RN1 × RN2 and x = (x 1 , x 2 ) ∈ RN with x 1 ∈ RN1 , x 2 ∈ RN2 , we set Ω" = {x = ( y1 , " y2 ) : ( y1 , y2 ) ∈ Ω}.

(5.27)

Then, if N2 ≤ 9, there exists "0 = "0 (N , Ω) > 0 such that if " < "0 , u∗" is smooth. Remark 5.2.1 Let Ω = B in dimension N ≥ 11 and let Ω" be the ellipsoid defined by (5.27) with N2 = 1. Combining Theorems 5.1.1 and 5.2.1 we can say that for " close to 1 (round ellipsoid), u∗ is singular, while for " close to 0 (flat ellipsoid), u∗ is regular. Proof of Theorem 5.2.1. We follow [73]. We assume by contradiction that for a sequence " j & 0, we have u∗" j 6∈ L ∞ (Ω" j ). Let M > 0 be a constant to be fixed later. By continuity, we can select a number λ j with 0 < λ j < λ∗" j such that the minimal solution u j of (5.26) with parameter λ j satisfies max u j = M . Ω" j

110

(5.28)

Chapter 5. Singular stable solutions Define v j ( y1 , y2 ) = u j ( y1 , " j y2 ). Then v j is defined in Ω and satisfies ¨

−(" 2j ∆ y1 + ∆ y2 )v j = " 2j λ j e v j in Ω, vj = 0

on ∂ Ω,

(5.29)

where ∆ yi denotes the Laplacian with respect to the variables yi , i = 1, 2. For some constant C0 we have λ∗" ≤

C0 "2

.

(5.30)

Indeed, let µ" = λ1 (−∆; Ω" ) denote the principal eigenvalue for −∆ in Ω" with Dirichlet boundary conditions and ϕ" > 0 an associated eigenfunction, that is, −∆ϕ" = µ" ϕ" in Ω" , ϕ" = 0 on ∂ Ω" . We normalize ϕ" so that kϕ" k L 2 (Ω" ) = 1. Multiplying (5.26) by ϕ" and integrating by parts we find ˆ ˆ ∗ ∗ ∗ µ" u" ϕ " d x = λ " eu" ϕ" d x. Ω"

Ω"

Since eu ≥ u for all u ∈ R, it follows that λ∗" ≤ µ" . But by changing variables (x 1 , x 2 ) = ( y1 , " y2 ) we find ´ ´ 1 2 2 2 Ω" |∇ϕ| d x Ω |∇ y1 ψ| + " 2 |∇ y2 ψ| d x ´ ´ 2 = inf . µ" = inf 2 ψ∈Cc1 (Ω) ϕ∈Cc1 (Ω" ) Ω" ϕ d x Ωψ dx ϕ6=0

ψ6=0

C

Fixing ψ ∈ Cc1 (Ω), ψ 6= 0 we deduce µ" ≤ "20 . Note that C0 = C0 (Ω, N ) does not depend on M . Next we show that for some constant C independent of j k∇v j k L ∞ (Ω) ≤ C. 111

(5.31)

Stable solutions of elliptic PDEs For this, using the uniform convexity of Ω, find R > 0 large enough so that for any y0 ∈ ∂ Ω there exists z0 ∈ RN such that the ball BR (z0 ) satisfies Ω ⊂ BR (z0 ) and y0 ∈ ∂ BR (z0 ). For convenience write for " > 0 L" = " 2 ∆ y1 + ∆ y2 . Define ζ( y) = R2 − | y − z0 |2 so that ζ ≥ 0 in Ω and −L" ζ = 2"N1 + 2N2 (this can be computed easily by shifting so that z0 is at the origin and writing |( y1 , y2 )|2 = | y1 |2 + | y2 |2 ). From (5.30) we have the uniform bound " 2j λ j ≤ C. It follows from (5.29) and the maximum principle that v j ≤ Cζ with C independent of j and y0 . Since v j ( y0 ) = ζ( y0 ) = 0, this in turn implies that |∇v j ( y0 )| ≤ C

∀ j, y0 ∈ ∂ Ω.

(5.32)

Recall that the minimal solution u j is strictly stable, that is, λ1 (−∆ − λ j eu j ; Ω" j ) > 0. By changing variables, the same holds true for the linearization of (5.29) at v j , that is, the operator w 7→ −L" j w − " 2j λ j e v j w has a positive principal eigenvalue. This implies that we have the maximum principle in the form: if w ∈ C 2 (Ω) satisfies −L" j w − " 2j λ j e v j w = 0 in Ω then max |w| ≤ max |w|. ∂Ω

Ω

Applying this to the partial derivatives of v j and using (5.32), we deduce (5.31). By (5.28), we can find subsequences, denoted (5.31), and (5.30) for simplicity v j , " j , and λ j , such that v j → v uniformly in Ω and " 2j λ j → λ0 ≥ 0. Multiplying (5.29) by ϕ ∈ Cc1 (Ω) and integrating by parts twice we find ˆ ˆ 2 2 − v j (" j ∆ y1 ϕ + ∆ y2 ϕ) d x = " j λ j e v j ϕ d x. Ω

Ω

Letting j → ∞ we obtain ˆ ˆ − v∆ y2 ϕ d x = λ0 e v ϕ d x Ω

Ω

∀ϕ ∈ Cc1 (Ω).

Writing v y1 ( y2 ) := v( y1 , y2 ) for ( y1 , y2 ) ∈ RN1 × RN2 ∩ Ω, we see that for each nonempty slice Ω y1 = { y2 ∈ RN2 : ( y1 , y2 ) ∈ Ω}, 112

Chapter 5. Singular stable solutions we have ¨

−∆ y2 v y1 = λ0 e v y1 in Ω y1 v y1 = 0

on ∂ Ω y1 .

(5.33)

Let y j ∈ Ω denote the point of maximum of v j , that is, v j ( y j ) = maxΩ v j = M . For a subsequence, y j → y0 ∈ Ω as j → ∞ and since v j converges uniformly to v, we have M = v j ( y j ) → v( y0 ). Since v|∂ Ω = 0, we must have y0 ∈ Ω. Write y0 = (a, b) and observe that Ωa is nonempty since y0 ∈ Ω. Then va ( y2 ) = v(a, y2 ) solves (5.33) in Ωa . Moreover maxΩa va = M and va is stable, that is, ˆ ˆ va 2 (5.34) λ0 e ϕ dx ≤ |∇ϕ|2 d x, ∀ϕ ∈ Cc1 (Ωa ). Ωa

Ωa

To see this, let ϕ ∈ Cc1 (Ωa ) and χ ∈ Cc1 (RN1 ) be such that χ ≡ 1 in a neighborhood of a and supp (χ( y1 )ϕ( y2 )) ⊂ Ω. By stability of u j and changing variables we have ˆ ˆ vj 2 2 2 " 2j ϕ( y2 )2 |∇χ( y1 )|2 + χ( y1 )2 |∇ϕ( y2 )|2 d x. " j λ j e χ( y1 ) ϕ( y2 ) d x ≤ Ω

Ω

Letting j → ∞ yields ˆ ˆ v 2 2 λ0 e χ( y1 ) ϕ( y2 ) d x ≤ χ( y1 )2 |∇ϕ( y2 )|2 d x. Ω

Ω

Choosing a sequence χk ∈ Cc1 (RN1 ) such that χk ≡ 1 in a neighborhood of a and supp (χk ) ⊂ B1/k (a) we obtain (5.34). (1) (1) (1) Let ymin = min{ y1 : Ω y1 6= ;}, ymax = max{ y1 : Ω y1 6= ;}. For any ymin < (1) y1 < yma the slice Ω y1 is a smooth open nonempty set and hence for the x problem ¨ −∆ y2 v = λe v in Ω y1 , (5.35) v=0 on ∂ Ω y1 , there exists an extremal parameter 0 < λ∗y1 < ∞. In particular, • If 0 ≤ λ < λ∗y1 then (5.35) has a unique minimal solution v y1 ,λ . Moreover v y1 ,λ is smooth and characterized as the unique stable solution to (5.35), that is, the unique solution satisfying ˆ ˆ v y1 ,λ 2 λ e ϕ dx ≤ |∇ϕ|2 d x, ∀ϕ ∈ Cc1 (Ω y1 ). Ω y1

Ω y1

113

Stable solutions of elliptic PDEs • If λ > λ∗y1 then (5.35) has no weak solution. • If λ = λ∗y1 then (5.35) has a unique weak solution v ∗y1 and v ∗y1 = limλ%λ∗y v y1 ,λ . 1

• If N2 ≤ 9 (recall that Ω y1 ⊂ RN2 ) then v ∗y1 is bounded. We claim that for any λ > 0 there exists Mλ > 0 depending only on Ω and λ (1) (1) such that for any ymin < y1 < ymax we have max v y1 ,λ ≤ Mλ .

(5.36)

Ω y1

That is, we assert that if we have some a priori control on λ, the boundedness (1) (1) . of v y1 ,λ is uniform when ymin < y1 < ymax Using (5.30) we have the bound λ0 ≤ C0 . Hence, choosing M = Mλ0 + 1 at the beginning of the proof, (5.36) contradicts (5.28). Proof of (5.36). The argument is the same as in the proof of Theorem 4.2.1 (1) (1) but we shall emphasize that the bound does not depend on ymin < y1 < ymax . For simplicity we write v = v y1 ,λ . Let 0 < α < 2 and multiply Equation (5.35) by e2αv − 1. Integrating in Ω y1 we find ˆ ˆ 2αv 2 e(2α+1)v − e v d x. (5.37) e |∇v| d x = λ 2α Ω y1

Ω y1

Using (5.34) with eαv − 1 yields ˆ ˆ v αv 2 2 e (e − 1) d x ≤ α λ

Ω y1

Ω y1

Combining (5.37) and (5.38) gives ˆ ˆ α (2α+1)v (1 − ) e dx ≤ 2 e(α+1)v d x ≤ 2 Ω y1 Ω y1 ˆ ≤2

Ω y1

For 0 < p < 5 we deduce the bound ke v k L p (Ω y ) ≤ C 1

114

e2αv |∇v|2 d x.

e(2α+1)v d x

α+1

2α+1

(5.38)

1− α+1 Ω 2α+1 . y1

Chapter 5. Singular stable solutions (1)

(1) with C independent of ymin < y1 < ymax . In dimension N2 ≤ 9, we thus have ke v k L p (Ω y ) ≤ C for some p > N2 /2. 1 Recalling (5.35), this shows that kvk L ∞ (Ω y ) ≤ C and the constant is indepen1

(1)

(1) dent of ymin < y1 < yma , as can be seen using Moser’s iteration technique x (as in Section B.4) and working on a large ball U such that Ω y1 ⊂ U for all (1) (1) ymin < y1 < yma , considering all functions on Ω y1 to be extended by zero in x U \ Ω y1 .

5.3

Partial regularity of stable solutions in higher dimensions

In dimension N ≥ 10, the extremal solution to the Gelfand problem (4.14) can be singular or regular, depending on the geometry of Ω. This is what we have learned in the two previous sections. In case the stable solution is singular, one may wonder how large its singular set can be: a point, a curve, a higher dimensional manifold? This question has recently been addressed in [221]. We present here a partial regularity theorem for the Lane-Emden nonlinearity (see also [220] for related results). Theorem 5.3.1 ([76]) Let N ≥ 3 and Ω ⊂ RN an open set. Suppose p > and u ∈ H 1 (Ω) ∩ L p+1 (Ω), u ≥ 0, is a stable weak solution to −∆u = u p

in Ω.

N +2 N −2

(5.39)

Then, u ∈ C ∞ (Ω \ Σ), where Σ is a closed set which the Hausdorff dimension is bounded above by p+γ , Hd im (Σ) ≤ N − 2 p−1 p with γ = 2p + 2 p(p − 1) − 1. Remark 5.3.1 In Exercise 4.2.1, we proved that the extremal solution to the problem −∆u = λ(1 + u) p in Ω, u=0 on ∂ Ω, is regular if 1 < p < pc (N ), where pc (N ) is given by (4.16). By Theorem 5.3.1, whenever N ≥ 11 and p ≥ pc (N ), the singular set Σ of the extremal solution has 115

Stable solutions of elliptic PDEs its Hausdorff dimension Hdim (Σ) ≤ N − 2

p+γ p−1

,

p where γ = 2p+2 p(p − 1)−1. Note that for p = pc (N ), this implies Hdim (Σ) = 0. This is precisely the case when Ω is the unit ball (see Exercise 5.1.1): the extremal solution is singular at one point.

5.3.1

Approximation of singular stable solutions

In this section, we extend the approximation procedure discussed in Section 3.2.2, in the context of (5.39). Lemma 5.3.1 Suppose u ∈ H 1 (Ω) ∩ L p+1 (Ω), u ≥ 0, is a stable weak solution to (5.39), that is, ˆ ˆ ∇u∇ϕ d x = u p ϕ d x, for all ϕ ∈ Cc∞ (Ω), Ω

Ω

and (1.5) holds. Then, there exists a sequence of stable solutions un ∈ C ∞ (Ω) to (5.39), such that un % u a.e. and in H 1 (Ω). Proof. Given c > 0, consider the function − 1 φc (t) = c + t −(p−1) p−1 ,

defined for t > 0.

We also set φc (0) = 0. Then, φc is increasing, concave, and smooth for t > 0. In addition, φc (t) % t as c & 0+ , and φc (t) ≤ t, for all t ≥ 0. Also, if c > 0, then φc , φc0 are uniformly bounded. We have φc0 (t)

=

φc (t) p tp

∀t > 0.

Let w c denote the unique solution to −∆w c = 0 in Ω, w c = φc (u) on ∂ Ω. Then, w c ≥ 0, w c ∈ L ∞ (Ω)∩H 1 (Ω). Moreover, w c is nonincreasing with respect to c. We claim that w c → w in H 1 (Ω) as c → 0, where w is the solution to −∆w = 0 in Ω, w = u on ∂ Ω. 116

Chapter 5. Singular stable solutions To see this, consider the problem −∆v = (v + w c ) p in Ω, v=0 on ∂ Ω.

(5.40)

Since w c ∈ L ∞ (Ω), (5.40) has a minimal solution vc , which can be constructed by the method of sub- and supersolutions, as follows. Note that v = 0 is a subsolution, since w c ≥ 0. Moreover, by Kato’s inequality (Lemma 3.2.1), v = φc (u) − w c is a bounded supersolution: −∆(φc (u) − w c ) = −∆φc (u) ≥ −φc0 (u)∆u = φc (u) p = (φc (u) − w c + w c ) p . In particular, (5.40) has a minimal solution vc . This minimal solution is bounded and by elliptic regularity, vc belongs to C 1,α (Ω). Moreover, vc is stable in the sense that ˆ ˆ p−1 2 p (vc + w c ) ϕ d x ≤ |∇ϕ|2 d x, for all ϕ ∈ Cc1 (Ω). Ω

Ω

Since vc is minimal and w c is nonincreasing with respect to c, we deduce that vc is also nonincreasing with respect to c. It follows that v(x) = limc→0 vc (x) is well defined for all x ∈ Ω. Since vc ∈ C 1 (Ω), we have ˆ ˆ ˆ 2 p |∇vc | d x = (vc + w c ) vc d x ≤ u p+1 d x. Ω

Ω

Ω

In particular, vc is bounded in H01 (Ω). It follows that vc * v weakly in H01 (Ω). Multiplying (5.40) by ϕ ∈ Cc∞ (Ω), integrating, and passing to the limit as c → 0, we see that v is a weak solution to −∆v = (v + w) p in Ω, (5.41) v=0 on ∂ Ω. Let ϕk ∈ Cc0,1 (Ω) be a sequence such that ϕk → v in H01 (Ω). Since v ≥ 0 we can assume ϕk ≥ 0. We can also assume that ϕk → v a.e. in Ω. Multiplying (5.41) by ϕk and integrating, we obtain ˆ ˆ ∇v∇ϕk d x = (v + w) p ϕk d x. Ω

Ω

By Fatou’s lemma, ˆ ˆ ˆ p (v + w) v d x ≤ lim inf ∇v∇ϕk d x = |∇v|2 d x. Ω

k→∞

Ω

117

Ω

Stable solutions of elliptic PDEs By monotone convergence, ˆ ˆ ˆ 2 p lim |∇vc | d x = lim (vc + w c ) vc d x = (v + w) p v d x. c→0

c→0

Ω

Hence,

ˆ

Ω

ˆ |∇vc | d x = 2

lim c→0

Ω

Ω

Since vc * v weakly in ˆ

ˆ (v + w) v d x ≤ p

Ω

Ω

|∇v|2 d x.

H01 (Ω),

Ω

the reverse inequality ˆ 2 |∇v| d x ≤ lim inf |∇vc |2 d x c→0

Ω

H01 (Ω).

also holds, which proves that vc → v in We claim that u = v+w, from which Lemma 5.3.1 follows. By construction, v = lim vc ≤ lim(φc (u) − w c ) = u − w. We need thus only prove that u ≤ v + w. Note that v˜ = u − w solves (5.41). Let z = v˜ − v ≥ 0. Then, z ∈ H01 (Ω), and since u is stable, ˆ ˆ p−1 2 p (˜ v + w) (˜ v − v) d x ≤ |∇(˜ v − v)|2 d x. (5.42) Ω

Ω

Now, v˜ − v satisfies ˆ ˆ v + w) p − (v + w) p )ϕ d x, ∇(˜ v − v)∇ϕ d x = ((˜ Ω

Ω

∀ϕ ∈ Cc∞ (Ω).

We would like to take ϕ = v˜ − v. First, we claim that we can take ϕ ∈ H01 (Ω) ∩ L ∞ (Ω). These functions can be approximated in H01 (Ω) by functions in Cc∞ (Ω) with a uniform bound. Then, take ϕ = min(˜ v − v, t), t > 0, which belongs to H01 (Ω) ∩ L ∞ (Ω). We get ˆ ˆ 2 |∇(˜ v − v)| d x = ((˜ v + w) p − (v + w) p ) min(˜ v − v, t) d x. [˜ v −v≤t]

Ω

Now let t → ∞. Then, ˆ ˆ 2 |∇(˜ v − v)| d x = ((˜ v + w) p − (v + w) p )(˜ v − v) d x. Ω

Ω

Combined with (5.42) we find ˆ (˜ v − v) p(˜ v + w) p−1 (˜ v − v) − (˜ v + w) p + (v + w) p d x ≤ 0. Ω

But p(˜ v + w) p−1 (˜ v − v) − (˜ v + w) p + (v + w) p ≥ 0 with strict inequality, unless v˜ ≡ v. 118

Chapter 5. Singular stable solutions

5.3.2

Elliptic regularity in Morrey spaces

The next ingredient in the proof of Theorem 5.3.1 is a so-called "-regularity result for weak solutions of (5.39) in Morrey spaces. Definition 5.3.1 Let Ω be a bounded open set of RN , N ≥ 1. Given p > 1 and λ ∈ [0, N ], the Morrey space L p,λ (Ω) is the set of functions u in L p (Ω) such that the following norm is finite: ˆ p −λ kuk L p,λ (Ω) = sup r |u| p d x < ∞. x 0 ∈Ω, r>0

B(x 0 ,r)∩Ω

Remark 5.3.2 Observe that L p,0 (Ω) = L p (Ω), while L p,N (Ω) = L ∞ (Ω). We begin by proving the following regularity theorem. Theorem 5.3.2 ([172]) Let N ≥ 3, p > N /(N − 2), and let u ≥ 0 be a weak 2 , then u is regular in solution to (5.39). If u ∈ L p,λ (Ω) for some λ > N − 2 − p−1 Ω. The proof relies on two useful technical lemmata. Lemma 5.3.2 ([172]) Let N ≥ 3, B = B(x, r) ⊂ RN and p > 1. Assume that 2 f ∈ L 1,λ (B) for some λ ∈ (N − 2 − p−1 , N − 2). For x ∈ B, let ˆ v(x) = |x − y|2−N f ( y) d y. B

Then, there exists a constant C = C(p, N , λ) > 0, such that N

kvk L p (B) ≤ C r p

−(N −2−λ)

k f k L 1,λ (B) .

(5.43)

2 Proof. Fix µ ∈ (N − 2 − p−1 , λ). Then, for x ∈ B, ˆ +∞ ˆ X −µ |x − y|−µ f ( y) d y ≤ |x − y| f ( y) d y −k r<|x− y|<2−k+1 r] B B∩[2 k=0 ˆ +∞ X kµ −µ ≤ 2 r f ( y) d y −k+1 k=0 +∞ X

≤2

B∩[|x− y|<2

2−k(λ−µ) r λ−µ k f k L 1,λ (B)

k=0 λ−µ

≤ Cr

r]

k f k L 1,λ (B) .

119

Stable solutions of elliptic PDEs We deduce that ˆ p p 2−N +µ −µ |v(x)| = |x − y| |x − y| f ( y) d y B

ˆ

ˆ

(2−N +µ)p−µ

|x − y| | f ( y)| d y |x − y| | f ( y)| d y B ˆ p−1 (2−N +µ)p−µ ≤C |x − y| | f ( y)| d y r (λ−µ)(p−1) k f k L 1,λ (B) .

≤

−µ

p−1

B

B

Integrating over B and recalling that µ > N − 2 − 2/(p − 1), we obtain ˆ p p−1 (2−N +µ)p−µ+N kvk L p (B) ≤ C r | f ( y)| d y r (λ−µ)(p−1) k f k L 1,λ (B) ˆ B p−1 (2−N +λ)p−λ+N ≤ Cr | f ( y)| d y k f k L 1,λ (B) ≤ Cr

(2−N +λ)p+N

kf

B p k L 1,λ (B) .

Lemma 5.3.3 ([49]) Let 0 < λ < N and c > 0. Assume that φ : R+ → R+ is a nondecreasing function such that for all 0 < ρ < R, we have N ρ λ φ(ρ) ≤ c φ(R) + R . (5.44) RN Then, there exists a constant C = C(λ, N , R, φ(R)), such that φ(ρ) ≤ Cρ λ ,

for all ρ ∈ (0, R).

Proof. Set ρ = τR, with τ ∈ (0, 1). By (5.44), φ(τR) ≤ c τN φ(R) + Rλ . Now fix γ ∈ (λ, N ) and τ ∈ (0, 1) such that cτN = τγ . Then, φ(τR) ≤ τγ φ(R) + cRλ . Iterating the above formula, we deduce that for k ∈ N, φ(τk R) ≤ τkγ φ(R) + cτλ(k−1) Rλ

k−1 X j=0

120

τ j(γ−λ) ≤ Cτkλ .

(5.45)

Chapter 5. Singular stable solutions Now take any ρ ∈ (0, R). There exists an integer k such that τk+1 R < ρ ≤ τk R. Since φ is nondecreasing, (5.45) follows. Proof of Theorem 5.3.2. Let u be a nonnegative weak solution to (5.39). Fix a ball B = B(x, R) ⊂ Ω, and let f ( y) = u p ( y), for y ∈ B. Also let ˆ v(x) = Γ(x − y) f ( y) d y, B

where Γ is the fundamental solution to the Laplace equation, given by (A.6). Then, −∆(u − v) = 0, in B, and so w = u − v is smooth in B. By the mean-value formula (A.10), for all y ∈ B/4, we can write w( y) =

w(z) dz. B( y,R/4)

Integrating over B(x, ρ), 0 < ρ < R/4, we deduce that

ˆ p

|w| d y ≤ C B(x,ρ)

ρN

ˆ |w| p dz.

RN

B(x,R)

Since u = v + w in B, we deduce that ˆ ˆ

ˆ |w| p d y +

up d y ≤ C B(x,ρ)

B(x,ρ)

≤C

ρ

N

RN

≤C

ρN RN

vp d y B(x,ρ)

ˆ

ˆ

|w| dz + p

ˆ

B(x,R)

ˆ

p

v dz B(x,ρ)

u dz + p

B(x,R)

p,N −2

p

v dz

.

B(x,R)

We claim that u ∈ L l oc (Ω). If λ ≥ N − 2, this is obvious. Otherwise, we may apply (5.43) and obtain for 0 < ρ < R/4, Nˆ ˆ ρ p p N −p(N −2−λ) u dy ≤C u dz + R . RN B(x,R) B(x,ρ) 121

Stable solutions of elliptic PDEs Choosing the constant C larger if necessary, the above inequality remains true for all ρ ∈ (0, R). Applying Lemma 5.3.3, we deduce that ˆ u p d y ≤ Cρ N −p(N −2−λ) B(x,ρ) p,λ

and so u ∈ L l oc 1 (Ω), with λ1 = N −p(N −2−λ). If λ1 ≥ N −2, the claim follows. p,λ Otherwise, we iterate the above estimate and conclude that u ∈ L loc k+1 (Ω), whenever λk < N − 2 and where, for k ≥ 1, λk+1 = N − p(N − 2 − λk ), that is, λk = p

k

λ−

(N − 2)p − N

p−1

+

(N − 2)p − N p−1

.

Since λ > N − 2 − 2/(p − 1) = ((N − 2)p − N )/(p − 1), we see that there exists k such that λk ≥ N − 2 and the claim follows. Applying once more (5.43) with λ → (N − 2)− , we deduce that v ∈ L q (B) for all 1 < q < +∞, hence q u ∈ L l oc (Ω) for all 1 < q < +∞. By standard elliptic regularity, we conclude that u is regular. Theorem 5.3.2 can be further refined as follows. p+1

Theorem 5.3.3 ([174]) Let N ≥ 3, p > 1, and λ = N − 2 p−1 . Also let B(x 0 , r0 ) be a ball. There exists " = "(N , p) > 0 such that for any weak solution u ∈ H 1 (B(x 0 , r0 )) ∩ C(B(x 0 , r0 )), u ≥ 0, to (5.39) satisfying kuk L p+1,λ (B(x 0 ,r0 )) ≤ ", there holds kuk L ∞ (B(x 0 ,r0 /2)) ≤

4

(5.46) 2 p−1

r0

.

Proof. We follow [128]. Without loss of generality, we may assume that x 0 = 0. For y ∈ B r0 , let 2

Φ( y) = (r0 − | y|) p−1 u( y). Let y0 ∈ B r0 denote a point of maximum of Φ in B r0 , ρ0 = | y0 |, and y1 a point of maximum of u in Bρ0 . Then, for y ∈ B r0 , Φ( y) ≤ Φ( y0 ) and so u( y) ≤

r0 − ρ0 r0 − | y|

2 p−1

u( y0 ) ≤ 122

r0 − ρ0 r0 − | y|

2 p−1

u( y1 ).

(5.47)

Chapter 5. Singular stable solutions We claim that u( y1 ) ≤

2

2 p−1

,

r0 − ρ0

(5.48)

from which the theorem follows. Assume by contradiction that (5.48) fails. Then, r0 − ρ0 p−1 ρ1 := u( y1 )− 2 ≤ , 2 so that B( y1 , ρ1 ) ⊂ B(0, (r0 + ρ0 )/2) ⊂ B(0, r0 ). Let 2

v(x) = ρ1p−1 u(ρ1 x + y1 ),

for x ∈ B1 .

Then, v solves (5.39) in B1 , and v(0) = 1. In addition, it follows from (5.47) that for all y ∈ B(0, (r0 + ρ0 )/2), u( y) ≤

r0 − ρ0

2 p−1

r0 − | y|

u( y1 ) ≤

hence

r0 − ρ0

(r0 − ρ0 )/2

2 p−1

u( y1 ),

2

kvk L ∞ (B1 ) ≤ 2 p−1 .

(5.49)

Now, Kato’s inequality (Lemma 3.2.1) implies that −∆v p+1 ≤ (p + 1)v p−1 v p+1 ≤ 4(p + 1)v p+1

in B1 ,

where we used (5.49) in the last inequality. By the mean-value inequality (Exercise A.1.5), we deduce that ˆ p+1 p+1 −λ 1 = v (0) ≤ C v d x = Cρ1 u p+1 d x. B( y1 ,ρ1 )

B1

This contradicts (5.46), provided " p+1 < 1/C.

5.3.3

Measuring singular sets

Theorem 5.3.2 shows that a weak solution u ≥ 0 of (5.39) is smooth in a neighborhood of a point x 0 ∈ Ω, provided it belongs to a suitable Morrey space in a neighborhood of x 0 . In this section, we estimate the Hausdorff 1 dimension of the set of points where a function u ∈ L loc (Ω) is locally large, provided it fails to belong to some Morrey space. 123

Stable solutions of elliptic PDEs 1 (Ω), Theorem 5.3.4 Let Ω denote an open set of RN , N ≥ 1, u a function in L loc and 0 ≤ s < N . Set ¨ « ˆ −s Es = x ∈ Ω : lim sup r |u( y)| d y > 0 . r→0+

Then,

B r (x)

H s (Es ) = 0,

where H s denotes the Hausdorff measure of dimension s. The proof of Theorem 5.3.4 relies on the following covering lemma. Lemma 5.3.4 Let Σ denote a bounded set of RN , N ≥ 1 and let r = r(x) be a function defined on Σ with values in (0, 1).Then, there exists a sequence of points x i ∈ Σ such that B(x i , r(x i )) ∩ B(x j , r(x j )) = ;

if i 6= j,

∪i B(x i , 3r(x i )) ⊃ Σ.

(5.50)

Proof. We follow [121]. Consider the family B1,1/2 = {B(x, r(x)) : x ∈ Σ ,

1 2

≤ r(x) < 1}.

Since Σ is bounded, there exists a finite subfamily of disjoint balls B1,1/2 = {B(x i , r(x i )) :

1 2

≤ r(x i ) < 1 1 ≤ i ≤ n1 },

which is maximal in the sense that each ball in B1,1/2 intersects at least one element in B1,1/2 . Once we have constructed x 1 , . . . , x n j , among the balls B(x, r(x)) with 2− j−1 ≤ r(x) < 2− j , which do not intersect any of the balls B(x i , r(x i )), i = 1, . . . , n j , we can find a finite family of balls, say, n j+1 − n j (possibly void) such that each B(x, r(x)) with 2− j−2 ≤ r(x) < 2− j−1 intersects at least one of the balls in {B(x i , r(x i )) : i = 1, . . . , n j+1 }. The sequence satisfies (5.50). In fact, the balls B(x i , r(x i )) are disjoint by construction; for x ∈ Σ, there exists x i such that B(x, r(x)) ∩ B(x i , r(x i )) 6= ; and 2r(x i ) ≥ r(x). Hence, |x − x i | ≤ r(x) + r(x i ) ≤ 3r(x i ), 124

Chapter 5. Singular stable solutions so that x ∈ B(x i , 3r(x i )).

Proof of Theorem 5.3.4. We follow [121]. It suffices to show that for each compact subset K ⊂⊂ Ω, H s (F ) = 0,

where F = Es ∩ K.

For n ≥ 1, set F (n) =

ˆ

¨

« |u( y)| > 1/n .

x ∈ F : lim sup r −s r→0+

B(x,r)

Obviously, F = ∪+∞ F (n) and it suffices to show that H s (F (n) ) = 0 for all n=1 n ≥ 1. Take a bounded open set Q such that K ⊂ Q ⊂ Q ⊂ Ω and d = min(1, d(x, ∂ Q)), where x ∈ F (n) . For all " > 0, 0 < " < d, and for all x ∈ F (n) , there exists r(x), 0 < r(x) < ", such that ˆ 1 −s r(x) |u( y)| d y ≥ . 2n B(x,r(x)) Let x i be the sequence in Lemma 5.3.4 corresponding to Σ = F (n) , and let ri = r(x i ). Then, ˆ X Xˆ s ri ≤ 2n |u( y)| d y = 2n |u( y)| d y. (5.51) i

B ri (x i )

i

∪B ri (x i )

Since s < N , this inequality implies ˆ X X N N −s s N −s ∪B (x ) ≤ |B | ri ≤ C" ri ≤ C n" |u( y)| d y. ri i 1 i

i

(5.52)

Q

From (5.52), it follows that the right-hand side of (5.51) converges to 0 as " → 0. Taking into account (5.50) and the definition of the Hausdorff measure H s (F (n) ), we deduce that H s (F (n) ) = 0.

5.3.4

A monotonicity formula

Consider a weak solution u to (5.3.1), a ball B(x 0 , r0 ) ⊂ Ω, and let µ = p+1 N − 2 p−1 . By the approximation lemma (Lemma 5.3.1) and the "-regularity theorem (Theorem 5.3.3), u is smooth in a neighborhood of x 0 provided the quantity ˆ −µ r u p+1 d y (5.53) B(x,r)

125

Stable solutions of elliptic PDEs is uniformly small at every scale r ∈ (0, r0 ) and for all x in a neighborhood of x 0 . For fixed x, by Theorem 5.3.4, the above quantity is indeed small for r smaller than some r(x) > 0, unless x belongs to a set of zero H µ Hausdorff measure. In this section, we prove a monotonicity formula, which will serve as a bridge between these two results: If (5.53) is small at some point x 0 and some scale r0 , it remains small at all scales r < r0 and for all x near x 0 . Theorem 5.3.5 ([173]) Let u ∈ C 2 (Ω), u ≥ 0, denote a solution to the LaneEmden equation, Equation (5.39). Given x ∈ Ω and r > 0 such that B(x, r) ⊂ Ω, consider the energy

ˆ Eu (x, r) = r

−µ B(x,r)

1 2

1

2

|∇u| −

p+1

where µ= N −2

u

p+1

p+1 p−1

dx +

r −µ−1 p−1

ˆ u2 dσ, ∂ B(x,r)

(5.54)

.

Then, Eu (x, r) is a nondecreasing function of r. The aforementioned monotonicity formula remains valid for weak solutions. Corollary 5.3.1 ([173]) Let u ∈ H 1 (Ω) ∩ L p+1 (Ω), u ≥ 0, denote a stable weak solution to the Lane-Emden equation (5.39). For x ∈ Ω, r > 0, such that B(x, r) ⊂ Ω, consider the energy Eu (x, r) given by (5.54). Then, • Eu (x, r) is nondecreasing in r. • Eu (x, r) is continuous in x ∈ Ω and r > 0. Proof. Using Theorem 5.3.5 and the approximation lemma (Lemma 5.3.1), we easily see that Eu (x,´r) is nondecreasing in r. Also, the continuity of Eu (x, r) reduces to that of ∂ B(x,r) u2 dσ. Let un ∈ C ∞ (Ω) denote the approximating sequence given by Lemma 5.3.1. Multiply (5.3.1) by un and integrate on B(x, r) ⊂ Ω: ˆ ˆ ∂ un un dσ = (|∇un |2 − unp+1 ) d y. ∂ n B(x,r) ∂ B(x,r) In other words, ∂ ∂r

∂ B(x,r)

u2n dσ = 2

B(x,r)

126

(|∇un |2 − unp+1 ) d y.

(5.55)

Chapter 5. Singular stable solutions Fix x 0 ∈ Ω and r0 > 0 such that B(x 0 , 2r0 ) ⊂ Ω. Integrate (5.55) between r and r0 . Then, for x ∈ B(x 0 , r0 ), ˆ r 2 2 un dσ + 2 (|∇un |2 − unp+1 ) d y d t. un dσ = ∂ B(x,r0 )

∂ B(x,r)

r0

B(x,t)

Passing to the limit as n → +∞, we deduce that ˆ r 2 2 u dσ = u dσ + 2 (|∇u|2 − u p+1 ) d y d t, ∂ B(x,r)

∂ B(x,r0 )

´

r0

B(x,t)

and so we just need to prove that x → ∂ B(x,r0 ) u2 dσ is continuous at x 0 . We may always assume that x 0 = 0. Then, ˆ ˆ ˆ ˆ 1 d u2n dσ = u2n dσ + u2n dσ d t d t ∂ B(x,r0 ) ∂ B(0,r0 ) 0 ∂ B(t x,r0 ) ˆ ˆ 1 ˆ = u2n dσ + 2 un ∇un · x dσ d t ∂ B(0,r0 )

0

ˆ =

∂ B(0,r0 )

u2n

dσ + 2|x|

∂ B(0,r0 )

∂ B(t x,r0 )

0

ˆ =

∂ B(t x,r0 )

ˆ 1 ˆ ˆ 1 ˆ

u2n

dσ + 2|x|

∂ B(0,r0 )

∂n

B(t x,r0 )

0

(|∇un |

dσ

dt

− unp+1 )

dy

d t.

(|∇u| − u 2

B(t x,r0 )

0

2

Passing to the limit as n → +∞, we deduce that ˆ ˆ ˆ 1 ˆ 2 2 u dσ = u dσ + 2|x| ∂ B(x,r0 )

un

∂ un

p+1

)dy

d t,

which is a continuous quantity of x, as desired.

Proof of Theorem 5.3.5. To simplify notation, we write B(x, r) = B r . We begin by applying Pohozaev’s identity. By (8.8): ˆ ˆ ˆ ˆ 2 N −2 r ∂u 2 2 ∆u(x ·∇u) d x = |∇u| d x − |∇u| dσ+r dσ. 2 2 ∂ Br ∂r Br Br ∂ Br Since u solves (5.39), we also have ˆ ˆ ∆u(x · ∇u) d x = − u p (x · ∇u) d x = Br

Br

ˆ

u p+1

N Br

127

p+1

ˆ dx − r

∂ Br

u p+1 p+1

dσ.

Stable solutions of elliptic PDEs Combining the above two equalities, we obtain the Pohozaev identity:

ˆ

N −2

r

2

|∇u| d x −

2

2

Br

ˆ

ˆ

2

∂u

2

dσ = |∇u| dσ + r ∂r ∂ Br ˆ ˆ N r p+1 u dx − u p+1 dσ. (5.56) p + 1 Br p + 1 ∂ Br

∂ Br

Now multiply the Lane-Emden equation (5.39) by u and integrate over B r . Then, ˆ ˆ ˆ ∂u p+1 2 u dx − u dσ. (5.57) |∇u| d x = ∂ Br ∂ r Br Br Differentiating the above equation, we obtain

ˆ

ˆ |∇u| d x = 2

∂ Br

u

p+1

∂ Br

ˆ

d

dx −

dr

u ∂ Br

∂u ∂r

(5.58)

dσ.

Plugging (5.57) and (5.58) in (5.56), we deduce that 1

N −2

−

2

r

ˆ

N

u

p+1

dx −

¨ˆ

u

2 dr

∂ Br

∂u ∂r

1 2

Br

1 d

=

p+1

−

«

ˆ

1

p+1 ˆ

dσ −

u p+1 dσ

∂ Br 2

∂u ∂r

∂ Br

+

N −2u∂u

r∂r

2

dσ.

p+1

Letting µ = N − 2 p−1 , this identity can be rewritten as p−1 d

ˆ

¨ r

p + 1 dr

«

−µ

u

p+1

dx

+

Br

1 2

= r −µ

r

−µ

d

¨ˆ

dr ˆ

u ∂ Br

∂u

2

∂r

∂ Br

∂u

« dσ

∂r +

N −2u∂u 2

dσ. (5.59)

r∂r

The second term on the left-hand side of this identity can be rewritten as r

−µ

d dr

¨ˆ u ∂ Br

∂u ∂r

« dσ =

d dr

ˆ

¨ r

−µ

u ∂ Br

∂u ∂r

ˆ

« dσ + µr

−µ−1

u ∂ Br

∂u ∂r

dσ. (5.60)

128

Chapter 5. Singular stable solutions In addition, d

ˆ

¨ r

dr

−µ

u ∂ Br

=

∂u ∂r

¨

1 d2

r

2 d r2

« dσ =

ˆ

−µ

¨

1 d

r

2 dr «

2

∂ Br

u dσ −

−µ

|∂ B r |

p+3 p−1

r

«

d

2

d r ∂ Br ˆ

u dσ. u2

2

−µ

+

p − 1 r2

∂ Br

u∂u

dσ

r∂r

Collecting this identity and (5.60) and plugging them in (5.59), we find 1 p−1 d

ˆ

¨

2 p + 1 dr

r

−µ

« u

p+1

+

dx

ˆ

¨

1 d2

«

−µ

2

u dσ r 4 d r2 ∂ Br ˆ 2 ∂u p + 3 u2 p+7 u∂u −µ + =r + dσ. ∂r 2(p − 1) r ∂ r (p − 1)2 r 2 ∂ Br Br

This can be rewritten as d dr

¨

ˆ

p−1

r

2(p + 1)

−µ

u p+1 d x+ Br

+

ˆ

1 d 4 dr

r

−µ

2

−

u dσ ∂ Br

=r

ˆ

1

r 4 ˆ

«

−µ−1

2

u dσ ∂ Br

−µ

∂u ∂r

∂ Br

+

2

u

2 dσ ≥ 0.

p−1 r

So, the quantity Fu (x, r) = p−1 2(p + 1)

ˆ r

−µ

u Br

p+1

dx +

1 d 4 dr

ˆ

¨ r

«

−µ

2

∂ Br

u dσ −

1 4

ˆ r

−µ−1

u2 dσ ∂ Br

(5.61) is nondecreasing. Using (5.57) and (5.58), we see that Eu (x, r) = Fu (x, r). Remark 5.3.3 Eliminating the terms r −µ−1 129

´ ∂ Br

u2 dσ between (5.54) and

Stable solutions of elliptic PDEs (5.61), we obtain the equivalent formulation for the energy Eu (x, r): Eu (x, r) = ˆ ˆ 1 p − 1 −µ 1 d 1 2 p+1 −µ 2 r |∇u| + u r d y+ u dσ . p+3 2 p+1 p + 3 dr B(x,r) ∂ B(x,r) (5.62)

5.3.5

Proof of Theorem 5.3.1

Our last ingredient is the following crucial capacitary estimate. Proposition 5.3.1 ([97]) Let Ω be an open set of RN , p > 1. Let u ∈ C 2 (Ω) denote a stable solution to −∆u = |u| p−1 u

in Ω.

p Then, for any γ ∈ [1, 2p + 2 p(p − 1) − 1), any ψ ∈ Cc1 (Ω), 0 ≤ ψ ≤ 1, and n o p+γ any integer m ≥ max p−1 , 2 , there exists a constant C p,m,γ > 0 such that ˆ ˆ 2 2 p+γ γ−1 2m p+γ ψ d x ≤ C p,m,γ ∇ψ p−1 d x. ∇ |u| 2 u + |u| Ω

Ω

Proof. Following [97], we split the proof into steps. Step 1. For any ϕ ∈ Cc1 (Ω), we have ˆ 2 ˆ γ+1 2 (γ + 1) |u| γ−1 |u| p+γ ϕ 2 − ∇ · ∇ϕ 2 d x. ∇ |u| 2 u ϕ 2 d x = 4γ γ + 1 Ω Ω (5.63) Multiply the equation by |u|γ−1 uϕ 2 and integrate by parts. Then, ˆ ˆ ˆ 2 γ−1 2 γ−1 2 γ|∇u| |u| ϕ d x + |u| u∇u · ∇ϕ d x = |u| p+γ ϕ 2 d x, Ω

Ω

Ω

and therefore

ˆ ˆ ˆ 2 |u|γ+1 γ−1 2 2 |u| p+γ ϕ 2 d x. · ∇ϕ d x = ∇ |u| 2 u ϕ d x + ∇ γ+1 2 γ+1 Ω Ω Ω γ

2

130

Chapter 5. Singular stable solutions Identity (5.63) then follows by multiplying the above equation by the factor γ+1 2

2

.

γ

Step 2. For any ϕ ∈ Cc1 (Ω) and any " > 0, there exists a constant C = C(", γ) such that !ˆ 2 ˆ γ+1 p+γ 2 p− −" |u| ϕ d x ≤ C |u|γ+1 |∇ϕ|2 d x. (5.64) 4γ Ω Ω γ−1

The function ψ = |u| 2 uϕ belongs to Cc1 (Ω), and thus it can be used as a test function in the stability inequality (1.5).

ˆ p Ω

|u|

p+γ

ˆ ˆ 2 2 γ−1 γ−1 2 |u| 2 u |∇ϕ|2 d x+ ϕ d x ≤ ∇(|u| 2 u) ϕ d x + Ω Ω ˆ γ−1 γ−1 2∇(|u| 2 u) · ∇ϕ|u| 2 uϕ d x. (5.65) 2

Ω

By Young’s inequality, given η > 0, there exists a constant C = C(η) such that ˆ γ−1 γ−1 2∇(|u| 2 u) · ∇ϕ|u| 2 uϕ d x Ω ˆ ˆ 2 γ−1 2 ≤ η ∇(|u| 2 u) ϕ d x + C |u|γ+1 |∇ϕ|2 d x. (5.66) Ω

Ω

Using this in (5.65), we obtain ˆ ˆ ˆ 2 γ−1 p+γ 2 2 p |u| ϕ d x ≤ (1 + η) ∇(|u| 2 u) ϕ d x + C |u|γ+1 |∇ϕ|2 d x. Ω

Ω

Ω

(5.67) Using (5.63) and (5.66), we also have ˆ 2 γ−1 2 ∇ |u| u ϕ 2 d x ≤ Ω ˆ ˆ (γ + 1)2 p+γ 2 (1 + η) |u| ϕ d x + C |u|γ+1 |∇ϕ|2 d x. (5.68) 4γ Ω Ω From the above inequality and (5.67), we obtain

ˆ p Ω

|u|

p+γ

ϕ d x ≤ (1 + η) 2

2

(γ + 1)2 4γ

ˆ Ω

131

ˆ |u|

p+γ

ϕ dx +C 2

Ω

|u|γ+1 |∇ϕ|2 d x,

Stable solutions of elliptic PDEs which, letting η = c", with c > 0 small, p gives the identity (5.64). Step 3. For any γ ∈ [1, 2p + 2 p(p − 1) − 1) and any integer m ≥ p+ γ max p−1 , 2 , there exists a constant C = C(p, m, γ), such that

ˆ ˆ p+γ γ−1 2 2 2m p+γ |u| + |∇(|u| 2 u)| ψ d x ≤ C |∇ψ| p−1 d x,

(5.69)

Ω

Ω

for all test functions ψ ∈ Cc1 (Ω) satisfying |ψ| ≤ 1 in Ω. Note that for p γ ∈ [1, 2p + 2 p(p − 1) − 1), there exists " > 0 sufficiently small so that 2 (γ+1) p − 4γ − " > 0. Apply (5.64) with ϕ = ψm , m ≥ 1. Then,

ˆ

ˆ Ω

|u|

p+γ

ψ

2m

dx ≤ C

Ω

|u|γ+1 ψ2m−2 |∇ψ|2 d x

ˆ ≤C

Ω

|u|

p+γ

ψ

γ+1 ˆ p+γ

p+γ

(2m−2) γ+1

dx Ω

At this point, we notice that m ≥ max Since |ψ| ≤ 1 in Ω, we deduce that

p+γ ,2 p−1

Ω

|u|

p+γ

ψ

2m

dx ≤ C

p+γ

dx

.

p+γ

p+γ

2 p−1

Ω

|∇ψ|

implies (2m − 2) γ+1 ≥ 2m.

ˆ

ˆ

p−1

p+γ

2 p−1

|∇ψ|

d x.

Going back to (5.68), (5.69) follows.

Proof of Theorem 5.3.1. Suppose u ∈ H 1 (Ω)∩ L p+1 (Ω), u ≥ 0 is a stable weak solution to (5.39). Given " > 0, define

ˆ

¨ Σ" =

x ∈ Ω : ∀r > 0

(u p+1 + |∇u|2 ) d x ≥ "r

p+1

N −2 p−1

« .

B(x,r)

Step 1. There exists a fixed value of " > 0 such that for every x 6∈ Σ" , u is bounded (hence smooth) in a neighborhood of x. To see this, let x 0 6∈ Σ" : there exists r0 > 0 such that

ˆ −µ r0

B(x 0 ,r0 )

(u p+1 + |∇u|2 ) d x < ", 132

Chapter 5. Singular stable solutions p+1

where µ = N − 2 p−1 . By (5.54), for r < r0 ,

ˆ Eu (x 0 , r) ≤ r

1

−µ

ˆ

B(x 0 ,r) 2

|∇u| d y +

1

≤ r −µ

r −µ−1

2

p−1

|∇u|2 d y +

r

−µ−1

ˆ u2 dσ ∂ B(x 0 ,r)

ˆ

p − 1 ∂ B(x 0 ,r) B(x 0 ,r0 ) 2 −µ −µ−1 ˆ " r r ≤ + u2 dσ. 2 r0 p − 1 ∂ B(x 0 ,r)

u2 dσ

Integrating between r = r0 /2 and r0 , and recalling that Eu (x, r) is nondecreasing in r, we deduce that r0 2

Eu (x 0 , r0 /2) ≤ 2µ−2 "r0 +

ˆ

1

p−1 ˆ

r0

ˆ r −µ−1

r0 /2

−µ−1

≤ C"r0 + C r0

u2 dσ

dr

∂ B(x 0 ,r)

u2 d y B(x 0 ,r0 )

−µ−1

ˆ

≤ C"r0 + C r0

2 p+1

u p+1 d y B(x 0 ,r0 )

2 N (1− p+1 )

r0

< C"r0 . Hence, Eu (x 0 , r0 /2) < C". Since Eu is continuous in x, there exists r1 < r0 /2 such that Eu (x, r0 /2) < 2C", for x ∈ B(x 0 , r1 ). Since Eu is nonincreasing in r, we deduce that for all x ∈ B(x 0 , r1 ) and all r < r1 , Eu (x, r) < 2C".

(5.70)

Now take an approximating sequence un given by Lemma 5.3.1. Integrating (5.62) between 0 and r2 < r1 , we find p−1 p+3

ˆ

ˆ

r2

r 0

−µ B(x,r)

1 2

|∇un | + 2

1 p+1

unp+1

−µ

dy

d r+

r2

p+3

ˆ ∂ B(x,r2 )

u2n dσ

≤ r2 Eun (x, r2 ). 133

Stable solutions of elliptic PDEs It follows that

ˆ

r2

C r2 Eu (x, r2 ) ≥

ˆ

r

0

ˆ ≥

−µ

u

p+1

dy

dr

B(x,r)

ˆ

r2

r

−µ

u

r2 /2

p+1

dy

d r.

B(x,r)

By the fundamental theorem of calculus, we deduce that there exists r3 ∈ (r2 /2, r2 ) such that ˆ ˆ −µ −µ p+1 CEu (x, r2 ) ≥ r3 u d y ≥ r2 u p+1 d y. B(x,r3 )

Now apply (5.70). Then,

B(x,r2 /2)

ˆ r

−µ

u p+1 d y ≤ C", B(x,r)

for all x ∈ B(x 0 , r1 ) and all r < r1 /2. Taking " sufficiently small, it follows from Theorem 5.3.3 that (un ) is uniformly bounded near x 0 , and so u is smooth in a neighborhood of x 0 . Step 2. For all γ ≥ 1, there exists " 0 > 0 such that « ¨ ˆ p+γ N −2 p−1 ˜ "0 := x ∈ Ω : ∀r > 0 u p+γ d x ≥ " 0 r . Σ" ⊆ Σ B(x,r)

˜ "0 . Then, Indeed, suppose x 6∈ Σ ˆ u p+γ d x < " 0 r

p+γ

N −2 p−1

B(x,r)

for some r > 0. By Hölder’s inequality,

ˆ

ˆ u

p+1

p+1

p+γ

dx ≤ C

B(x,r)

u

p+γ

dx

p+1

r

N (1− p+γ )

B(x,r)

p+1 p+γ p+1 p+1 p+1 N −2 p−1 p+γ N (1− p+γ ) N −2 p−1 < C "0 r r = C(" 0 ) p+γ r . (5.71) Take a function ϕ ∈ Cc2 (Ω) and multiply the Lane-Emden equation (5.39) by uϕ 2 . Then, ˆ ˆ ˆ 2 2 2 |∇u| ϕ d x + u∇u · ∇ϕ d x = u p+1 ϕ 2 d x, Ω

Ω

Ω

134

Chapter 5. Singular stable solutions that is,

ˆ

ˆ |∇u| ϕ d x = 2

Ω

2

u

p+1

Ω

ϕ dx + 2

1 2

ˆ Ω

u2 ∆ϕ 2 d x.

Now choose ϕ such that ϕ = 1 in B(x, r/2), ϕ = 0 outside B(x, r), and |∆ϕ 2 | ≤ C/r 2 . Then, ˆ ˆ ˆ C 2 p+1 |∇u| d x ≤ C u dx + 2 u2 d x. r B(x,r/2) B(x,r) B(x,r) We estimate 1 r2

ˆ 2

u dx ≤ B(x,r)

C

ˆ

r2

u

p+γ

2 p+γ

dx

r

2 1− p+γ

B(x,r)

<

2 p+1 2 C 0 n−2 p+γ p+γ 1− 2 N −2 p−1 p−1 " r r p+γ = C(" 0 ) p+γ r . 2 r

Using (5.71), we deduce that ˆ p+1 2 N −2 p−1 (u p+1 + |∇u|2 ) d x < C(" 0 ) p+γ r . B(x,r/2) 2

˜ " 0 ⊃ Σ" . Choosing " 0 such that C(" 0 ) p+γ ≤ ", we deduce that x 6∈ Σ" . And so, Σ p+γ Step 3. By the capacitary estimate (Proposition 5.3.1), u ∈ L loc (Ω) if γ ∈ p [1, 2p + 2 p(p − 1) − 1). By Theorem 5.3.4, it follows that for " 0 > 0 small, p+γ

H

N −2 p+1

˜ "0 ) = 0. (Σ

p This being true for all γ ∈ [1, 2p + 2 p(p − 1) − 1), Theorem 5.3.1 follows.

135

Chapter 6 Liouville theorems for stable solutions In this chapter, we explore Equation (1.4) when Ω = RN . We begin with the study of radial solutions.

6.1

Classifying radial stable entire solutions

In this section, we focus our attention on bounded radial stable solutions of −∆u = f (u)

in RN .

(6.1)

The following Liouville-type theorem is sharp (see, for example, [97, Theorem 5] for counter-examples). Theorem 6.1.1 ([38, 216]) Let 1 ≤ N ≤ 10 and f ∈ C 1 (R). Then, every bounded radial stable solution u ∈ C 2 (RN ) of (6.1) is constant. The proof of Theorem 6.1.1 uses two technical lemmata, which we present next. Lemma 6.1.1 ([216]) Let N ≥ 1 and u be a stable nonconstant radial solution to (6.1). Then, there exists K > 0 such that

ˆ

+∞ r

ds s N −1 u2r (s)

p

≤ K r −2

N −1

137

,

for all r ≥ 1.

(6.2)

Stable solutions of elliptic PDEs Proof. We prove the lemma for N ≥ 2 and refer the reader to [216] for the case N = 1. Step 1 ([38]) u r > 0 in RN \ {0}. Assume by contradiction that u r (R) = 0 for some R > 0. Differentiate (6.1) with respect to r. Then, u r solves −∆u r +

N −1 r

2

u r = f 0 (u)u r ,

for r > 0.

(6.3)

Take a cutoff function ζ" ∈ H 1 (RN ) defined by  0 if |x| < " 2 ,   ln |x| ζ" (x) = 2 − if " 2 ≤ |x| < ",  ln "  1

if |x| ≥ ",

if N = 2, and ζ" (x) = ζ(x/"), with ζ ∈ C 1 (RN ) such that ζ ≡ 0 in B1 and 1 ζ ≡ 1 in RN \ B2 , if N ≥ 3. One easily checks that ζ" → 1 in H loc (RN ). Multiply (6.3) with ζ" u r , integrate, and let " → 0. We deduce ˆ ˆ 2 u2r 0 2 d x < 0, Q u (u r χBR ) = ∇u r − f (u)u r d x = −(N − 1) 2 BR BR r a contradiction with the stability of u, since u r χBR ∈ H01 (BR ) is a valid test function in (1.5) (see Remark 1.1.1). Step 2 ([38]) For all η ∈ H 1 (RN ), ˆ ˆ 2 u2r η2 (N − 1) dx ≤ u2r ∇η d x. (6.4) 2 RN |x| RN This is a restatement of the geometric Poincaré formula, Equation (4.25), in the radial setting. By density, it suffices to consider η ∈ Cc1 (RN \ {0}). We calculate Q u (u r η): ˆ ∇(u η) 2 − f 0 (u)(u η)2 d x 0 ≤ Q u (u r η) = r r N ˆR 2 2 = u2r ∇η + ∇η2 · u r ∇u r + η2 ∇u r − f 0 (u)u2r η2 d x N ˆR 2 2 2 2 2 0 2 2 = u r ∇η − η ∇ · (u r ∇u r ) + η ∇u r − f (u)u r η d x N ˆR 2 = u2r ∇η − η2 (u r ∆u r + f 0 (u)u2r ) d x. RN

138

Chapter 6. Liouville theorems for stable solutions Using (6.3), we obtain (6.4). Step 3. Fix R > r ≥ 1 and consider the function  1  p   t − N −1   p  ˆ R r − N −1 ds η(t) =  ´  R N −1 ds  u2r (s)  t s  r s N −1 u2 (s)  r 0

if 0 ≤ t ≤ 1, if 1 < t ≤ r, if r ≤ t ≤ R, if R < t.

In particular,

ˆ (N − 1) 0

+∞

ˆ u2r η2

t

N −3

1

d t ≥ (N − 1) 0

u2r t N −3 d t+ ˆ r p + (N − 1) u2r t −2 N −1+N −3 d t. 1

In addition, ˆ +∞ 0

ˆ u2r (η0 )2 t N −1 d t = (N − 1)

r

1

p

p u2r t −2 N −1+N −3

r −2 dt + ´ R

N −1 ds

.

r s N −1 u2 r

Applying Step 2, we conclude that p ˆ 1 −2 N −1 r (N − 1) u2r t N −3 d t ≤ ´ R ds . 0

r s N −1 u2 r

Finally, since R > r is arbitrarily large, we obtain (6.2) with K = −1 ´1 (N − 1) 0 u2r t N −3 d t . Lemma 6.1.2 ([216]) Let N ≥ 2 and let u denote a nonconstant stable radial solution to (6.1). Then, there exists a constant M > 0 such that |u(2r) − u(r)| ≥ M r −N /2+

p

N −1+2

Proof. Fix r ≥ 1 and consider the functions − 2 1−N α(s) = s 3 u r (s) 3 , 2 β(s) = u r (s) 3 , 139

,

for all r ≥ 1.

s ∈ (r, 2r), s ∈ (r, 2r).

(6.5)

Stable solutions of elliptic PDEs By Lemma 6.1.1, we have 2p

1

kαk L 3 (r,2r) ≤ K 3 r − 3

N −1

,

where K > 0 is independent of r. In addition, since u r > 0 in RN \ {0}, it follows that 2 kβk 23 = |u(2r) − u(r)| 3 . L (r,2r)

Applying Hölder’s inequality to α and β, we deduce that r

4−N 3

ˆ

2

t

1−N 3

d t = ks

1−N 3

1

k L 1 (r,2r) ≤ kαk L 3 (r,2r) kβk

3

L 2 (r,2r) 2p

1

≤ K 3 r− 3

N −1

2

|u(2r) − u(r)| 3 .

We may now prove Theorem 6.1.1. Proof. Case 1. 2 ≤ N ≤ 9. Let r ≥ 1. There exists m ∈ N and 1 ≤ r1 < 2 such that r = 2m−1 r1 . From the monotonicity of u and Lemma 6.1.2, we obtain X m−1 u(2k r ) − u(2k−1 r ) − u(r ) ≥ |u(r)| ≥ u(r) − u(r1 ) − u(r1 ) = 1 1 1 k=1

≥M

m−1 X

(2k−1 r1 )−N /2+

k=1

=M

r −N /2+

p

p

N −1+2

− u(r1 ) =

N −1+2

− r1

2−N /2+

Since r1 ∈ [1, 2), u is continuous and −N /2 + |u(r)| ≥ M r −N /2+

p

p −N /2+ N −1+2

p

p

N −1+2

N −1+2

−1

! − u(r1 ) .

N − 1 + 2 > 0, we finally have − M2 ,

contradicting the assumption that u isp bounded. Case 2. N = 10. This time, −N /2 + N − 1 + 2 = 0 and working as above, we obtain M (ln r − ln r1 ) |u(r)| ≥ M (m − 1) − u(r1 ) = − u(r1 ) , ln 2 contradicting again the assumption that u is bounded. 140

Chapter 6. Liouville theorems for stable solutions

6.2

Classifying stable entire solutions

6.2.1

The Liouville equation

We now turn to possible nonradial solutions. At the time of writing of this book, the theory is yet incomplete. We begin our presentation with the nonlinearity f (u) = eu (see, for example, [88], [89] for more general results). That is, we study stable solutions of −∆u = eu

in RN ,

(6.6)

which we shall refer to as the Liouville equation. Stable solutions are classified as follows. Theorem 6.2.1 ([95]) For 1 ≤ N ≤ 9, there is no stable solution u ∈ C 2 (RN ) of (6.6). Remark 6.2.1 The theorem is sharp: For every N ≥ 10, there exists a radial stable solution to (6.6) (see Chapter 2). The proof of Theorem 6.2.1 relies on the following capacitary estimate. Proposition 6.2.1 ([95]) Assume that N ≥ 1 and let Ω denote an open set of RN . Let u ∈ C 2 (Ω) denote a stable solution to −∆u = eu

in Ω.

Then, for any integer m ≥ 5 and any α ∈ C(m, α) > 0 such that ˆ ˆ (2α+1)u 2m e ψ dx ≤ C Ω

Ω

(0, 2), there exists a constant C = ∇ψ 2(2α+1) d x,

(6.7)

for all ψ ∈ Cc1 (Ω) such that 0 ≤ ψ ≤ 1. Proof. The first step is very similar to the proof of Theorem 4.2.1: given ϕ ∈ Cc1 (Ω), we multiply the equation by e2αu ϕ 2 on the one hand, and use the stability inequality (1.5) with test function eαu ϕ on the other hand. The former computation is developed next. ˆ ˆ ˆ 2αu 2 2αu 2 ∇u∇ e ϕ d x + e ∇u∇ ϕ d x = e(2α+1)u ϕ 2 d x. Ω

Ω

Ω

141

Stable solutions of elliptic PDEs Hence, ˆ ˆ ˆ 2 2 (2α+1)u 2 αu 2 2 e ϕ dx = |∇ (e )| ϕ d x + ϕ∇ (eαu ) eαu ∇ϕ d x α α Ω Ω Ω ˆ ˆ 2 2 αu 2 2 ≥ −" |∇ (e )| ϕ d x − C" e2αu ∇ϕ d x, α Ω Ω where we used the Cauchy-Schwarz inequality and where " > 0 is chosen so small that 2/α − " > 1. So, there exists β = (2/α − ")−1 < 1 such that ˆ ˆ ˆ 2 αu 2 2 (2α+1)u 2 |∇ (e )| ϕ d x ≤ β e ϕ dx + C e2αu ∇ϕ d x. (6.8) Ω

Ω

Ω

Next, we use stability (1.5) with test function eαu ϕ and obtain

ˆ

ˆ

e

(2α+1)u

Ω

ˆ 2 ϕ d x ≤ |∇ (e )| ϕ d x + ∇ϕ e2αu d x + Ω Ω ˆ + 2 eαu ∇ (eαu ) ϕ∇ϕ d x ≤ Ω ˆ ˆ 2 αu 2 2 ≤ (1 + ") |∇ (e )| ϕ d x + C" ∇ϕ e2αu d x, αu

2

2

2

Ω

Ω

where " > 0 is chosen so small that γ = β(1 + ") < 1. Plugging (6.8) in the latter inequality yields ˆ ˆ ˆ 2 (2α+1)u 2 (2α+1)u 2 ϕ d x + C ∇ϕ e2αu d x. e ϕ dx ≤ γ e Ω

Ω

Ω

That is,

ˆ e

(2α+1)u

Ω

ˆ 2 ϕ d x ≤ C ∇ϕ e2αu d x. 2

(6.9)

Ω

The next step consists in iterating the above formula: set ϕ = ψm , where ψ ∈ Cc1 (Ω), 0 ≤ ψ ≤ 1 and m ≥ 5. We obtain

ˆ e Ω

(2α+1)u

ψ

2m

ˆ 2 d x ≤ C ∇ψ ψ2(m−1) e2αu d x ˆ

≤C

Ω

Ω

∇ψ 2(2α+1) d x

1 2α+1

142

ˆ Ω

ψ

(2α+1)u 2(m−1) 2α+1 2α

e

dx

2α 2α+1

.

Chapter 6. Liouville theorems for stable solutions Notice that for m ≥ 5 and α ∈ (0, 2), there holds (m − 1) 2α+1 ≥ 2m, hence α 2α+1

ψ(m−1) α ≤ ψ2m and (6.7) follows. We are now ready for the proof of the theorem. Proof of Theorem 6.2.1. Assume by contradiction that equation (6.6) admits a stable solution for some N ≤ 9. Fix an integer m ≥ 5 and choose α ∈ (0, 2) such that N − 2(2α + 1) < 0. This is always possible since N ≤ 9. Consider a standard cutoff φR , defined for R > 0 and x ∈ RN by φR (x) = φ(|x| /R), where φ ∈ Cc1 (R) is such that 0 ≤ φ ≤ 1 and 1 if |t| ≤ 1, φ(t) = (6.10) 0 if |t| ≥ 2. Apply (6.7) with ψ = φR and Ω = RN to get ˆ e(2α+1)u d x ≤ CRN −2(2α+1) , BR

where C > 0 is independent of R. ´ (2α+1)u d x = 0, a contradiction. RN e

Letting R → +∞, we obtain

Exercise 6.2.1 ([97]) In this exercise, we classify stable solutions to the LaneEmden equation −∆u = |u| p−1 u in RN . (6.11) Using the capacitary estimate in Proposition 5.3.1, prove that if u ∈ C 2 (RN ) is a stable solution to (6.11), with 1 < p < pc (N ), where pc (N ) is given by (4.16), then u ≡ 0.

6.2.2

Dimension N = 2

Is Theorem 6.2.1 true for general nonlinearities? In addition to stability, we must impose further restrictions on u in order to obtain a Liouville theorem for −∆u = f (u) in RN . (6.12) Indeed, in the simple case f ≡ 0, all solutions are stable, but the Liouville theorem fails without assuming, say, that u is bounded. This being said, we have the following statement. Definition 6.2.1 A function u : RN → R is said to be one-dimensional if up to a rotation of space, u is a function of x N only, that is, if there exists τ ∈ SN −1 and g : R → R such that u(x) = g(τ · x) 143

for all x ∈ RN .

Stable solutions of elliptic PDEs Theorem 6.2.2 ([99]) Let f ∈ C 1 (R) and let u ∈ C 2 (R2 ) be a stable solution to (6.12) such that |∇u| ∈ L ∞ (R2 ). Then, u is one-dimensional. Remark 6.2.2 Theorem 6.2.2 remains valid if f is only assumed to be locally Lipschitz continuous (but the proof is quite different). See [99]. A difficult open problem is to decide whether Theorem 6.2.2 still holds in dimension N = 3. Proof. Since |∇u| is bounded and N = 2, there exists a constant C > 0 independent of R > 0 such that ˆ (6.13) |∇u|2 d x ≤ CR2 . BR

Since u is stable, there exists a solution v > 0 to the linearized equation −∆v = f 0 (u)v

in R2 .

(6.14)

Note that ∂ u/∂ x j also solves (6.14). Now recall that the principal eigenvalue of an elliptic operator on a bounded domain is simple. By analogy, we can hope that ∂ u/∂ x j is a constant multiple of v. Next, we prove that this is indeed the case. Let 1 ∂u (6.15) σj = v ∂ xj for j = 1, 2. Then, since v and ∂ u/∂ x j both solve the linearized equation (6.14), it follows that −∇ · v 2 ∇σ j = 0 in R2 . (6.16) ∞ Proposition 6.2.2 ([17], [7]) Let N ≥ 1 and let v ∈ L loc (RN ) denote a positive 1 function. Suppose that σ ∈ H loc (RN ) satisfies

−σ∇ · v 2 ∇σ ≤ 0

in D 0 (RN ).

Assume that there exists a constant C > 0 such that for every R > 1, ˆ v 2 σ2 d x ≤ CR2 . BR

Then, σ is constant. 144

(6.17)

(6.18)

Chapter 6. Liouville theorems for stable solutions Proof. Take a standard cutoff function ϕ0 such that ϕ0 = 1 on B1 and ϕ0 = 0 on RN \ B2 . Test (6.17) with ϕ 2 , where ϕ(x) = ϕ0 (x/R). We obtain

ˆ

ˆ 2

ϕ v |∇σ| d x ≤ −2 2 2

RN

ˆ

RN

ϕv 2 σ∇ϕ∇σ d x 2

ϕ v |∇σ| d x 2 2

≤2

[R<|x|<2R]

1/2 ˆ RN

ˆ

≤C

2 v σ ∇ϕ d x 2

2

1/2

[R<|x|<2R]

ϕ 2 v 2 |∇σ|2 d x

1

1/2

ˆ

R2

1/2 v 2 σ2 d x

.

B2R

By (6.18), it follows that

ˆ

ˆ

RN

ϕ 2 v 2 |∇σ|2 d x ≤ C

In particular,

´ RN

1/2

[R<|x|<2R]

ϕ 2 v 2 |∇σ|2 d x

.

(6.19)

ϕ 2 v 2 |∇σ|2 d x ≤ C 2 and letting R → +∞, we obtain that ˆ v 2 |∇σ|2 d x < +∞. RN

Hence, the right-hand side of (6.19) converges to zero as R → +∞ and so, ˆ v 2 |∇σ|2 d x = 0. RN

Since v is positive, we conclude that σ is constant. Thanks to (6.13), we may apply Proposition 6.2.2 to σ j given by (6.15), and deduce that σ j is constant, that is, there exists a constant C j such that ∂u ∂ xj

= C j v.

In particular, the gradient of u points in a fixed direction, that is, u is onedimensional.

6.2.3

Dimensions N = 3, 4

In higher dimensions, only partial information is currently available. We present here a Liouville theorem that applies to any positive nonlinearity. 145

Stable solutions of elliptic PDEs Theorem 6.2.3 ([88]) Assume that f ∈ C 1 (R), f ≥ 0 and 1 ≤ N ≤ 4. Assume that u ∈ C 2 (RN ) is a bounded, stable solution to (6.12). Then, u is constant. Proof. For R > 0, let BR denote the ball of radius R centered at the origin. We begin by proving that there exists a constant C > 0 independent of R > 0 such that ˆ |∇u|2 d x ≤ CRN −2 . (6.20) BR

Let M ≥ kuk∞ , ϕ ∈ Cc2 (RN ), ϕ ≥ 0, and multiply (6.12) by (u − M )ϕ: ˆ ˆ −∆u(u − M )ϕ d x = f (u)(u − M )ϕ d x. RN

RN

Integrating by parts and recalling that f ≥ 0, it follows that ˆ ˆ ˆ 2 |∇u| ϕ d x + (u − M )∇u∇ϕ d x = f (u)(u − M )ϕ d x RN

RN

RN

≤ 0, hence, ˆ

ˆ |∇u| ϕ d x ≤ −

1

2

RN

ˆ

RN

≤ 2M

2

ˆ ∇(u − M ) ∇ϕ d x =

2 RN

2

RN

(u − M )2 2

∆ϕ d x

∆ϕ d x.

Let ϕ0 denote any nonnegative test function such that ϕ0 = 1 on B1 and apply the above inequality with ϕ(x) = ϕ0 (x/R). We obtain (6.20). Since u is stable, there exists a solution v > 0 of the linearized equation −∆v = f 0 (u)v Let σ j =

1 ∂u v ∂ xj

in RN .

(6.21)

for j = 1, . . . , N . Then, since v and ∂ u/∂ x j both solve the

linearized equation (6.21), it follows that −∇ · v 2 ∇σ j = 0

in RN .

(6.22)

Apply Proposition 6.2.2 to (6.22). By (6.20), we deduce that if N ≤ 4, then σ j is constant, that is, there exists a constant C j such that ∂u ∂ xj

= C j v.

146

Chapter 6. Liouville theorems for stable solutions In particular, the gradient of u points in a fixed direction, that is, u is onedimensional and solves −u00 = f (u)

in R.

Since f ≥ 0 and u is bounded, this is possible only if u is constant and f (u) = 0.

6.3

Classifying solutions that are stable outside a compact set

Much less is known about the classification of solutions that are stable outside a compact set. Power nonlinearities were first treated by [97], followed by the exponential nonlinearity by [71] and more general convex nonlinearities by [88]. In this section, we return to the Liouville equation, Equation (6.6). The results are substantially different in the cases N = 2 (critical), 3 ≤ N ≤ 9 (supercritical range) and N ≥ 10 (twice supercritical).

6.3.1

The critical case

Theorem 6.3.1 ([95]) Let N = 2. Let u ∈ C 2 (R2 ) denote a solution to (6.6). Then, u is stable outside a compact set if and only if u is of the form     32λ2  u(x) = ln  2 2  ,  4 + λ2 x − x 0

λ > 0,

x 0 ∈ R2 .

(6.23)

Proof. Step 1. Any function given by (6.23) is stable outside a large ball of R2 . Clearly, it is not enough to prove this claim for x 0 = 0 and λ > 0. To this end, we observe that there exists R = R(λ) > 1 such that eu(x) ≤

1 4 |x|2 ln2 (|x|)

for |x| > R, and that for all ψ ∈ Cc1 (R2 \ B(0, R)) we have ˆ 2 ψ2 ∇ψ − d x ≥ 0. 4 |x|2 ln2 (|x|) [|x|>R] 147

Stable solutions of elliptic PDEs The latter follows immediately from the fact that the function ϕ(x) = ln1/2 (|x|) ∈ C 2 (R2 \ B(0, 1)) is a positive solution to −∆ϕ − 4|x|2 ln1 2 (|x|) ϕ = 0 in R2 \ B(0, 1). Combining these two properties, we obtain the desired conclusion. Step 2. Conversely, if u is stable outside a compact set, there exists R0 > 0 such that u is stable in Ω = RN \ B(0, R0 ). For every R > R0 + 3 and every x ∈ R2 , consider a function ψR ∈ Cc1 (R2 \ B(0, R0 )) satisfying ψR (x) =

ξ(x) φR (x)

if |x| ≤ R0 + 3, if |x| ≥ R0 + 3,

(6.24)

where φR (x) = φ(|x| /R), with φ ∈ Cc1 (R), 0 ≤ φ ≤ 1, satisfying (6.10), and where ξ is any function belonging to C 1 (R2 ) such that 0 ≤ ξ ≤ 1 in R2 , ξ ≡ 0 inside the ball B(0, R0 +1) and ξ ≡ 1 outside the ball B(0, R0 +2). u is stable on the support of ψR , so Q u (ψR ) ≥ 0. In particular, letting R → +∞, we deduce that ˆ eu d x < +∞. R2

We then conclude that u is of the form (6.23), using Theorem 6.3.2 below. Theorem 6.3.2 ([58]) Let N = 2. Let u ∈ C 2 (R2 ) denote a solution to (6.6). If ˆ eu d x < +∞, (6.25) R2

then u is of the form (6.23). The proof of Theorem 6.3.2 uses the moving-planes device and the following information on the asymptotics of solutions. Lemma 6.3.1 ([58]) Assume that u ∈ C 2 (R2 ) is a solution to (6.6) satisfying the integrability condition (6.25). Then, lim

|x|→+∞

u(x) ln |x|

=−

1 2π

Proof. Step 1 ([30]). u is bounded above. 148

ˆ R2

eu d x ≤ −4.

Chapter 6. Liouville theorems for stable solutions Since u satisfies the integral condition (6.25), there exists R > 0 such that for every |x 0 | ≥ R, ˆ eu d x ≤ π/2.

(6.26)

B(x 0 ,1)

In the ball B(x 0 , 1), u = v + w, where v, w, respectively solve −∆v = eu in B(x 0 , 1), v = 0 on ∂ B(x 0 , 1),

(6.27)

and

−∆w = 0 in B(x 0 , 1), w = u on ∂ B(x 0 , 1).

(6.28)

We begin by estimating w. Since u is superharmonic, w ≤ u in B(x 0 , 1). By the mean-value formula (Proposition A.1.2), for any x ∈ B(x 0 , 1/2), w(x) = B(x,1/2)

eu d y ≤ 2,

udy ≤

w dy ≤ B(x,1/2)

(6.29)

B(x,1/2)

where we used (6.26). Next, we estimate v. Using the maximum principle (Proposition A.2.2) and the expression of the fundamental solution of the Laplace operator (A.7), we first note that v ≤ v˜ in B(x 0 , 1), where ˆ 1 v˜(x) = − ln |x − y| f ( y) d y, 2π R2 and f ( y) = eu( y) for y ∈ B(x 0 , 1), f ( y) = 0 otherwise. Now, ˆ ˆ 2v e dx ≤ e2˜v d x = B(x 0 ,1) B(x ,1) ˆ ˆ 0 1 ln |x − y| f ( y) d y d x = exp − π R2 B(x 0 ,1) ˆ ˆ k f k L 1 (R2 ) f ( y) exp − ln |x − y| d y d x. π k f k L 1 (R2 ) B(x 0 ,1) B(x 0 ,1) Set δ = k f k L 1 (R2 ) /π and note that δ ≤ 1/2, by (6.26). Using Jensen’s inequality with probability measure dµ = f /k f k L 1 (R2 ) d y, it follows that ˆ ˆ f ( y) 2v e dx ≤ |x − y|−δ d y dx k f k L 1 (R2 ) B(x 0 ,1) B(x 0 ,1)×B(x 0 ,1) ˆ ≤ |x|−δ d x < +∞. (6.30) B(0,2)

149

Stable solutions of elliptic PDEs Combining (6.29) and (6.30),´ we deduce that there exists ´ a constant C > 0 independent of x 0 such that B(x 0 ,1/2) e2u d x ≤ C and B(x 0 ,1) v d x ≤ C. By elliptic regularity applied to (6.27), we deduce that v is bounded in B(x 0 , 1/4) by a constant C independent of x 0 . By (6.29), we deduce that u is bounded above in B(x 0 , 1/4) by a constant C independent of x 0 . Step 1 follows. Step 2 ([86]). ˆ eu d x ≥ 8π. (6.31) R2

For t ∈ R, let Ω t = {x ∈ RN : u(x) > t}. Then, ˆ ˆ ˆ u e dx = − ∆u = |∇u| dσ. Ωt

Ωt

∂ Ωt

Using the coarea formula, we have for almost every t ˆ dσ d . − Ω t = dt ∂ Ω t |∇u| Applying the Cauchy-Schwarz inequality and the isoperimetric inequality, we also have ˆ ˆ 2 dσ |∇u| dσ ≥ ∂ Ω t ≥ 4π Ω t . ∂ Ω t |∇u| ∂ Ω t Hence,

ˆ

d − Ω t dt

≥ 4π Ω t .

u

e dx Ωt

And so, using the coarea formula once more, d dt

ˆ

2

= 2e t

eu d x Ωt

ˆ d u t Ω e d x ≤ −8πe Ωt . t dt Ωt

Integrating from −∞ to +∞ gives ˆ −

u

ˆ

2

e dx R2

≤ −8π

which implies (6.31). Step 3. lim

|x|→+∞

w(x) ln |x|

=

1 2π

150

eu d x, R2

ˆ eu d x, R2

Chapter 6. Liouville theorems for stable solutions where w(x) =

1

ˆ

2π

R2

(ln x − y − ln y )eu( y) d y,

for x ∈ R2 .

´ Since eu is bounded and R2 eu d x < +∞, w is well defined for all x ∈ R2 . We want to show that ˆ ln |x − y| − ln | y| − ln |x| u I := e d y → 0, as |x| → +∞. ln |x| R2 Given " > 0, choose R > 0 so large that ˆ eu d y < ". | y|>R

Write I = I1 + I2 + I3 , where I1 , I2 , and I3 are the integrals on the respective regions D1 = { y ∈ R2 : | y − x| ≤ 1}, D2 = { y ∈ R2 : | y − x| > 1 and | y| ≤ R}, D3 = { y ∈ R2 : | y − x| > 1 and | y| > R}. I1 is estimated as follows ˆ ˆ 1 u eu d y | ln |x − y||e d y + C |I1 | ≤ ln |x| |x− y|≤1 |x− y|≤1 ˆ ˆ u ke k L ∞ (R2 ) ≤ | ln |x|| d x + C eu d y → 0, ln |x| | y|≥|x|−1 B(0,1)

as |x| → +∞.

To ´ estimate I2 , we note that | ln |x − y| − ln |x|| ≤ C for | y| ≤ R and | y|≤R | ln | y|| d y < +∞. We deduce that |I2 | ≤

keu k L ∞ (R2 ) ln |x|

ˆ ln |x − y| − ln |x| + | ln | y|| d y → 0, as |x| → +∞. | y|≤R

To estimate I3 , we note that for |x − y| > 1 and | y| > R, ln |x − y| − ln | y| − ln |x| ≤ C. ln |x| 151

Stable solutions of elliptic PDEs Hence,

ˆ eu d y ≤ C".

|I3 | ≤ C | y|>R

We deduce that lim sup|x|→+∞ |I| ≤ C". Since " > 0 is arbitrary, Step 3 follows. Step 4. End of proof. Clearly, ∆w = eu in R2 . So, the function v = u + w is harmonic and by Steps 1 and 3, v(x) ≤ C + C1 ln(|x| + 1), for some constants C, C1 > 0. This implies |v(x)| ≤ |v(x) − C + C1 ln(|x| + 1)| + C + C1 ln(|x| + 1) ≤ 2(C + C1 ln(|x| + 1)) − v(x). Since v is harmonic, so is ∂i v. Using the mean-value formula (Proposition A.2.2), we deduce that for any R > 0, ˆ C 1 |v| dσ |∂i v(x)| = ∂i v d y = vni dσ ≤ B ∂ B(x,R) R ∂ B(x,R) B(x,R) R

≤ =

C R C R

≤C

∂ B(x,R)

2(C + C1 ln(| y| + 1)) − v dσ( y)

∂ B(x,R)

2(C + C1 ln(| y| + 1)) dσ( y) − v(x)

(1 + ln(|x| + R + 1)) − v(x)

, R where we used the mean-value formula again in the penultimate line. Letting R → +∞, we deduce that v is constant, so that ˆ 1 u(x) lim = eu d x. |x|→+∞ ln |x| 2π R2 Applying (6.31), the lemma follows.

Proof of Theorem 6.3.2. By Lemma 6.3.1, u achieves its maximum at some point x 0 ∈ R2 . Replacing u by u(x + x 0 ) if needed, we may always assume that x 0 = 0. Our goal is to apply the moving-planes device to prove that u is radially symmetric. Since for given α ∈ R, the solution to the ODE  1 0 00 u  −u − r u = e , u0 (0) = 0,  u(0) = α, 152

Chapter 6. Liouville theorems for stable solutions is unique, (6.23) will follow. So, we are left with proving that u is radially symmetric. It suffices to prove that u is symmetric in any given direction. Choose one. Rotating space if necessary, we may work with the x 1 direction and show only that u(x 1 , x 2 ) ≥ u(−x 1 , x 2 ),

whenever x 1 ≤ 0.

(6.32)

Given λ > 0, set Tλ = {x = (x 1 , x 2 ) ∈ R2 : x 1 = λ}

Σλ = {x ∈ R2 : x 1 < λ}.

and

For x = (x 1 , x 2 ) ∈ R2 , let x λ denote the reflection of x with respect to the plane Tλ , that is, x λ = (2λ − x 1 , x 2 ). Step 1. There exists R1 > 0 such that u(x) > u(x λ ) in Σλ for λ ≥ R1 . To see this, given λ > 0, we note that vλ (x) = u(x λ ) − u(x) satisfies −∆vλ − Vλ (x)vλ = 0

in Σλ ,

(6.33)

´1 where Vλ (x) = 0 e tu(x λ )+(1−t)u(x) d t. Since |x λ | > |x| for all x ∈ Σλ , Lemma 6.3.1 implies that for some R0 > 1 independent of λ, there holds Vλ (x) ≤

1 |x|(1 + |x| ) ln(|x| − 1) 2

,

in Σλ \ BR0 .

In particular, w(x) := ln(|x| − 1) satisfies ( −∆w − Vλ (x)w ≥ 0

in Σλ \ BR0 ,

w>0

in Σλ \ BR0 .

(6.34)

(6.35)

By Lemma 6.3.1, lim|x|→+∞ u(x) = −∞. Then there exists R1 > R0 such that max|x|≥R1 u < min|x|≤R0 u. And so, for λ ≥ R1 , vλ > 0

in BR0 .

(6.36)

It remains to prove that vλ ≥ 0 in Σλ \ BR0 . To this end, we observe that 

−∆vλ − Vλ (x)vλ = 0

in Σλ \ BR0 ,

 

vλ > 0 vλ (x) lim = 0. |x|→+∞ w(x)

on ∂ BR0 .

 

153

(6.37)

Stable solutions of elliptic PDEs The first equation follows from (6.33), the second from (6.36), and the third from the expression of w and Lemma 6.3.1. Using the maximum principle (Proposition A.7.2) and the strong maximum principle (Corollary A.5.1), Step 1 follows. Step 2. For λ ∈ Λ = {λ > 0 : ∀ µ ≥ λ, u(x) > u(x µ ) in Σµ }, there holds ∂u ∂ x1

<0

in R2 \ Σλ .

Indeed, for µ ≥ λ, vµ solves  −∆vµ = Vµ (x)vµ ≥ 0  vµ > 0  vµ = 0.

(6.38)

in Σµ , in Σµ , on Tµ = ∂ Σµ .

By the boundary point lemma (Proposition A.4.1), we deduce that 2

∂u ∂ x1

(µ, x 2 ) =

∂ vµ ∂ x1

(µ, x 2 ) < 0,

as desired. Step 3. inf Λ = 0. Assume by contradiction that λ0 = inf Λ > 0. By continuity of u, vλ0 ≥ 0 in Σλ0 and by the strong maximum principle, either vλ0 ≡ 0, or vλ0 > 0 in Σλ0 . In the former case, the function v(x) = u(λ0 + x) is even. Using Step 2, we deduce that ∂ u/∂ x 1 (0) = −∂ u/∂ x 1 (2λ0 , 0) > 0. This contradicts our initial assumption that 0 is a point of maximum of u. So, vλ0 > 0 in Σλ0 . By definition of λ0 , there exists a sequence λn % λ0 and a sequence x n ∈ Σλn such that vλn (x n ) ≤ 0. It follows that vλn ( yn ) ≤ 0 for some sequence yn ∈ Σλn ∩ BR0 . Otherwise, we would have vλn > 0 in Σλn ∩ BR0 and by the strong maximum principle applied to (6.37) with λ = λn , we would deduce that vλn > 0 in Σλn , contradicting vλn (x n ) ≤ 0. Since vλn ( yn ) ≤ 0, there exists a point zn in the segment joining yn to ynλn such that ∂ u/∂ x 1 (zn ) ≥ 0. Finally, since ( yn ) is bounded, so is (zn ) and a subsequence znk → z0 , with z0 ∈ Tλ0 , vλ0 (z0 ) = 0, and ∂ u/∂ x 1 (z0 ) ≥ 0. This contradicts the boundary point lemma. Equation (6.32) follows.

6.3.2

The supercritical range

Theorem 6.3.3 ([71]) Let 3 ≤ N ≤ 9. There is no solution u ∈ C 2 (RN ) of (6.6), which is stable outside a compact set. 154

Chapter 6. Liouville theorems for stable solutions Remark 6.3.1 As observed in Remark 6.2.1, the above theorem ceases to hold for N ≥ 10. Proof. Take a standard cutoff function φ ∈ Cc1 (R), that is, 0 ≤ φ ≤ 1 in R and (6.10) holds. For s > 0, also let θs ∈ Cc1 (R), such that 0 ≤ θs ≤ 1 in R and θs (t) =

if |t| ≤ s + 1, if |t| ≥ s + 2.

0 1

(6.39)

The proof of the theorem is by contradiction and proceeds in four steps. Let us suppose that the equation admits a C 2 solution, which is stable outside a compact set. Then, Step 1. There exists R0 > 0 such that • For every α ∈ (0, 2) and every r > R0 + 3 we have

ˆ [R0 +2<|x|
e(2α+1)u d x ≤ A + Br N −2(2α+1),

(6.40)

where A, B are positive constants depending only on α, N , R0 , but not on r. • For every α ∈ (0, 2) and every open ball B = B( y, 2R) such that B ⊂ {x ∈ RN : |x| > R0 }, we have

ˆ e(2α+1)u d x ≤ CRN −2(2α+1) ,

(6.41)

B

where C is a positive constant depending only on α, N , R0 , but neither on R nor y. To prove this, since u is stable outside a compact set, there exists R0 > 0 such that Proposition 6.2.1 holds in Ω = RN \ B(0, R0 ). We fix m = 5 and, for every r > R0 + 3, consider the test function ξ r ∈ Cc1 (RN ) defined by 

R0 (|x|)  θ |x| ξ r (x) = φ r

155

if |x| ≤ R0 + 3, if |x| ≥ R0 + 3.

(6.42)

Stable solutions of elliptic PDEs Applying Proposition 6.2.1 with ψ = ξ r yields

ˆ

ˆ e

(2α+1)u

[R0 +2<|x|
dx ≤

RN \B(0,R

ˆ

≤C ˆ ≤C

[|x|≤R0 +2]

RN

0)

e(2α+1)u d x ≤

∇ξ 2(2α+1) d x ≤ r

∇θ 2(2α+1) d x + R0

ˆ [r≤|x|≤2r]

∇ξ 2(2α+1) d x r

≤ C1 + C2 r N −2(2α+1) , for all r > R0 + 3. Equation (6.40) follows. The (6.41) is obtained estimate | x− y | in Proposition 6.2.1. similarly by using the test function ψR, y (x) = φ R Step 2. Let η > 0. Then, there exists R1 = R1 (N , η, u) > 0 such that ˆ N N e 2u dx ≤ η2 .

(6.43)

[|x|>R1 ]

Let α1 =

N −2 4

∈ (0, 2). By (6.40) we infer that, for all r > R0 + 3,

ˆ e [R0 +2<|x|
So,

´

N 2

ˆ u

dx =

[R0 +2<|x|
e(2α1 +1)u d x ≤ A + Br N −2(2α1 +1) .

N

[|x|>R0 +2]

e 2 u d x < +∞, which immediately yields (6.43).

Step 3. lim |x|2 eu(x) = 0.

|x|→+∞

N Set ε = 1/10 and observe that 2−ε ∈ (1, 5). Thus, there exists α2 = α2 (N ) ∈ N (0, 2) such that 2α2 + 1 = 2−ε . Here we have used the assumption 3 ≤ N ≤ 9. Next, we fix η > 0 and observe that w = eu satisfies

−∆w − eu w ≤ 0

in B( y, 2R).

The Serrin-Trudinger inequality (Theorem B.4.2) for positive subsolutions to the equation −∆w − eu w = 0 asserts that, for any t > 1, N

kwk L ∞ (B( y,R)) ≤ CST R− t kwk L t (B( y,2R)) , 156

(6.44)

Chapter 6. Liouville theorems for stable solutions where CS T is a positive constant depending on N , t and Rε keu k

N

L 2−ε (B( y,2R)

.

N y > In order to apply the above result, we consider points y ∈ R such that 10R1 and set R = y /4, t = N /2 > 1. Here, R1 > R0 is defined by (6.43) of Step 2. This choice yields B( y, 2R) ⊂ {x ∈ RN : |x| > R1 } ⊂ {x ∈ RN : |x| > R0 }. In addition, (6.43) holds. So, Rε keu k

N L 2−ε (B( y,2R)

= Rε

ˆ

= Rε

ˆ e

N u 2−ε

2−ε N

=

B( y,2R)

e(2α2 +1)u

2−ε N

2−ε ≤ Rε CRN −2(2α2 +1) N ≤ C 0 Rε R2−ε R−2 = C 0 ,

B( y,2R)

where in the latter we have used (6.41). This proves that the constant CST is independent of both y and R. Actually, it depends only on N and R0 . Now, using t = N /2 in (6.44) and Step 2, we are led to −2 −2 eu( y) ≤ C R−2 keu k N /2 keu k N /2 η, ST L (B( y,2R)) ≤ 16CST y L (B( y,2R)) ≤ C2 y which proves Step 3. Step 4. End of the proof. By Step 3, there exists R2 > 0 such that the function v = v(|x|), defined as the mean-value of the solution u over spheres of radii |x| > 0, satisfies 1 for all r > R2 . −∆v ≤ 2 2r Hence, the radial function v verifies v 0 (r) ≥

C(N ) r N −1

−

1 2(N − 2)r

and thus v 0 (r) ≥ −

for all r > R2

1

for all r > R3 , r for some R3 > R2 . Integrating the latter and taking the exponential, we get r 2 e v(r) ≥ C r

for all r > R3 , 157

(6.45)

Stable solutions of elliptic PDEs where C is a positive constant independent of r. Finally, we observe that (6.45) contradicts Step 3. Indeed, by Jensen’s inequality, we have for all r > R3 , 2 u(x)

max |x| e |x|=r

= r max e 2

|x|=r

u(x)

≥

ˆ

r2 |[|x| = r]|

[|x|=r]

eu dσ ≥ r 2 e v(r) ≥ cr,

which clearly contradicts the conclusion of Step 3.

6.3.3

Flat nonlinearities

The previous classification results remain valid for a greater class of nonlinearities (see [97] for the Lane-Emden nonlinearity, as well as [88] for more general results). However, the following example teaches us that no Liouville theorem can hold for general f (even in the restricted class of convex functions), if the solution u is only assumed to be stable outside a compact set. Example 6.3.1 ([89]) Let N ≥ 3 and 1 0 such that the equation −∆u = [(u − β)+ ] p

N +2 . N −2

in

For every R > 0 there

RN

(6.46)

admits a solution u = uR ∈ C 2 (RN ) satisfying: (i) u is positive, radially symmetric, and strictly radially decreasing. (ii) u(R) = β, lim r→+∞ u(r) = 0, where r = |x|. (iii) The Morse index of u is finite and positive. −

2

Furthermore one has that β(R, p, N ) = β(1, p, N )R p−1 where β(1, p, N ) = ´ 1 p 1 φ1 (s)s N −1 ds > 0 and φ1 is the unique positive radial solution to N −2 0 ¨

p

−∆φ1 = φ1 φ1 (1) = 0.

in

B1 ,

In particular, for every β > 0, Equation (6.46) admits a solution u satisfying the above properties. 158

Chapter 6. Liouville theorems for stable solutions Proof. Fix R > 0. Since N ≥ 3 and 1 0 in BR ,   0

φR < 0

Now we consider the function ¨ vR = vR (r) =

in BR \ {0}.

φR (r) αr

2−N

(6.47)

for −β

0 ≤ r ≤ R, for

r ≥ R,

´R p with α = N 1−2 0 φR (s)s N −1 ds > 0 and β = αR2−N . Set hR (r) = αr 2−N − β. The above function vR is clearly continuous, nonnegative, radially symmetric, strictly radially decreasing, and satisfies lim r→+∞ vR (r) = −β < 0. Furthermore vR belongs to C 2 (RN ) and solves the equation −∆v = (v + ) p

in RN .

(6.48)

To check that vR is of class C 2 we observe that, by integrating the ODE satisfied by φR , we get ˆ R 0 0 p 1−N φR (R) = −R φR (s)s N −1 ds = (2 − N )αR1−N = hR (R). 0

Moreover, 00

p

φR (R) = −φR (R) −

(N − 1) R

0

00

φR (R) = 0 + (1 − N )(2 − N )αR−N = h (R).

Therefore vR is in C 2 (RN ) and solves Equation (6.48). The desired function u is then obtained by setting u := uR = vR + β. Then, u belongs to C 2 (RN ), solves Equation (6.46) and, by making use of the properties of vR , it satisfies (i) and (ii). To prove (iii) we first show that the Morse index of uR is at least 1. To this end we multiply Equation (6.46) by the function vR+ ∈ Cc0,1 (RN ) (note that vR+ = (u − β)+ > 0) and integrate by parts to obtain: ˆ ˆ ˆ + 2 + |∇vR | d x = ∇u∇vR d x = [(u − β)+ ] p+1 d x = N N N R R R ˆ ˆ + p−1 + 2 = [(u − β) ] (vR ) d x < p[(u − β)+ ] p−1 (vR+ )2 d x. (6.49) RN

RN

159

Stable solutions of elliptic PDEs Hence,

ˆ Q u (vR+ )

=

ˆ

RN

|∇vR+ |2

dx −

RN

p[(u − β)+ ] p−1 (vR+ )2 d x < 0.

The latter proves that the Morse index of u is at least 1. To prove that u has finite Morse index, we work as follows. Since ind RN (u) = sup ind Bn (u),

(6.50)

n∈N?

it is enough to bound, independently of n, each one of the quantities ind Bn (u). To do so, we first observe that ind Bn (u) is the number of negative Dirichlet eigenvalues of the operator −∆ − p[(u − β)+ ] p−1 in Bn and next we invoke the Cwikel-Lieb-Rozenbljum formula (see [64], [142], [187], as well as [141] for the version of the formula used here). This formula says that the number of negative Dirichlet eigenvalues of the operator −∆ − V in Bn is bounded by ˆ N 4e N 2 −1 (ωN −1 ) V 2 d x, (6.51) N (N − 2) Bn N

provided N ≥ 3 and V is a nonnegative potential belonging to L 2 (Bn ). The desired conclusion then follows by observing that ˆ ˆ N (p−1) N (p−1) N N + p 2 φR 2 ∀ n>R p 2 [(u − β) ] 2 = BR

Bn

= C = C(p, N , φR ) < +∞, (6.52) where C is independent of n since (u − β)+ is supported in BR . −

2

To conclude the proof we observe that φR (r) = R p−1 φ1 (R−1 r) and that ´R p − 2 β = β(R, p, N ) = R2−N N 1−2 0 φR (r)r N −1 d r = β(1, p, N )R p−1 by making the change of variable s = Rr. Remark 6.3.2 (i) The function f (t) = [(t − β)+ ] p , β > 0 is nonnegative, nondecreasing, convex, and of class C 1 . It is of class C 2 for p > 2. The latter is always possible in dimension N = 3, 4, 5 (recall that p is subcritical). (ii) Let β > 0. Any solution to  −∆w = [(w − β)+ ] p in RN ,  w>0 in RN , (6.53)  lim w(x) = 0, |x|→+∞

160

Chapter 6. Liouville theorems for stable solutions must be one of the functions uR built in Example 6.3.1 (up to translation). Proof. Indeed, (up to translation) by a theorem of Gidas, Ni, and Nirenberg, w must be radially symmetric and strictly radially decreasing. Let R = R(β) > 0, the only value for which w(R) = β (such a value always exists since a bounded nonconstant solution to (6.53) must satisfy: sup w > β, otherwise it would be harmonic and hence constant). Now, it is clear that w = uR(β) . Indeed, w − β is radial and harmonic for r ≥ R(β) and hence w − β = Ar 2−N + B, where A and B are constants. The assumption lim|x|→+∞ w(x) = 0 implies B = β and w = Ar 2−N for r ≥ R(β) and A > 0. In addition, w − β is a solution to (6.47) and thus it must be equal to φR(β) on BR(β) . Combining this information and 0 using the continuity of w and w , we have w = uR(β) . (iii) In particular, for every β > 0 problem (6.53) admits a unique solution (up to translation). This solution necessarily coincides with uR(β) . Clearly the value R(β) can be explicitly calculated by using the formula for β given in Example 6.3.1. (iv) Note that any solution to (6.46) (not necessarily positive) converging uniformly to some constant γ < β is automatically stable outside a compact set of 0 RN . This follows by observing that f (and thus f ) is zero on (−∞, β). Arguing as in the above proof one can prove that such u must have finite Morse index.

161

Chapter 7 A conjecture of De Giorgi 7.1

Statement of the conjecture

In this chapter, we discuss a celebrated conjecture, due to De Giorgi ([81]): Conjecture 7.1.1 ([81]) Let N ≥ 1. Let u ∈ C 2 (RN , [−1, 1]) satisfy −∆u = u − u3

∂u

and

∂ xN

>0

in all of RN .

(7.1)

Then, the level sets of u are hyperplanes, at least if N ≤ 8. Note that if the level sets of u are hyperplanes, then they must be parallel (no two level sets can cross). So the conclusion of the conjecture is equivalent to requesting that u is one-dimensional (that is, u is a function of one variable only; see Definition 6.2.1). One-dimensional solutions of (7.1) are easily classified, as the following proposition demonstrates. Proposition 7.1.1 Let N ≥ 1 and u ∈ C 2 (RN , [−1, 1]). Assume that u is a onedimensional solution to (7.1). Then, there exists a unit vector τ ∈ S N −1 and a constant α ∈ R such that u(x) = g0 (τ · x + α) , p where g0 (s) = tanh(s/ 2).

for all x ∈ RN ,

Proof. Since u is one-dimensional, there exists a vector τ ∈ S tion h ∈ C 2 (R) such that u(x) = h(τ · x). In addition, −h00 = h − h3

and 163

h0 > 0

in R.

N −1

and a func(7.2)

Stable solutions of elliptic PDEs Note that since h is increasing, there exists constants m− , m+ ∈ [−1, 1] such that lim±∞ h = m± . In particular, lim inf±∞ h0 = 0. By the mean-value theorem, we deduce that lim inf±∞ h00 = 0. Using (7.2), we deduce that m± − m3± = 0.

(7.3)

Multiply (7.2) by 2h0 and integrate: there exists a constant c ∈ R such that (h0 )2 =

2 1 1 − h2 − c 2

(7.4)

2 and so c = 12 1 − m2± . In particular, m2+ = m2− and since h is increasing, it follows, using (7.3), that m− = −1, m+ = 1 and so c = 0. Let m = h(0). By direct integration, we see that the unique increasing solution to (7.4) such that h(0) = m is given by ˆ h(t) p ds = t. 2 1 − s2 m One can easily check that h(t) = g0 (t +α), where α ∈ R is such that g0 (α) = m is the desired solution.

7.2

Motivation for the conjecture

In this section, we give a possible motivation for the conjecture of De Giorgi 7.1.1. We warn the reader that full proofs will not be given; a thorough investigation would be beyond the scope of this book. This should however provide us enough insight on the conjecture to move on to its proof in dimensions N = 2 and N = 3, which, as we are about to discover, uses stability as a central tool.

7.2.1

Phase transition phenomena

The Allen-Cahn equation appearing in (7.1) arises as a crude model in the theory of phase transitions. Consider a pure body, contained in a bounded region Ω of space, which state may change from one (say, thermodynamical) phase to another. To each of these phases, we assign a given value, say u = −1 and u = +1, while the transient state of the body will be assigned a value u ∈ (−1, +1). In the informal discussion that follows, we are interested in the description of the interface between these two states. We argue that such an interface should be “close” to a surface of minimal area. 164

Chapter 7. A conjecture of De Giorgi The system tends to minimize an energy functional, which should favor the states ±1. Take W , a double-well potential having minimal value at u = ±1, for example, 1 W (u) = (1 − u2 )2 . (7.5) 4 We might be tempted ´ at first to model our problem by minimizing an energy of the form ´E(u) = Ω W (u) d x among all density distributions u of prescribed total mass Ω u d x = m. One quickly realizes that any solution to such a minimization problem must be piecewise constant. In addition, there are infinitely many solutions, with no restriction on the interface between the sets [u = −1] and [u = +1]. In particular, there is no way to recover the physically reasonable criterion that the interface has minimal area. Accordingly, an interfacial energy must be added. This can be done by considering the Ginzburg-Landau energy ˆ ˆ 1 2 |∇u| d x + W (u) d x, (7.6) EΩ (u) = 2 Ω Ω which the Euler-Lagrange equation is given by −∆u = u − u3

in Ω.

(7.7)

It can be argued that the interfacial energy should be relatively small, compared to the potential term, that is, we should rather consider the rescaled energy ˆ ˆ 1 " 2 |∇u| d x + W (u) d x. (7.8) E",Ω (u) = 2 Ω " Ω Note that u minimizes EΩ/" if and only if u" (x) := u(x/") minimizes E",Ω . Using Young’s inequality and the coarea formula, there holds E",Ω (u" ) ≥

ˆ p Ω

ˆ 2W (u" ) ∇u" d x =

1

−1

p

2W (s)H N −1 ([u" = s]) ds,

for any u" ∈ C 2 (Ω, [−1, 1]). In addition, the inequality is an equality, provided 1p ∇u = 2W (u" ). " "

(7.9)

So, heuristically, if (7.9) holds and if the level sets [u" = s] are surfaces of minimal area, then u" should be a local minimizer of (7.8). Now, (7.9) implies that ∇u" has constant length along any given level set and so the level sets 165

Stable solutions of elliptic PDEs of u" must be parallel. Assume that the level set Γ = [u" = 0] is a smooth hypersurface and let dΓ denote the distance to Γ. Writing u" (x) = g(dΓ (x)), (7.9) reduces to 1p g0 = 2W (g). " Working as in Proposition 7.1.1, we deduce that g(s) = g0 "s + α for some p α ∈ R, where g0 (s) = tanh(s/ 2). Unfortunately, the above discussion is not completely correct: the level sets of u" need not be surfaces of minimal area. It can be shown however that these level sets are close to a surface of minimal area: for a sequence (u"k ), there 1 exists a set E of minimal perimeter such that u"k → χ E − χ E c in L loc , as "k → 0 (see [156]).

7.2.2

Monotone solutions and global minimizers

In the previous section, we saw how the Allen-Cahn equation (7.7) appears naturally in the description of phase transition phenomena. But why did De Giorgi state his conjecture for monotone solutions? To gain further insight, let us return to the study of minimizers of E",Ω . Recall that u minimizes EΩ/" , given by Equation (7.6), if and only if u" (x) := u(x/") minimizes E",Ω , given by Equation (7.8). So, to find a minimizer of E",Ω for arbitrary " > 0, it suffices to exhibit a function u ∈ C 2 (RN ) that minimizes EΩ/" for all " > 0. Definition 7.2.1 A function u ∈ C 2 (RN ) is said to be a global minimizer of the Ginzburg-Landau energy functional if EΩ (u) ≤ EΩ (u + ϕ) for every ϕ ∈ Cc1 (Ω) and for every bounded domain Ω ⊂ RN , where EΩ is given by (7.6). Global minimizers and monotone solutions are related through the following theorem. Theorem 7.2.1 ([3]) Let N ≥ 1 and let u ∈ C 2 (RN ; [−1, 1]) denote a monotone solution to (7.1). In addition, assume that lim u(x 0 , x N ) = ±1,

x N →±∞

for every x 0 ∈ RN −1 .

Then, u is a global minimizer of EΩ , defined by (7.6). 166

(7.10)

Chapter 7. A conjecture of De Giorgi Remark 7.2.1 Note that the additional requirement (7.10) is compatible with the De Giorgi conjecture (Conjecture 7.1.1). Note also that in (7.10), we do not require the limits to be uniform in the x 0 variable. Proof. We use a foliation argument of Weierstraß (see [122]). Consider the family of functions uτ defined for τ ∈ R by uτ (x) = u(x 0 , x N + τ),

for all x = (x 0 , x N ) ∈ RN .

Since u is monotone, the graphs of the family (uτ )τ form a foliation of RN ×[−1, 1], that is, the graphs are (strictly) ordered and they fill RN ×[−1, 1]. Also observe that by standard elliptic regularity, (uτ ) converges locally uniformly to 1, as τ → +∞. Now fix a bounded domain Ω ⊂ RN and let us prove that u is the unique solution to ¨ −∆v = v − v 3 in Ω, (7.11) v=u on ∂ Ω. The following exercise will then imply that u is a global minimizer. Exercise 7.2.1 • Prove that every solution to (7.11) satisfies −1 < v < 1 in Ω. • Check that there exists an absolute minimizer of the energy (7.6) subject to the boundary condition v = u on ∂ Ω. It remains to be proven that u is the unique solution to (7.11). Assume by contradiction that there exists a solution v 6= u of (7.11). Then, −1 < v < 1 in Ω and so the set Λ = {τ > 0 : v ≤ uτ in Ω} is nonempty. Let τ0 = inf Λ. Since v 6≡ u but v = u on ∂ Ω, there exists a point x 0 ∈ Ω such that v(x 0 ) 6= u(x 0 ). Say v(x 0 ) > u(x 0 ) (the reverse inequality can be treated similarly). In particular, τ0 > 0. In addition, by definition of τ0 , v ≤ u τ0 in Ω, and there exists a point x 1 ∈ Ω such that v(x 1 ) = uτ0 (x 1 ). Since uτ0 ∂ Ω > u|∂ Ω = v|∂ Ω , we deduce that x 1 ∈ Ω. By the strong maximum principle, we must have v ≡ uτ0 , which contradicts uτ0 ∂ Ω > v|∂ Ω . Theorem 7.2.1 can be combined to useful energy estimates that hold for global minimizers. 167

Stable solutions of elliptic PDEs Corollary 7.2.1 Let N ≥ 1. Assume that u ∈ C 2 (RN ; [−1, 1]) is a global minimizer of the Ginzburg-Landau energy (7.6). Then, there exists a constant C > 0, depending on N only, such that given any R > 1, ˆ 1 1 2 2 2 EBR (u) = |∇u| + (1 − u ) d x ≤ CRN −1 . (7.12) 2 4 BR Proof. Given R > 1, take a cutoff function ϕR ∈ Cc1 (BR+1 ) such that ϕR ≡ 1 in BR , 0 ≤ ϕR ≤ 1 and ∇ϕR ≤ C. Let v = ϕR + u(1 − ϕR ). Since u is a global minimizer, ˆ 1 1 2 2 2 |∇v| + (1 − v ) d x ≤ CRN −1 . EBR (u) ≤ EBR (v) = 2 4 BR \BR−1 Note that Corollary 7.2.1 is optimal: inequality (7.12) is an equality if u is a one-dimensional solution. The following monotonicity formula completes the description of the energy EBR (u). Theorem 7.2.2 ([157]) Let N ≥ 1. Let u ∈ C 3 (RN ) denote a bounded solution to the Allen-Cahn equation −∆u = u − u3 Then,

in RN .

(7.13)

Φ(R) = R1−N EBR (u)

(7.14)

is a nondecreasing function of R. Proof. We exploit Pohozaev’s identity. By (8.8),

ˆ ∆u(x·∇u) d x = BR

N −2

ˆ

2

2

|∇u| d x− BR

R 2

ˆ

ˆ

2

∂ BR

|∇u| dσ+R

∂ BR

∂u ∂r

2 dσ.

Since u solves (7.13), we also have

ˆ

ˆ ∆u(x · ∇u) d x = − BR

(u − u3 )(x · ∇u) d x = BR ˆ ˆ −N W (u) d x + R BR

168

∂ BR

W (u) dσ,

Chapter 7. A conjecture of De Giorgi where W (u) is given by (7.5). Combining the above two equalities, we obtain the Pohozaev identity:

ˆ (N − 2) |∇u|2 + 2N W (u) d x = BR

ˆ

ˆ

R ∂ BR

|∇u|2 + 2W (u) dσ − 2R

∂u

2

∂r

∂ BR

dσ. (7.15)

Now differentiate Φ given by (7.14): 2Φ0 (R) = − (N − 1)R

−N

ˆ ˆ 2 1−N |∇u| + 2W (u) d x + R

|∇u|2 + 2W (u) dσ.

∂ BR

BR

Using (7.15), we deduce that

ˆ ˆ 2 2W (u) − |∇u| d x + 2R 2R Φ (R) = N

0

∂ BR

BR

∂u

2

∂r

dσ.

To complete the proof, we apply Theorem 7.2.3 below.

Theorem 7.2.3 ([158]) Let N ≥ 1. Let u ∈ C 3 (RN ) denote a bounded solution to the Allen-Cahn equation (7.13). Then, |∇u|2 ≤ 2W (u)

in RN ,

(7.16)

where W (u) is given by (7.5). Proof. We want to show that the function P(x) := |∇u|2 − 2W (u) is nonpositive. Since u is bounded, it follows that P is bounded (by standard elliptic regularity) and that infRN |∇u| = 0. In particular, given δ > 0, we may assume (up to a translation of space) that |∇u|2 (0) < δ.

(7.17)

Step 1. Set f (u) = −W 0 (u) = u − u3 . Then, P satisfies the inequality |∇u|2 ∆P ≥

1 2

|∇P|2 − 2 f (u)∇u · ∇P 169

in RN .

(7.18)

Stable solutions of elliptic PDEs Indeed, by definition of P, we have for i = 1, . . . , N , X ∂i P = 2 ∂ j u∂i j u + 2 f (u)∂i u.

(7.19)

j

In particular, !2 X

∂i P − 2 f (u)∂i u

2

=4

i

≤4

X

X

i

j

∂ j u∂i j u

X X (∂ j u)2 (∂i j u)2 j

i, j 2

2

= 4 |∇u| |Hu| ,

(7.20)

where Hu denotes the Hessian matrix of u. Differentiating once more (7.19) and using (7.9), we obtain X X ∆P = 2 (∂i j u)2 + 2 ∂ j u∂ j ∆u + 2 f 0 (u) |∇u|2 + 2 f (u)∆u i, j

j 2

= 2 |Hu| − 2 f (u) . 2

Using (7.20), it follows that |∇u|2 ∆P ≥ =

1X 2 1 2

∂i P − 2 f (u)∂i u

2

− 2 f (u)2 |∇u|2

i

|∇P|2 − 2 f (u)∇u · ∇P,

as claimed. Step 2. We assume temporarily that given " > 0, R > 0, there exists a radial cutoff function η(x) = η",R (|x|) ∈ C 2 (RN ) having the following properties: η(R) = 1,

η > 0,

η0 < 0

and

lim η(r) = 0.

r→+∞

lim η(r) = 1

for all r ≥ R 0 0 η2 2η M 0 (N − 1)η " − η − η00 − ≤ , 0 2 (η ) η " r L

(7.22)

"→0+

where

M = sup 2 f (u) |∇u| , RN

170

for r ≥ R,

L = sup 2 |∇u|2 . RN

(7.21)

(7.23)

(7.24)

Chapter 7. A conjecture of De Giorgi Step 3. Set v = ηP. Then, v(x) ≤ max(", max P), |x|=R

for all |x| ≥ R.

(7.25)

Equation (7.25) is obvious if sup|x|≥R v ≤ 0, so we may assume that v is positive somewhere. Since P is bounded and lim+∞ η = 0, lim|x|→+∞ v = 0. So, either v reaches its maximum on |x| = R and (7.25) follows, or v achieves its then maximum at some point x 0 such that x 0 > R. At x 0 , 0 = ∇v = η∇P + P∇η, ∇η hence ∇P = −P η . Using (7.18), we deduce that at x 0 , 1 |∇u|2 ∆v ≥ |∇u|2 P∆η + 2 |∇u|2 ∇P · ∇η + η |∇P|2 − 2η f (u)∇u · ∇P 2   2 2 2 2 ∇η P |∇u| ∇η + 2 f (u)∇u · ∇η P + . = |∇u|2 ∆η − 2 η 2η Now, ∆v(x 0 ) ≤ 0 since x 0 is an interior point of maximum. Furthermore, P(x 0 ) > 0, since η(x 0 ) > 0 and v(x 0 ) > 0. So, at x 0 , 2 |∇u|2 ∇η P |∇u|2 ≤2 − 2 f (u)∇u · ∇η − |∇u|2 ∆η. (7.26) 2η η If |∇u|2 (x 0 ) ≤ ", then since η ≤ 1 and W ≥ 0, we have v(x) ≤ v(x 0 ) ≤ P(x 0 ) ≤ |∇u|2 (x 0 ) ≤ " for |x| ≥ R and so (7.25) holds. If |∇u|2 (x 0 ) > ", since η0 < 0, (7.24) and (7.26) imply that at x 0 ,   ∇η 2 M ∇η 2 2η2 2  − ∆η v = ηP ≤ + 2 |∇u|  η ∇η |∇u|2   2 2 ∇η M ∇η η2 ≤ L + − ∆η .  η " ∇η 2 Recalling (7.23), we conclude that v(x 0 ) ≤ " and (7.25) is proven. Step 4. Now, we may let " → 0 in (7.25). Then, P(x) ≤ max(0, max P), |x|=R

for |x| ≥ R.

Now let R → 0+ . Using (7.17), we obtain P(x) ≤ max(0, P(0)) < δ 171

for all x ∈ RN .

Stable solutions of elliptic PDEs Since δ > 0 is arbitrary, P ≤ 0 as requested. So, we are left with the proof of Step 2. Set ˆ 1 −"/Ls e g" (t) = ds, for 0 ≤ t ≤ 1 s2 t and

ˆ h",R (t) =

Take at last

t

e−(M /")s

R

s N −1

ds,

η(r) = η",R (r) = g"−1 (c h",R (r)),

for t ≥ R. for r ≥ R,

where c = g" (0)/h",R (+∞). Then, (7.21) is obvious. For (7.22), it suffices to observe that h",R → 0 pointwise as " → 0 and g"−1 (0+ ) → 1. For (7.23), differentiate, take the log and differentiate again with respect to r the equality ˆ r −(M /")s ˆ 1 −"/Ls e e ds = ds. 2 s s N −1 R η Thanks to Theorem 7.2.1, it is natural to study monotone entire solutions of the Allen-Cahn equation, Equation (7.1). However, it is not clear at this stage that the conclusion of Conjecture 7.1.1 has a chance to hold. Also, why should we restrict to dimension N ≤ 8? We discuss this in the next section.

7.2.3

From Bernstein to De Giorgi

In this section, we continue our discussion of the De Giorgi conjecture 7.1.1 in its weak form. That is, we make the additional assumption that (7.10) holds. The following result shows that the level sets of u are “flat at infinity.” Theorem 7.2.4 ([3]) Let 1 ≤ N ≤ 8 and let u ∈ C 2 (RN ; [−1, 1]) denote a monotone solution to (7.1) such that (7.10) hold. Then, there exists a sequence ("k ) converging to zero and a unit vector τ ∈ S N −1 such that ˆ 2 N −1 lim "k |∇u|2 − ∂τ u d x = 0. (7.27) k→+∞

B1/"k

Moreover, there exists a half-space E ⊂ RN orthogonal to τ such that u"k (x) = u(x/"k ) → χ E − χ E c in L l1oc (RN ), as k → +∞. 172

(7.28)

Chapter 7. A conjecture of De Giorgi To prove this theorem, one is led to establishing first that (7.28) holds for some set E that is a local minimizer of perimeter. This is where Theorem 7.2.1 and the energy estimates Corollary 7.2.1 and Theorem 7.2.2 are used. In addition, since u is monotone, it follows from the implicit function theorem that any level set Eλ := {x ∈ Ω : u(x) > λ} lies above the graph of a function ψλ : Ωλ ⊂ RN −1 → R, that is, Eλ = {(x 0 , x N ) ∈ Ωλ × R : x N > ψλ (x 0 )}. Hence, E should lie above an entire graph ψ : RN −1 → R, that is, E = {(x 0 , x N ) ∈ RN × R : x N > ψ(x 0 )}. Since E has a locally minimal perimeter, ψ solves (in a weak sense) the minimal surface equation in all of RN −1 . Since global minimal graphs are flat in dimension N − 1 ≤ 7 due to Bernstein-type theorems (see [127]), one finally obtains the desired conclusion. In fact, much more can be said. The following difficult results, which we state without proof, completely settle the weak form of the De Giorgi conjecture. Theorem 7.2.5 ([194]) Let 1 ≤ N ≤ 8. Assume that u ∈ C 2 (RN ; [−1, 1]) is a monotone solution to (7.1) such that (7.10) holds. Then, u is one-dimensional. Theorem 7.2.6 ([84]) Let N ≥ 9. There exists a monotone solution u ∈ C 2 (RN ; [−1, 1]) of (7.1) such that (7.10) holds and u is not one-dimensional. Let us turn now to the full De Giorgi conjecture, which we shall prove in dimensions N = 2 and N = 3. For the state of the art on what is known in dimensions 4 ≤ N ≤ 8, we refer the reader to [104].

7.3

Dimension N = 2

Theorem 7.3.1 ([120]) Conjecture 7.1.1 holds true if N = 2. Proof. Let u denote a solution to (7.1). Since u is monotone, u is stable. Since u is bounded, so is |∇u| (by applying elliptic regularity in any given ball B(x 0 , 1), x 0 ∈ RN ). We then simply apply Theorem 6.2.2. 173

Stable solutions of elliptic PDEs

Dimension N = 3

7.4

Theorem 7.4.1 ([7]) Conjecture 7.1.1 holds true if N = 3. Proof. We follow [96]. Step 1. Assume that u = u(x 1 , x 2 , x 3 ) satisfies (7.1). Then, the function u(x 1 , x 2 ) = lim u(x 1 , x 2 , x 3 )

(7.29)

x 3 →+∞

is a bounded stable solution to −∆u = u − u3

in R2 .

(7.30)

For every t ∈ R, consider the function u t defined by u t (x 1 , x 2 , x 3 ) = u(x 1 , x 2 , x 3 + t)

for all (x 1 , x 2 , x 3 ) ∈ R3 .

(7.31)

Clearly, u t solves (7.1) and by standard elliptic regularity, ku t kC 2 (R3 ) ≤ C, for some constant C independent of t. Since ∂ u/∂ x 3 > 0, we deduce that u t → u in Cl1oc (R3 ), as t → +∞. Therefore, u is a bounded solution to (7.30). To see that u is stable, recall that u t is monotone, hence stable. In particular, taking ϕ ∈ Cc1 (R3 ) of the form ϕ(x 1 , x 2 , x 3 ) = φ(x 1 , x 2 )ψR (x 3 ), with φ ∈ Cc1 (R2 ) and ψR ∈ Cc1 (R) such that 0 ≤ ψR ≤ 1, 0 ≤ ψ0R ≤ 2, ψR = 0 in R \ [R, 2R + 2] and ψR = 1 in [R + 1, 2R + 1], the stability of u t implies that ˆ ˆ ∇(φ(x , x )ψ (x )) 2 d x ≥ (1 − 3(u t )2 )φ(x 1 , x 2 )2 ψR (x 3 )2 d x. 1 2 R 3 R3

Now,

R3

(7.32)

∇(φ(x , x )ψ (x )) 2 = ψ2 ∇φ 2 + (ψ0 )2 φ 2 1 2 R 3 R R

and so ˆ ∇(φ(x , x )ψ (x )) 2 d x = 1 2 R 3 R3 ˆ ˆ

ˆ

∇φ 2 d x+ ψ2R d x 3 [R,R+1]∪[2R+1,2R+2] R2 ˆ (ψ0R )2 d x 3 φ2 d x 2 [R,R+1]∪[2R+1,2R+2] R ˆ ˆ 2 ≤ (R + 2) ∇φ d x + 8 φ 2 d x. (7.33)

ψ2R d x 3 + [R+1,2R+1] ˆ +

R2

174

R2

Chapter 7. A conjecture of De Giorgi Similarly,

ˆ R3

(1 − 3(u t )2 )φ(x 1 , x 2 )2 ψR (x 3 )2 d x = ˆ ˆ = + ≥ [R+1≤x 3 ≤2R+1] [R<x 3
R2

Collecting (7.32), (7.33), and (7.34), we deduce that R+2 R

ˆ ˆ 12 ∇φ 2 d x + φ2 d x ≥ 2 2 R R R ˆ R2

[R+1≤x 3 ≤2R+1]

(1 − 3(u t )2 ) d x 3

φ 2 d x. (7.35)

Since for every (x 1 , x 2 ) ∈ R2 , we have

(1 − 3(u ) ) d x 3 − (1 − 3u ) ≤ 2

t 2

[R+1≤x 3 ≤2R+1]

≤3

[R+1≤x 3 ≤2R+1]

2 u − (u t )2 d x ≤ 3

≤ 6[u(x 1 , x 2 ) − u t (x 1 , x 2 , R)] → 0

as R → +∞,

we may pass to the limit R → +∞ in (7.35), by dominated convergence. We deduce that u is stable, as claimed. Step 2. There exists τ = (τ1 , τ2 ) ∈ S 1 and h ∈ C 2 (R) such that u(x 1 , x 2 ) = h(τ1 x 1 + τ2 x 2 ),

for all (x 1 , x 2 ) ∈ R2 .

(7.36) p Furthermore, either h ≡ 1 or there exists α ∈ R such that h(t) = tanh(t/ 2 + α) for every t ∈ R. To see this, we apply Theorem 7.3.1. Thus, (7.36) holds and h is a monotone solution to −h00 = h − h3 in R. (7.37) Using the strong maximum principle (Corollary A.5.1), we see that either h is constant or h is a strictly monotone solution to (7.37). If h is constant, then h ∈ {−1, 0, 1}. The case h = 0 is excluded since u is stable. The case h = −1 is also ruled out by the monotonicity assumption ∂ u/∂ x 3 > 0. Hence, h = 1. 175

Stable solutions of elliptic PDEs When h is not constant, it is easily seen that h can always be chosen strictly increasing (if h were strictly decreasing, simply replace τ by −τ and h(t) by h(−t) in (7.36)). To conclude the proof of Step 2, simply apply Proposition 7.1.1. Step 3. The Dirichlet energy of u satisfies ˆ |∇u|2 d x ≤ CR2 , for all R > 1, (7.38) BR

for some positive constant C independent of R. For every R > 1, set ˆ 1 t 2 1 2 t 2 t d x, ∇u + ((u ) − 1) ER (u ) = 2 4 BR 1 where u t is the function defined in (7.31). Since u t → u in Cloc (R3 ) we get

lim ER (u t ) = ER (u),

for all R > 1.

t→+∞

In addition,

ˆ

(7.39)

ˆ

∂ t ER (u ) =

∇u · ∇(∂ t u ) d x +

t

t

((u t )3 − u t )∂ t u t d x.

t

BR

BR

Since ∂ u/∂ x 3 > 0, we have ∂ t u t > 0. Since ku t kC 1 R3 ≤ C for some C independent of t, it follows that ˆ ˆ ∂ ut t t ∂ t ER (u ) = ∂ t u dσ ≥ −C ∂ t u t dσ. ∂ n ∂ BR ∂ BR Hence, for every T > 0 and every R > 1, we have ˆ T T ER (u) = ER (u ) − ∂ t ER (u t ) d t 0 ˆ T ˆ ≤ ER (u T ) + C 0

ˆ = ER (u ) + C

ˆ

= ER (u T ) + C

∂ BR

∂ t u t dσ

dt

T

∂ t u t d t dσ

T

ˆ∂ BR

∂ BR

0

(u T − u)dσ ≤ ER (u T ) + C1 R2 ,

176

Chapter 7. A conjecture of De Giorgi for some constant C1 independent of T and R. Letting T → +∞ and using (7.39), we obtain ER (u) ≤ ER (u) + C1 R2 ˆ R 1 0 2 1 2 2 2 ≤ C2 R (h ) + (h − 1) d t + C1 R2 , 2 4 −R where C2 is independent of R. Due to the expression of h found in Step 2, ˆ 1 0 2 1 2 2 (h ) + (h − 1) d t < +∞ 4 R 2 and the desired conclusion follows. Step 4. Let σ j = ∂∂ xu / ∂∂xu for j = 1, 2. Then, since ∂ u/∂ x 3 and ∂ u/∂ x j both j

3

solve the linearized equation

it follows that

−∆v = (1 − 3u2 )v

in R3 ,

−∇ · v 2 ∇σ j = 0

in R3 .

(7.40)

Apply Proposition 6.2.2 for (7.40). By (7.38), we deduce that σ j is constant, that is, there exists a constant C j such that ∂u ∂ xj

= Cj

∂u ∂ x3

.

In particular, the gradient of u points in a fixed direction, that is, u is onedimensional.

177

Chapter 8 Further readings 8.1

Stability versus geometry of the domain

So far, we have mostly dealt with stable solutions of (1.3) for two specific types of domains: Ω is bounded or Ω = RN . In this section, we review a number of results applying to other geometries.

8.1.1

The half-space

The Lane-Emden nonlinearity N We begin discussing (1.3) for Ω = R+ in the case of the Lane-Emden nonlinearity. ¨ N −∆u = |u| p−1 u in R+ , (8.1) N u=0 on ∂ R+ .

When p is subcritical, no positive solution exists. Theorem 8.1.1 ([124]) Let N ≥ 3 and 1 < p ≤ pS (N ) =

N +2 N −2

(or N = 2 and

N p < +∞). If u ∈ C (R+ ) is a nonnegative solution to (8.1), then, u ≡ 0. 2

When restricting to bounded nonnegative solutions, the above theorem can be extended to any exponent below the second critical exponent pc (N − 1), defined by Equation (4.16), in dimension N − 1. Theorem 8.1.2 ([97]) Let N ≥ 2 and 1 < p < pc (N − 1) (where pc (N ) is given N by (4.16)) . If u ∈ C 2 (R+ ) is a nonnegative bounded solution to (8.1), then, u ≡ 0. 179

Stable solutions of elliptic PDEs N . Let us Proof. By the strong maximum principle either u = 0 or u > 0 in R+ prove that the second possibility does not happen. Suppose to the contrary N . Then, u is monotone (using a similar—but more delicate— that u > 0 in R+ strategy as that of Lemma 1.2.1, see [70]), hence stable. The boundedness of u, standard elliptic estimates, and the monotonicity of u with respect to the variable x N , imply that the function

v(x 1 , . . . , x N −1 ) := lim u(x) x N →+∞

is a positive smooth solution to the Lane-Emden equation in RN −1 . Furthermore, mimicking Step 1 in the proof of Theorem 7.4.1, we deduce that v is stable. At this point, an application of Exercise 6.2.1 to the solution v in RN −1 , gives v = 0 in RN −1 . This result clearly contradicts v > 0 in RN −1 . Hence, u = 0, which completes the proof. In the proof above, we used the fact that bounded positive solutions of the equation are monotone (hence stable). A similar classification result is available for solutions that are stable outside a compact set (possibly unbounded and/or sign-changing), but for a smaller range of p. N Theorem 8.1.3 ([97]) Let N ≥ 2 and p > 1. Assume that u ∈ C 2 (R+ ) is a solution to (8.1) that is stable outside a compact set and 1 < p < pc (N ), where pc (N ) is given by Equation (4.16). Then, u ≡ 0.

None of the previous three theorems are known to be optimal. General nonlinearity In this section, we consider bounded positive solutions to  N −∆u = f (u) in R+ ,  N , u>0 in R+  N u=0 on R+ ,

(8.2)

where f ∈ C 1 (R). A useful criterion for obtaining Liouville-type theorems in this context is the following. N Theorem 8.1.4 ([16]) Let N ≥ 2, f ∈ C 1 (R) and let u ∈ C 2 (R+ ) be a solution to (8.2). Assume that

f (M ) ≤ 0,

where M = sup u. N R+

Then, u is monotone and one-dimensional. 180

Chapter 8. Further readings As a consequence, we obtain the following. N Theorem 8.1.5 ([16, 17, 70, 102]) Let N ≥ 2, f ∈ C 1 (R) and let u ∈ C 2 (R+ ) be a solution to (8.2).

• If N = 2, then u is monotone and one-dimensional. • If N = 3 and f (0) ≥ 0, then u is monotone and one-dimensional. • If N ≤ 5 and f ≥ 0, then u is monotone and one-dimensional. • If N ≥ 2 and f (0) ≥ 0, then u is monotone. Remark 8.1.1 It is not known whether the assumption f (0) ≥ 0 is necessary to conclude that (strictly) positive solutions are monotone. Note, however, that u = 1 − cos(x N ) is a nonnegative solution to (8.2) for f (u) = u − 1, but u is clearly not monotone. Proof of Theorem 8.1.5. Using a similar strategy as that of Lemma 1.2.1, see [17], u must be monotone. The boundedness of u, standard elliptic estimates, and the monotonicity of u with respect to the variable x N , imply that the function v(x 1 , . . . , x N −1 ) := lim u(x) x N →+∞

solves

−∆v = f (v)

in RN −1 .

Furthermore, working as in Step 1 in the proof of Theorem 7.4.1, we deduce that v is stable. Using Theorem 6.2.2 if N − 1 ≤ 2, or Theorem 6.2.3 if N − 1 ≤ 4, we deduce that v = 0 and so ! 0 = ∆v = f (v) = f

sup u

.

N R+

We may then apply Theorem 8.1.4.

8.1.2

Domains with controlled volume growth

We return to the model Lane-Emden equation, posed this time in an arbitrary proper domain Ω ⊂ RN . ¨ −∆u = |u| p−1 u in Ω, (8.3) u=0 on ∂ Ω. 181

Stable solutions of elliptic PDEs The following theorem shows that Liouville theorems for stable solutions extend to any domain having controlled volume growth. Theorem 8.1.6 ([97]) Let p > 1 and let Ω denote a proper C 2,α domain of RN . Let u ∈ C 2 (Ω) be a stable p solution to (8.3). Suppose that there exists a point N x 0 ∈ R and γ ∈ [1, 2p + 2 p(p − 1) − 1) such that lim inf R→+∞

Ω ∩ B(x , R) 0 p+γ 2 p−1

= 0.

(8.4)

R

Then, u ≡ 0. It is worth observing that the volume growth condition (8.4) is automatically satisfied in many interesting cases. Proposition 8.1.1 ([97]) Condition (8.4) is satisfied in any of the following cases • N ≥ 2, p > 1 and Ω has finite volume |Ω| < +∞. • N ≤ 10, p > 1 and Ω is any domain of RN . • N ≥ 11, 1 1 and Ω ⊂ RK × ω, where 1 ≤ K ≤ 10, ω ⊂ RN −K is any domain with finite (N − K) dimensional Lebesgue measure. For the classification of solutions that are stable outside a compact set, the volume growth condition is no longer sufficient: just think of the case of a bounded domain, where all (classical) solutions are stable outside a compact set. However, if Ω is a smooth unbounded proper domain, satisfying ∃X ∈ RN ,

|X | = 1

:

n(x) · X ≥ 0,

n(x) · X 6≡ 0 on ∂ Ω,

(8.5)

where n is the normal unit vector to ∂ Ω pointing outward, for example, if Ω is a smooth epigraph, then the following result holds. Theorem 8.1.7 ([97]) Let Ω be a proper unbounded C 2,α domain of RN , N ≥ 2 satisfying condition (8.5). Let u ∈ C 2 (Ω) be a solution to (8.3) that is stable (p < +∞ if N = 2). Then, outside a compact set, with 1 < p ≤ pS (N ) = NN +2 −2 u ≡ 0. 182

Chapter 8. Further readings

8.1.3

Exterior domains

As the next theorem demonstrates, even when working in unbounded domains, geometric conditions on Ω such as (8.5) cannot be avoided to obtain a Liouville-type result for solutions that are stable outside a compact set. Theorem 8.1.8 ([72]) Let N ≥ 3 and let D ⊂ RN be a smoothly bounded open +2 set, such that Ω = RN \ D is connected. Then, there exists a number p0 > NN −2 such that for any

N +2 N −2

0  u=0

p

in Ω, in Ω, on ∂ Ω,

having fast decay at infinity, that is, u(x) = O (|x|2−N ), as |x| → +∞. In particular, u is stable outside a compact set. Remark 8.1.2 When D is a ball, the theorem remains valid for any p > NN +2 . −2 N +2 In [72], the authors also construct slow-decay solutions, for any p > N −2 . More precisely, such solutions satisfy 2 − p−1

u = c p |x|

2 p−1

2 p−1

(1 + o(1)),

as |x| → +∞,

1 p−1

N −2− where c p = . By the optimality of Hardy’s inequality, every such solution is unstable outside every compact set if p < pc (N ). One should also note that if a solution decays at least like |x|−2/(p−1) , then it must satisfy the following alternative. +2 }, and Ω = RN \ B(0, 1). Theorem 8.1.9 ([20]) Let N ≥ 3, p ∈ (1, +∞) \ { NN −2 Let u ∈ C 2 (Ω) denote a solution to −∆u = u p in Ω, u>0 in Ω, 2 − p−1

such that u(x) = O (|x| that

), as |x| → +∞. Then, either there exists γ > 0 such lim |x|N −2 u(x) = γ,

|x|→+∞

183

Stable solutions of elliptic PDEs or there exists a positive solution w ∈ C 2 (S

N −1

−∆S N −1 w = w p − λw with λ = c pp−1 =

2 p−1

N−

2p p−1

) to in S

N −1

,

, such that 2

lim r p−1 u(r, ·) = w(·),

r→+∞

in the C k (S

8.2

N −1

) topology, for any k ∈ N.

Symmetry of stable solutions

We have seen in Proposition 1.3.4 that stable solutions defined on the unit ball are always radial. One may wonder what symmetry properties stable solutions have when working in more general domains. Also, what can be said of solutions with a higher Morse index? In this section, we point out several recent results in this direction of research.

8.2.1

Foliated Schwarz symmetry

When working with solutions of positive Morse index on the unit ball, radial symmetry may fail. For example, if f (u) = λ2 u, where λ2 = λ2 (−∆; B), all solutions have index one. Furthermore, in this case, any solution to (1.3) takes the form u(x) = φ(r) cos(θ ). In the above formula, r = |x|, cos(θ ) = and the function φ is explicit, namely, φ(r) = Ar

2−N 2

J( jr),

x |x|

· p, where p ∈ S

N −1

is arbitrary,

r ∈ (0, 1),

where A is an arbitrary constant, J is the Bessel function of the first kind of order N 2−2 , and j is its first zero. In particular, although u is not radial, u has the following partial symmetry property. Definition 8.2.1 Let N ≥ 2 and let B ⊂ RN be either a ball or an annulus centered at the origin. A function u ∈ C(B) is foliated Schwarz symmetric if there exists a unit vector p ∈ RN such that 184

Chapter 8. Further readings • u(x) = u(r, θ ) is a function of only two variables: the distance r = |x| of x the point x to the origin and the angle θ = arccos( |x| · p) formed by the vectors x and p. • u is nonincreasing in θ . It turns out that foliated Schwarz symmetry is a general property of solutions of low Morse index, as the following two results demonstrate. For the proof, we refer the reader to [176]. Theorem 8.2.1 ([175]) Let N ≥ 2, B denote either a ball or an annulus in RN . Assume that f ∈ C 1,α (R) is convex. Then, every solution to (1.18) with Morse index ind (u) = 1 is foliated Schwarz symmetric. Theorem 8.2.2 ([177]) Let N ≥ 2, B denote either a ball or an annulus in RN . Assume that f ∈ C 1,α (R) is such that f 0 is convex. Then, every solution to (1.18) with Morse index ind (u) ≤ N is foliated Schwarz symmetric. Whether such symmetry results hold for solutions of higher Morse index is an open problem. Partial results of Bouchez and VanSchaftingen indicate that there should exist solutions of index N + 2 or higher, which are not foliated Schwarz symmetric, see [24]. In this direction, we indicate the following symmetry breaking theorem. Theorem 8.2.3 ([23]) For every p > 1 sufficiently close to 1, there exists a rectangle Ω ⊂ R2 such that any least energy nodal solution to ¨ −∆u = |u| p−1 u in Ω, (8.6) u=0 on ∂ Ω, is neither symmetric nor antisymmetric with respect to the medians of Ω. Remark 8.2.1 A least energy nodal solution to (8.6) is a minimizer of the energy functional ˆ ˆ 1 1 2 EΩ (u) = |∇u| d x − |u| p+1 d x 2 Ω p+1 Ω over the nodal Nehari set M p defined by N p = {u ∈ H01 (Ω) \ {0} : DEΩ (u).u = 0},

M p = {u ∈ H01 (Ω) : u± ∈ N p }.

It can be shown that there exists a least energy nodal solution to (8.6) (see [54]) and that it has Morse index ind(u) = 1. 185

Stable solutions of elliptic PDEs

8.2.2

Convex domains

We have seen in Proposition 1.3.4 that a stable solution u to (1.18) in the ball is radial. In addition, u = u(r) is either constant, radially decreasing, or radially increasing. In other words, the level sets of u are hyperspheres. Suppose now that the domain Ω is convex. Is it true that the level sets of u are convex? Here is a result in this direction. Theorem 8.2.4 ([37, 109]) Let Ω ⊂ R2 be a smoothly bounded strictly convex domain. Suppose that f ≥ 0 is a smooth function and that u is a positive stable classical solution to (1.3). Then, u has a unique critical point and the level curves of u are convex in Ω. Note also that under more restrictive assumptions on the nonlinearity f , it is known that any positive solution (whether stable or not) of (1.3) in a convex domain has convex level sets. This is true, for example, if f (u) = u p , p ∈ (0, 1) and any dimension N . We refer the reader to [134, 135, 138] for more details.

8.3 8.3.1

Beyond the stable branch Turning point

In Section 3.3, we studied the branch of stable solutions to semilinear elliptic equations of the form (3.1). If the extremal solution is bounded, it can be shown that (λ∗ , u∗ ) is a turning point on the bifurcation diagram, that is, solutions (λ, u) to (3.1) close to the extremal solution belong to a curve of solutions bending back at λ∗ , in which the lower part is the stable branch and the upper part consists of solutions of higher Morse index (compare to Figure P.1). Theorem 8.3.1 ([63]) Assume N ≥ 1. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f ∈ C 3 (R), f > 0. In addition, assume that f is nondecreasing and that f is superlinear in the sense of (3.26). In addition, assume that the extremal solution u∗ to (3.1) is bounded. Then, there exists δ > 0 such that the solutions to (3.1) near (λ∗ , u∗ ) form a curve {(λ(s), u(s)) : s ∈ (−δ, δ)}, where • the map s 7→ (λ(s), u(s)) is twice continuously differentiable from (−δ, δ) to R × C02,α (Ω), • for s = 0, λ(0) = λ∗ , u(0) = u∗ , 186

Chapter 8. Further readings • for s ∈ (−δ, 0), λ(s) < λ∗ and u(s) is the minimal solution associated to λ(s), • for s ∈ (0, δ), λ(s) < λ∗ and u(s) is an unstable solution to (3.1), and • λ0 (0) = 0 and λ00 (0) < 0. Let us also mention that, at least for the Gelfand problem in an arbitrary domain, there are infinitely many bifurcation points on the solution curve whenever 3 ≤ N ≤ 9. In the case of the ball, Figure P.1 shows that these are in fact turning points. Theorem 8.3.2 ([71]) Assume 3 ≤ N ≤ 9. Let Ω ⊂ RN denote a smoothly bounded domain. Assume that f : R → R is analytic, f > 0, and f 0 (u) ∼ ceu , as u → +∞, for some c > 0. Then, there exists an unbounded connected curve Tˆ = {(λ(s), u(s)) : s ≥ 0} starting from (0, 0) such that ku(s)kC 1 (Ω) + λ(s) → +∞, as s → +∞. Moreover, −∆ − λ(s) f 0 (u(s)) : C02,α (Ω) → C α (Ω) is invertible except at isolated points, called bifurcation points, and given any bounded set S ⊂ C([0, 1]) × R, the set Tˆ \ S contains infinitely many bifurcation points. Remark 8.3.1 The bifurcation points can be turning points, as is the case for the Gelfand problem in the ball, see Figure P.1, but they can also be secondary bifurcation points, for example, if Ω is an annulus (see [165]).

8.3.2

Mountain-pass solutions

When the problem is subcritical (with respect to the usual Sobolev exponent +2 ), it is possible to continue the bifurcation branch beyond the pS (N ) = NN −2 neighborhood of the turning point given in Theorem 8.3.1. Theorem 8.3.3 ([31, 63]) Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and p ∈ (1, pS (N )]. Then, for every λ ∈ (0, λ∗ ), there exists a mountainpass solution Uλ to

−∆u = λ(1 + u) p in Ω, u=0 on ∂ Ω.

Remark 8.3.2 In nonconvex domains, unlike the case where Ω is a ball (recall Figure P.1), there can be more than one mountain-pass solution for a fixed value of λ (see, for example, [118]). However, in a left neighborhood of λ∗ , the mountain-pass solution is necessarily unique (see [154]). 187

Stable solutions of elliptic PDEs

8.3.3

Uniqueness for small λ

Take another look at Figure P.1 and observe that for N ≥ 3 and λ > 0 small enough, the Gelfand problem has a unique solution: the stable solution. In this section, we show that this uniqueness result can be extended to the case of quite general supercritical nonlinearities. Theorem 8.3.4 ([199]) Let N ≥ 3. Assume Ω ⊂ RN is smoothly bounded and star-shaped. Assume that f ∈ C 2 (R), f > 0, is supercritical in the following sense: F (t) 1 1 1 lim sup < ∗= − , (8.7) 2 2 N t→+∞ t f (t) ´t where F (t) = 0 f (s) ds. Then, there exists λu > 0 such that for all 0 < λ < λu , there exists at most one classical solution to (3.1). In order to prove Theorem 8.3.4, we first establish two auxiliary results. Lemma 8.3.1 Let N ≥ 2. Let Ω ⊂ RN denote a domain with C 1 boundary. Assume that Ω is star-shaped with respect to the origin, that is, for each x ∈ Ω, the line segment {t x : t ∈ [0, 1]} lies in Ω. Then,

for all x ∈ ∂ Ω,

x · n(x) ≥ 0

where n is the outward unit normal to the boundary of Ω. Proof. We follow [93]. Since ∂ Ω is C 1 , for every ε > 0, there exists δ > 0 y−x such that y − x < δ and y ∈ Ω imply n(x) · y−x ≤ ε. In particular, | | y−x ≤ 0. lim sup n(x) · y − x y→x y∈Ω

Let y = t x, t ∈ [0, 1]. Then, y ∈ Ω, since Ω is star-shaped. Thus, n(x) ·

x |x|

= − lim− n(x) · t→1

tx − x |t x − x|

≥ 0.

188

Chapter 8. Further readings Lemma 8.3.2 ([181]) Let N ≥ 2. Let Ω ⊂ RN denote a smoothly bounded domain and let w ∈ C 2 (Ω) ∩ C 1 (Ω). Then, ˆ ∆w (x · ∇w) d x = Ω ˆ ˆ ˆ N −2 1 2 2 |∇w| d x − |∇w| x · n dσ + (∇w · n)(∇w · x)dσ, (8.8) 2 2 ∂Ω Ω ∂Ω where n denotes the outward unit normal to ∂ Ω. If in addition, w is constant on the boundary of Ω, then ˆ ˆ ˆ N −2 1 2 ∆w (x · ∇w) d x = |∇w| d x + |∇w|2 x · n dσ. (8.9) 2 2 Ω Ω ∂Ω Proof. Using integration by parts, we have

ˆ Ω

∆w (x · ∇w) d x =

X ˆ i, j=1...N

Ω

w ii x j w j d x =

ˆ X ˆ − w i (x j w j )i d x + Ω

i, j=1...N

∂Ω

w i ni x j w j dσ

ˆ ˆ X ˆ = − w i δi j w j d x − w i w i j x j d x + Ω

i, j=1...N

ˆ

=−

2

1

Ω

ˆ

ˆ

∂Ω

w i ni x j w j dσ

2

|∇w| d x − ∇(|∇w| ) · x d x + (∇w · n) (∇w · x) dσ 2 Ω ∂Ω ˆ ˆ ˆ N −2 1 = |∇w|2 d x − |∇w|2 x · n dσ + (∇w · n) (∇w · x) dσ. 2 2 ∂Ω Ω ∂Ω Ω

If in addition, w is constant on ∂ Ω, then ∇w = |∇w| n on ∂ Ω and so (8.9) follows. Proof of Theorem 8.3.4. We follow [87]. Up to a translation of space, we may assume that Ω is star-shaped with respect to the origin. Assume that (3.1) has two solutions, u and w, for a given λ > 0. Without loss of generality, we may assume that u = uλ is the minimal solution to (3.1). In particular, w = u + v, for some v ≥ 0 and lim+ kuλ k L ∞ (Ω) = 0. (8.10) λ→0

The above equality follows, for example, from the fact that "ζ0 , where ζ0 solves (3.8), is a supersolution to (3.1) for λ > 0 sufficiently small. 189

Stable solutions of elliptic PDEs Hence, v = w − u solves −∆v = λ f (u + v) − f (u) ,

in Ω.

Multiply (8.11) by v and integrate. Then, ˆ ˆ 2 |∇v| d x = λ v f (u + v) − f (u) d x. Ω

(8.11)

(8.12)

Ω

Multiply (8.11) by x · ∇v and integrate. Using Pohozaev’s identity (Lemma 8.3.2), it follows that ˆ ˆ ˆ 1 N −2 2 2 |∇v| d x+ |∇v| (x·n) dσ = −λ [ f (u+v)− f (u)]x·∇v d x = 2 2 ∂Ω Ω Ω ˆ = −λ [ f (u + v)∇(u + v) − f (u)∇u − ( f (u)∇v + v f 0 (u)∇u)] · x d x+ Ω ˆ +λ f (u + v) − f (u) − f 0 (u)v x · ∇u d x Ω ˆ = −λ ∇[F (u + v) − F (u) − f (u)v] · x d x+ ˆ Ω +λ f (u + v) − f (u) − f 0 (u)v x · ∇u d x Ω ˆ = Nλ F (u + v) − F (u) − f (u)v d x+ Ω ˆ + λ (x · ∇u) f (u + v) − f (u) − f 0 (u)v d x. (8.13) Ω

F (t)

Set η = lim sup t→+∞ t f (t) . By (8.7), there exists η1 ∈ (η, 1/2∗ ). By (8.10) and elliptic regularity, given " > 0, we can choose λ > 0 sufficiently small so that |x · ∇u| ≤ " in Ω. Define h" (u, v) := N F (u + v) − F (u) − f (u)v + "| f (u + v) − f (u) − f 0 (u)v| − N η1 v f (u + v) − f (u) . Since f is C 2 and since (8.7) holds, the function h" (u, v)/v 2 is bounded above by some constant K, uniformly in ", provided " > 0 is sufficiently small. Since Ω is star-shaped, we also have x · n ≥ 0 on ∂ Ω, by Lemma 8.3.1. Using (8.12) and (8.13), we deduce that ˆ ˆ ˆ N −2 2 2 |∇v| d x ≤ N λK v d x + N η1 |∇v|2 d x. 2 Ω Ω Ω 190

Chapter 8. Further readings Applying Poincaré’s inequality, the condition 1 1 λ< − η1 λ1 (−∆; Ω)K 2∗ implies v = 0 and the desired result follows.

8.3.4

Regularity of solutions of bounded Morse index

We saw in Chapter 4 that in low dimensions, stable solutions of (1.3) are bounded for a quite general class of nonlinearities. We also saw that for the specific nonlinearities f (u) = eu and f (u) = (1 + u) p solutions of bounded Morse index can be uniformly controlled whenever N ≤ 9. For general nonlinearities, the regularity theory of solutions of bounded Morse index is still to be established (see [76] for partial results). We mention the following result, applying to quite general subcritical nonlinearities. Theorem 8.3.5 ([224]) Let Ω denote a smoothly bounded domain of RN , N ≥ 1. Let f ∈ C 1 (R) be such that • (superlinearity) there exists µ > 0 such that f 0 (u)u2 ≥ (1 + µ) f (u)u > 0

for |u| > M , and

• (subcritical growth) there exists θ ∈ (0, 1) such that 2N

F (u) ≥ (1 + θ ) f (u)u, N −2 ´u where F (u) = 0 f (s) ds.

for |u| > M

Then, given any solution u ∈ C 2 (Ω) to (1.3), kuk L ∞ (Ω) ≤ C(ind(u) + 1)β , where β depends on µ, N , θ only.

8.4

The parabolic equation

As discussed in Section 1.4, solutions to (1.3) can be seen as stationary states to the corresponding nonlinear heat equation (1.22). In addition, a solution 191

Stable solutions of elliptic PDEs must at least be stable, in the sense of Equation (1.5), in order to be asymptotically stable (as defined in Section 1.4). Next, we describe deeper connections between the elliptic equation and its parabolic counterpart. For f ∈ C 1 (R), u0 ∈ C02 (Ω), it is well known that (1.22) is well posed: there exists a maximal time T ∈ (0, +∞] and a unique solution v ∈ C 1 ([0, T ); C 2 (Ω)), see, for example, [184]. The first natural question is to determine whether the solution is global (T = +∞) or whether it blows up in finite time. This is very much related to the size of the initial datum u0 and to the existence of a stationary solution. Theorem 8.4.1 ([29]) Let N ≥ 1 and let Ω be a smoothly bounded domain of RN . Assume that f ∈ C 1 (R) is a nondecreasing convex function such that f (0) > 0. Assume that there exists an L 1 −weak solution u to (1.3). Then, for any initial condition u0 ∈ C02 (Ω) such that 0 ≤ u0 ≤ u, the solution to (1.22) is global. Conversely, if in addition, we have

ˆ

+∞ t0

dt f (t)

< +∞,

and if (1.22) has a global solution for some u0 ∈ C02 (Ω), u0 ≥ 0, then there exists a weak solution to (1.3). See also [14, 111, 112, 140] for earlier results. As an immediate corollary, see the following. Corollary 8.4.1 ([29]) Let N ≥ 1, Ω be a smoothly bounded domain of RN , and λ > 0. Assume that f ∈ C 1 (R) is a nondecreasing convex function such that f (0) > 0. In addition, assume that f is superlinear in the sense of (4.17). Then, the solution to  ∂v in Ω × (0, T ),  ∂ t − ∆v = λ f (v) (8.14) v=0 on ∂ Ω × (0, T ),  v(x, 0) = 0 for x ∈ Ω, is global if and only if λ ≤ λ∗ , where λ∗ is the extremal parameter associated to (3.1). Solutions may or may not continue to exist in the weak sense for t ≥ T ∗ . Theorem 8.4.2 ([150]) Make the same assumptions as in Theorem 8.4.1. In addition, assume that u0 ≥ 0 and −∆u0 ≥ f (u0 ) and that the solution v to (1.22) blows up in finite time T . Then, v blows up completely after T , that is, 192

Chapter 8. Further readings for any nondecreasing sequence ( f n ) of bounded continuous functions such that f n % f pointwise, the solution vn to (1.22) with nonlinearity f n satisfies vn (t, x) dΩ (x)

→ +∞,

uniformly for t ∈ [T + ", +∞),

where " > 0 is arbitrary and dΩ is the distance to the boundary of Ω. The hypothesis on the initial condition u0 implies that v is a monotone nondecreasing function of time. This turns out to be crucial: Theorem 8.4.3 ([107, 108]) Let 3 ≤ N ≤ 9 and B be the unit ball of RN . Fix λ ∈ (0, λ∗ ), where λ∗ is the extremal parameter associated to the Gelfand problem, Equation (2.1). Denote by uλk the k-th solution to (2.1) associated to the parameter λ (ordered by its L ∞ norm, see Figure P.1). Then, for any k ≥ 2, there exists a radial function u0 ∈ C02 (B) such that the solution to  ∂v v in B × (0, T ),  ∂ t − ∆v = λe (8.15) v=0 on ∂ B × (0, T ),  v(x, 0) = u0 (x) for x ∈ B, satisfies • v blows up in finite time. • v can be extended to a global L 1 -weak solution. 1 • v(t, ·) → u0λ as t → +∞ in Cloc ((0, 1]).

• v(t, ·) is well defined and smooth for all t ∈ (−∞, T ) and v(t, ·) → uλk , as t → −∞. v is called an L 1 connection between u0λ and uλk . In the above theorem, we used the following notion: v ∈ C([0, T ]; L 1 (Ω)) is an L 1 -weak solution to (1.22) if f (u) ∈ L 1 ((0, T ) × Ω) and ˆ ˆ tˆ ˆ tˆ s=t uϕ s=τ d x − uϕ t d x ds = (u∆ϕ + f (u)ϕ) d x ds, Ω

τ

Ω

τ

Ω

for all 0 ≤ τ < t < T and ϕ ∈ C 2 ([0, T ] × Ω) with ϕ = 0 on [0, T ] × ∂ Ω. For more results on the parabolic problem, see, for example, [184]. 193

Stable solutions of elliptic PDEs

8.5

Other energy functionals

8.5.1

The p-Laplacian

A natural generalization of the energy functional (1.1) considered throughout this book is the following ˆ ˆ 1 p |∇u| d x − F (u) d x, (8.16) EΩ (u) = p Ω Ω where p ∈ (1, +∞) and F ∈ C 2 (R). Working as in Chapter 1, one easily proves that EΩ is well defined on the space X = W 1,p (Ω) ∩ L ∞ (Ω) and that its critical points satisfy the Euler-Lagrange equation −∆ p u := −∇ · (|∇u| p−2 ∇u) = f (u) in the following weak sense: u ∈ W 1,p (Ω) and ˆ ˆ p−2 |∇u| ∇u · ∇ϕ d x = f (u)ϕ d x, Ω

in Ω

for all ϕ ∈ Cc1 (Ω).

Ω

(8.17)

In general, solutions to (8.17) need not be classical. Take for example u = 0 |x| p , where p0 is the conjugate exponent of p. Then, u solves (8.17) with constant right-hand side, yet u is not C 2 if p > 2. This lack of regularity can be understood through the loss of ellipticity of the p-Laplace operator near critical points of u. Still, letting Z := {x ∈ Ω : ∇u(x) = 0},

(8.18)

1,α it can be proven that any solution u to (8.17) belongs to Cloc (Ω) ∩ C 2 (Ω \ Z) (see [85, 143, 211]). Taking a test function ϕ ∈ Cc1 (Ω \ Z) supported away from the singular set Z, the function E(t) = EΩ (u + tϕ) is twice differentiable 2 2 and d E/d t t=0 is given by

ˆ Q u (ϕ) :=

ˆ ρ|∇ϕ| d x + (p − 2) 2

Ω

Ω

ρ

∇u |∇u|

ˆ

2 · ∇ϕ

dx −

Ω

f 0 (u)ϕ 2 d x,

where ρ = |∇u| p−2 . (8.19) This leads us to the following natural definition. Definition 8.5.1 Let p ∈ (1, +∞), α ∈ (0, 1), Ω an open set of RN , N ≥ 1, 1,α and f ∈ C 1 (R). A function u ∈ Cloc (Ω) solving (8.17) is stable away from its singular set if Q u (ϕ) ≥ 0, for all ϕ ∈ Cc1 (Ω \ Z). 194

Chapter 8. Further readings The notion of stability near the singular set is more delicate. When p ≥ 2, ρ = |∇u| p−2 ∈ C(Ω), since u ∈ C 1 (Ω). In particular, the function E(t) = EΩ (u + tϕ) is still twice differentiable, for any ϕ ∈ Cc1 (Ω). So, a stable solution to (8.17) should at least satisfy Q u (ϕ) ≥ 0, for all ϕ ∈ Cc1 (Ω). When p ∈ (1, 2), it is not clear in general what the natural notion of stability should be. However, if f > 0, then the singular set has zero Lebesgue measure: |Z| = 0, see [148]. So, ρ = |∇u| p−2 is well defined almost everywhere and measurable. This leads us to the following definition. Definition 8.5.2 Let p ∈ (1, +∞), α ∈ (0, 1), Ω an open set of RN , N ≥ 1, and 1,α f ∈ C 1 (R). If p < 2, assume that f > 0. A function u ∈ Cloc (Ω) solving (8.17) 1 is stable if Q u (ϕ) ≥ 0, for all ϕ ∈ Cc (Ω) such that ˆ ρ|∇ϕ|2 d x < +∞, where ρ = |∇u| p−2 . Ω

We note that in both cases p ∈ (1, 2) and p ≥ 2, it is not yet clear that the above definition includes enough test functions ϕ for practical purposes. For example, in order to derive the geometric Poincaré formula (Theorem 4.4.1), we used test functions of the form ϕ = |∇u|ψ. For positive solutions to (8.17), this is indeed a licit choice, thanks to the regularity results given in [67]. Bifurcation diagrams similar to Figure P.1 are given in [59,131]. Regularity results for stable solutions to (8.17) are addressed in [40, 42, 52, 113, 114, 190, 191]. Liouville results are discussed in [53, 66], the geometric Poincaré formula and the generalization of the De Giorgi conjecture in [99, 100, 103].

8.5.2

The biharmonic operator

Another possible generalization of the energy functional (1.1) is ˆ ˆ 1 2 EΩ (u) = |∆u| d x − F (u) d x. 2 Ω Ω

(8.20)

This energy arises, for example, when describing the deformations of an elastic thin plate, see [117]. Again, working as in Chapter 1, one easily proves that EΩ is well defined on the space X = H 2 (Ω) ∩ L ∞ (Ω) and that its critical points satisfy the Euler-Lagrange equation ∆2 u := −∆(−∆u) = f (u) The second variation of energy is given by ˆ ˆ 2 Q u (ϕ) := |∆ϕ| d x − f 0 (u)ϕ 2 d x, Ω

Ω

195

in Ω.

for all ϕ ∈ Cc2 (Ω).

(8.21)

(8.22)

Stable solutions of elliptic PDEs Hence, the following definition. Definition 8.5.3 Let Ω be an open set of RN , N ≥ 1, and f ∈ C 1 (R). A function u ∈ C 4 (Ω) solving (8.21) is stable if Q u (ϕ) ≥ 0, for all ϕ ∈ Cc2 (Ω). When working on bounded domains Ω, equation (8.21) must be complemented with boundary conditions. Among the most studied are: the Dirichlet boundary condition u = |∇u| = 0 on ∂ Ω, the Navier boundary condition u = ∆u = 0 on ∂ Ω, and the Steklov boundary condition u = ∆u − a ∂∂ un = 0 on ∂ Ω. Throughout this book we have used extensively the maximum principle (see Appendix A). Unfortunately, the maximum principle fails to be true for the biharmonic operator with Dirichlet boundary conditions (see, for example, [202]), unless stringent assumptions are made on the geometry of the domain, for example, Ω is a ball (see [21]). Additional difficulties arise with the spectral theory of the operator, even when one is solely interested in the principal eigenvalue. See the monograph [117]. Equation (8.21) with Navier boundary conditions can be rewritten as a system of equations in the unknown (u, v), where v = −∆u. If f is nondecreasing, the system is cooperative and so maximum principle tools are again available, at least for smooth domains. Still, other difficulties arise. For example, when proving regularity of stable solutions to the Gelfand problem (see Theorem 4.2.1), we used crucially the elementary calculus identity ∇u · ∇e2αu = α2 |∇eαu |2 , which has no counterpart when the nabla operator ∇ is replaced by the Laplace operator ∆. Nevertheless, stable solutions (in particular regularity theory) have been actively studied, perhaps because they appear naturally in a physical context: the study of micro-electro-mechanical devices (MEMS), see the monographs [91, 178]. Full bifurcation diagrams (like Figure P.1) have been established in [79, 80] for the exponential and for power-type nonlinearities. It should be noted that in the latter case and for negative exponents only, the diagrams are qualitatively different than for the Laplace operator. The question of regularity of the stable branch is addressed by [10, 15, 55, 60, 61, 77, 105, 106, 160, 161, 218]. Classification results in entire space appear in [9, 105, 116, 219, 223].

8.5.3

The fractional Laplacian

Consider again the energy functional (1.1) in the special case of Ω = RN . Using the Fourier transform, the energy can be rewritten as ˆ ˆ 1 2 2 |ξ| |ˆ u| dξ − F (u) d x, 2 RN RN 196

Chapter 8. Further readings ˆ is the Fourier transform of u. Yet another generalization of (1.1) that where u is being currently investigated consists of using a lower order Sobolev norm: ˆ ˆ 1 2s 2 E (u) = |ξ| |ˆ u| dξ − F (u) d x, (8.23) 2 RN RN where s ∈ (0, 1). The Euler-Lagrange equation associated to (8.23) reads (−∆)s u = f (u)

in RN ,

(8.24)

where, letting F −1 denote the inverse Fourier transform, ˆ). (−∆)s u := F −1 (|ξ|2s u Up to a constant multiplicative factor, the fractional Laplacian is also expressed through second-order difference quotients of u by ˆ u(x + y) − 2u(x) + u(x − y) s d y. (−∆) u = − | y|N +2s RN Indeed, ˆ ˆ e i y·ξ − 2 + e−i y·ξ u(x + y) − 2u(x) + u(x − y) ˆ(ξ) dy =u dy F | y|N +2s | y|N +2s RN RN ˆ cos( y · ξ) − 1 = 2ˆ u(ξ) dy | y|N +2s RN ˆ cos( y|ξ| · ξ ) − 1 |ξ| = 2ˆ u(ξ) dy N +2s | y| RN ˆ cos(z · e1 ) − 1 2s ˆ(ξ) = 2|ξ| u dz |z|N +2s RN ˆ(ξ). = −cN ,s |ξ|2s u Note in particular, that the fractional Laplacian is a nonlocal operator. There exists at least two other ways to define this operator: as the generator of a random walk with long jumps (see, for example, [213]), or as the Dirichletto-Neumann operator of a degenerate elliptic equation in N + 1 variables (see, for example, [47]). To illustrate the latter point of view, we present here the case s = 1/2. Given a function u defined over RN , let v = v(x, t) =: Eu denote its harmonic extension in RN × (0, +∞), that is, v solves ¨ ∆v = 0 in RN × (0, +∞), v(x, 0) = u(x)

for all x ∈ RN . 197

Stable solutions of elliptic PDEs Note that v is well defined and unique in the class of bounded functions if, say, u is smooth and bounded. Now, define the Dirichlet-to-Neumann operator Tu := −∂ t (Eu) t=0 . We claim that Tu = (−∆)1/2 u. To see this, let w = −∂ t Eu. Then, ∆w = −∂ t ∆Eu = 0

in RN × (0, +∞)

and w| t=0 = Tu. Hence, w = E(Tu). But then, T 2 u = − ∂ t E(Tu) t=0 = − ∂ t w t=0 = ∂ t t Eu t=0 . Since Eu is harmonic, 0 = ∆Eu = ∆ x Eu + ∂ t t Eu and so T 2 u = ∂ t t Eu t=0 = − ∆ x Eu t=0 = −∆u, that is, T is a square root of the Laplace operator. With the previous interpretation in mind, we can reformulate (8.24) as a boundary reaction problem for its harmonic extension v = Eu, in the case s = 1/2: ¨ ∆v = 0 in RN × (0, +∞), −∂ t v(x, 0) = f (v(x, 0))

for all x ∈ RN .

Working as in Chapter 1, we define stability as follows. Definition 8.5.4 A bounded solution u ∈ C 2 (RN ) to (8.24) is stable if ˆ ˆ s/2 2 |(−∆) ϕ| d x ≥ f 0 (u)ϕ 2 d x, for all ϕ ∈ Cc∞ (RN ). RN

RN

Seeing again the fractional Laplacian as a Dirichlet-to-Neumann operator over RN × (0, +∞), stability is equivalently defined through ˆ ˆ 1−2s 2 |∇ψ| d x d t ≥ f 0 (u)ψ2 (x, 0) d x, t RN ×(0,+∞)

RN

for all ψ ∈ Cc∞ (RN × [0, +∞)), see, for example, [90]. Regularity of stable solutions of boundary-reaction problems have been investigated by [78], and for the fractional Laplacian in [50]. Liouville theorems are studied by [41, 43, 90, 205]. See also [8, 45, 46, 195, 196] for the theory of nonlocal minimal surfaces. 198

Chapter 8. Further readings

8.5.4

The area functional

A classical problem in geometry is the following: Determine the graph of a smooth function u = u(x, y) over a two-dimensional bounded open domain Ω, having the least area among all graphs that assume given values at the boundary of Ω. The area of a graph u : Ω ⊂ RN → R can be computed as ˆ p EΩ (u) = 1 + |∇u|2 d x, Ω

and the corresponding (Euler-)Lagrange equation is the famous minimal surface equation:   ∇u  = 0, in Ω. (8.25) −∇ ·  p 2 1 + |∇u| From geometry’s point of view, this is equivalent to requesting that the mean curvature of the graph of u, seen as a nonparametric surface in RN +1 , vanishes identically. Solutions to (8.25) have been actively investigated. We mention in particular the following Bernstein theorem. Theorem 8.5.1 ([6, 19, 22, 82, 204]) Let u ∈ C 2 (RN , R) be a solution to the minimal surface equation (Equation (8.25)) on the entire space RN . Then, u is an affine function, that is, the graph of u is a plane in RN +1 , if and only if N ≤ 7. The proof of the above theorem when N = 2 follows from a Liouville-type theorem for elliptic operators (see, for example, [96]), while in higher dimensions one uses a connection between minimal graphs defined over RN and minimal hypercones in RN (see, for instance, [127, 203]). In fact, the stability of such cones (that is, the assumption that the second variation of area is nonnegative) is used crucially, see, for example, [39]. For more on minimal surfaces, see the monograph by [127]. For recent developments on the regularity of stable solutions to equations involving the minimal surface operator, see [151].

8.5.5

Stable solutions on manifolds

What is known when the ambient space RN is replaced by a manifold M in the definition of the energy functional (1.1)? More precisely, consider a complete, connected, smooth, m-dimensional manifold M without boundary, endowed with a smooth Riemannian metric g = (g i j ). Given a bounded open set Ω of M , we are interested in the energy functional defined for u ∈ C 2 (Ω) by ˆ ˆ 1 2 EΩ (u) = |∇ g u| dσ − F (u) dσ, 2 Ω Ω 199

Stable solutions of elliptic PDEs where ∇ g is the Riemannian gradient on M , dσ its volume element, and |X |2 = g(X , X ) for X ∈ T M the norm induced by the metric g. Using the notations of Section C.2, when M is a submanifold of the Euclidean space RN , |∇ g u|2 = ∇ T u · ∇ T u, where ∇ T u is the tangential gradient of u and X · Y the canonical dot product of two vectors X , Y ∈ RN . As in the Euclidean case, a critical point of EΩ is a function u ∈ C 2 (Ω) solving −∆ M u = f (u)

in Ω,

where ∆ M is the Laplace-Beltrami operator of M endowed with the metric g. Stable solutions verify ˆ ˆ 2 |∇ g ϕ| dσ ≥ f 0 (u)ϕ 2 dσ, for all ϕ ∈ Cc1 (Ω). Ω

Ω

We shall also need the following definition. Definition 8.5.5 A Riemannian manifold M is parabolic if for any x ∈ M , there exists a compact neighborhood Ω of x, such that for any " > 0, there exists ϕ" ∈ Cc∞ (M ) such that ϕ" ≡ 1 in Ω, and ˆ |∇ g ϕ" |2 dσ < ". M

For example, M = R endowed with the standard Euclidean metric is parabolic, while this is not the case of RN , N ≥ 3. We have the following Liouville-type theorems. 2

Theorem 8.5.2 ([101]) Let M be a connected manifold and let u ∈ C 2 (M ) be a stable solution to −∆ M u = f (u) in M . (8.26) Assume that the Ricci curvature of M is nonnegative. In addition, assume that • either M is compact, • or M is complete, parabolic, |∇ g u| ∈ L ∞ (M ), and Ric g does not vanish identically. Then, u is constant. The above theorem need not hold on a (noncompact) manifold with zero Ricci curvature. Indeed, M = R2 endowed with the standard Euclidean metric is parabolic,pwith identically zero Ricci tensor. However, the function u(x 1 , x 2 ) = tanh(x 1 / 2) is a stable nonconstant solution to the two-dimensional AllenCahn equation, see Chapter 7. The previous example motivates the following. 200

Chapter 8. Further readings Theorem 8.5.3 ([101]) Let M be a complete, connected Riemannian surface (that is, a complete, connected Riemannian manifold of dimension 2). Assume that the Ricci curvature of M vanishes identically. Let u be a stable solution of (8.26), with |∇ g u| ∈ L ∞ (M ). Then, any connected component of the level set of u on which ∇ g u does not vanish, is a geodesic.

201

Appendix A Maximum principles This section is devoted to the various versions of the maximum principle used in this book. We begin by reviewing the model operator ∆.

A.1

Elementary operator

properties

of

the

Laplace

Given an open set Ω ⊂ RN , a function u ∈ C 2 (Ω), and a point x 0 ∈ Ω, the Taylor expansion of u at x 0 is given by u(x 0 + h) = u(x 0 ) + Du(x 0 ).h +

1 2

D2 u(x 0 + th)(h, h),

(A.1)

where |h| is small, t is some number in the interval (0, 1) and where Du ∈ L (RN , R), D2 u ∈ B(Rn × RN , R) denote the first- and second-order differentials of u. The gradient of u at x 0 is defined as the unique vector in RN such that ∇u(x 0 ) · h = Du(x 0 ).h for all h ∈ RN (where a · b is the canonical inner product of a, b ∈ RN ). The Hessian matrix of u at x 0 is the unique matrix in RN × RN such that (Hu(x 0 ).h) · h = D2 u(x 0 )(h, h) for all h ∈ RN . In particular, if x 1 , x 2 , . . . , x N denote coordinates in an orthonormal basis of RN , (A.1) can be rewritten in the familiar form 1 u(x 0 + h) = u(x 0 ) + ∇u(x 0 ) · h + (Hu(x 0 + th).h) · h 2 N N X ∂u 1X ∂ 2u = u(x 0 ) + (x 0 )hi + (x 0 + th)hi h j . ∂ xi 2 i, j=1 ∂ x i ∂ x j i=1 203

Stable solutions of elliptic PDEs Definition A.1.1 The Laplacian of u ∈ C 2 (Ω) is defined as the trace of the Hessian matrix of u. In particular, if x 1 , x 2 , . . . , x N denote coordinates in an orthonormal basis of RN , then ∆u =

∂ 2u ∂ x 12

+ ··· +

∂ 2u ∂ x N2

.

We begin by listing the fundamental invariance properties of the Laplace operator: Proposition A.1.1 Consider a function u ∈ C 2 (RN ), N ≥ 1. The Laplace operator commutes with translations. More precisely, given a vector τ ∈ RN , ∆(u(· + τ)) = (∆u) (· + τ). (A.2) The Laplace operator commutes with orthogonal transformations. More precisely, if T ∈ L (Rn ) is such that |T x| = |x| for all x ∈ RN , then ∆(u ◦ T ) = (∆u) ◦ T.

(A.3)

The Laplace operator scales like a homogeneous polynomial of degree 2, that is, given λ ∈ R, ∆(u(λ ·)) = λ2 (∆u)(λ ·). (A.4) Proof. Equations (A.2) and (A.4) are straightforward. For (A.3), we simply observe that orthonormal bases are preserved by an orthogonal transformation and then apply Definition A.1.1. Let {ei }i=1,...,N denote an orthormal basis of RN and ei0 = Tei for i = 1, . . . , N . Then, |ei0 | = |Tei | = |ei | = 1. In addition, we have for i 6= j, |ei0

+

e0j |2

2 2 2 2 = ei0 + e0j + 2ei0 · e0j = ei + e j + 2ei0 · e0j .

Now, we also have 2 2 |ei0 + e0j |2 = |T (ei + e j )|2 = |ei + e j |2 = ei + e j , since ei · e j = 0. Hence, ei0 · e0j = 0 and {ei0 }i=1,...,N is orthonormal. The simplest equation involving the Laplace operator is the one satisfied by harmonic functions. Definition A.1.2 Let u ∈ C 2 (Ω). u is harmonic in Ω if ∆u = 0 in Ω. 204

Appendix A. Maximum principles Let us look for harmonic functions that have the same invariance properties as the Laplace operator. Translation invariant functions are simply the constant functions. A function u that is invariant under all orthogonal transformations is radial, that is, u is of the form u(x) = u(r), where r = |x|. Now, r 2 = x 12 + · · · + x N2 . So, for i = 1, . . . , N , r∂ r/∂ x i = x i . So, ∂u ∂ xi

du ∂ r

=

d r ∂ xi

=

du x i dr r

.

Differentiating once more, ∂ 2u ∂ x i2

=

d 2 u x i 2 d r2

r

+

du

dr

1 r

−

x i2 r3

.

Summing over i, we finally obtain ∆u =

d 2u dr

2

+

N − 1 du r

dr

.

(A.5)

If u is harmonic, it follows that v = du/d r satisfies a first-order differential equation, which can be solved explicitly, using separation of variables. We obtain that any radial harmonic function u must take the form u(r) = a + br 2−N , if N 6= 2 and u(r) = a + b ln r, if N = 2. Note that such functions are harmonic in Ω = RN \ {0} but not in RN , if b 6= 0. We shall be particularly interested in the so-called fundamental solution of the Laplace operator given for N 6= 2 by 1 r 2−N , (A.6) Γ(x) = Γ(r) = N (N − 2) |B| where |B| denotes the volume of the unit ball in RN , and by Γ(x) = Γ(r) = −

1 2π

ln r,

(A.7)

if N = 2. Using Green’s formula in Ω = B(0, R) \ B(0, "), with R > 0 large and " > 0 converging to 0, one can easily check (see, for example, Section 2.2 in [93]) that Γ solves −∆Γ = δ0 in D 0 (RN ), (A.8) where δ0 denotes the Dirac mass at the origin. Exercise A.1.1 Find harmonic functions that remain invariant under the transformation u → λ−2 u(λ·). 205

Stable solutions of elliptic PDEs Exercise A.1.2 Given N ≥ 1, find a function Γ2 solving ∆2 Γ2 := ∆(∆Γ2 ) = δ0

in D 0 (RN ).

We continue our discussion with the celebrated mean-value formula. Recall that Taylor’s formula (A.1) expresses the value u(x 0 + h) at a point x 0 + h close to x 0 in terms of values of u and its differentials at x 0 . Suppose now that instead of the value of u at x, one is interested in the mean of u over a given ball B(x 0 , R) ⊂⊂ Ω or its boundary ∂ B(x 0 , R). Proposition A.1.2 ([115]) Consider an open set Ω ⊂ RN , N ≥ 2 and a function u ∈ C 2 (Ω). For any ball B(x 0 , R) ⊂ Ω, there exists y0 , y1 ∈ B(x 0 , R) such that

∂ B(x 0 ,R)

u dσ = u(x 0 ) +

1 2N

R2 ∆u( y0 )

(A.9)

and u d x = u(x 0 ) + B(x 0 ,R)

1 2(N + 2)

R2 ∆u( y1 ).

(A.10)

Remark A.1.1 Much like Taylor expansions, higher-order expansions of the form

∂ B(x 0 ,R)

u dσ = u(x 0 ) +

k−1 X

a j R2 j ∆ j u(x 0 ) + ak R2k ∆k u( y0 )

j=1

hold true for u ∈ C 2k (Ω). A formula with an integral remainder term is also available. For all these results, see [168]. Proof. Working with v defined for x ∈ B = B(0, 1) by v(x) = u(Rx + x 0 ) if necessary, we may always assume that x 0 = 0 and R = 1. We also restrict to the case N ≥ 3, the case N = 2 being similar. Consider the fundamental solution of the Laplace operator, that is, the radial function Γ given by (A.6). For " > 0, integrate (−∆Γ)u on B \ B(0, "), apply Green’s identity and let " → 0. Then,

ˆ

ˆ

u(0) =

(−∆u)Γ d x + B

∂B

206

−

∂Γ ∂ν

u+Γ

∂u ∂ν

dσ.

Appendix A. Maximum principles By (A.6), −∂ Γ/∂ ν|∂ B = 1/ |∂ B|. It follows that ˆ ˆ ∂u u(0) = (−∆u)Γ d x + u dσ + Γ(1) dσ B ∂B ∂B ∂ ν ˆ u dσ = (Γ(x) − Γ(1)) (−∆u) d x + ∂B B ˆ ˆ 1 =

(Γ(r) − Γ(1))

∂ Br

0

ˆ

1

= 0

= =

1 N −2

∂ B r0

1 2N

(−∆u) dσ

dr +

r−r

N −1

ˆ

(−∆u) dσ 0

(−∆u)( y0 ) +

∂ Br

1

u dσ ∂B

1

N −2

(−∆u) dσ

r − r N −1

dr +

dr +

u dσ ∂B

u dσ ∂B

u dσ, ∂B

where r0 ∈ (0, 1), y0 ∈ B have been obtained using the first mean-value theorem for integration. We have just proved (A.9). Equation (A.10) follows by observing that ˆ 1 1 ∂ B u dx = u dσ d r r |B| 0 B ∂ Br and integrating (A.9) accordingly. The mean-value formulae are of particular interest when dealing with suband superharmonic functions. Definition A.1.3 A function u ∈ C 2 (Ω) is superharmonic in Ω if −∆u ≥ 0

in Ω.

Similarly, a function u ∈ C 2 (Ω) is subharmonic in Ω if −∆u ≤ 0 in Ω. Exercise A.1.3 Prove that a function u ∈ C 2 (Ω) is superharmonic in Ω if and only if for all balls B(x 0 , R) ⊂ Ω, u d x ≤ u(x 0 ).

(A.11)

B(x 0 ,R)

The notion of superharmonic function (respectively subharmonic) can be extended to solutions having weaker regularity. 207

Stable solutions of elliptic PDEs • Prove that u ∈ L l1oc (Ω) satisfies −∆u ≥ 0 in D 0 (Ω) if and only if (A.11) holds for almost all x 0 ∈ Ω and all R > 0 such that B(x 0 , R) ⊂ Ω. • Let u ∈ L l1oc (Ω) solve −∆u = 0 in D 0 (Ω). Prove that u ∈ C 2 (Ω). • Let u ∈ C 1 (Ω) solve −∆u = 0 in D 0 (Ω \ {0}). Prove that u ∈ C 2 (Ω) and u is harmonic in Ω. Exercise A.1.4 Let B be the unit ball of RN and B + = {x ∈ B : x N > 0}. + Assume that u ∈ C 2 (B + ) ∩ C 1 (B ) solves −∆u = 0 in B + u=0 on [x N = 0]. ˜ be the odd extension of u through [x N = 0], that is, Let u ¨ u(x) if x ∈ B + , ˜(x) = u −u(x 0 , −x N ) if x ∈ B \ B + . ˜ is harmonic in B. Prove that u Exercise A.1.5 Let λ > 0 and u ∈ C 2 (Ω), u ≥ 0, satisfy −∆u ≤ λu

in Ω. p

Prove that the function v defined for (x, t) ∈ Ω × R by v(x, t) = e λt u(x) is subharmonic. Deduce that there exists a constant C = C(N ) > 0, such that u(x 0 ) ≤ C e

p λR

u d x, B(x 0 ,R)

for every ball B(x 0 , R) ⊂ Ω.

A.2

The maximum principle

A crucial corollary of the mean-value formulae is the following strong maximum principle. Proposition A.2.1 Let Ω denote a domain of RN , N ≥ 1 and u ∈ C 2 (Ω) a superharmonic function. Then, u cannot achieve an interior point of minimum, unless u is constant. 208

Appendix A. Maximum principles Proof. Let m = infΩ u and assume that there exists a point x 0 ∈ Ω such that u(x 0 ) = m. Then, F = {x ∈ Ω : u(x) = m} is nonempty. Since u is continuous, F is relatively closed in Ω. Now take x 1 ∈ F and R > 0 so small that B(x 1 , R) ⊂ Ω. Apply the mean-value formula (A.11). Then, u d x ≤ u(x 1 ) = m.

m≤ B(x 1 ,R)

In particular, B(x 1 , R) ⊂ F so F is open and closed in Ω. Since Ω is connected, we deduce that F = Ω, that is, u ≡ m. The strong maximum principle immediately implies the following strong comparison principle. Proposition A.2.2 Let Ω denote a bounded domain of RN , N ≥ 1 and u ∈ C 2 (Ω) ∩ C 0 (Ω) a function satisfying −∆u ≥ 0 in Ω, (A.12) u≥0 on ∂ Ω. Then, either u ≡ 0, either u > 0 in Ω. Proof. Let m = minΩ u and assume that u is nonconstant (the remaining case being straightforward). By the strong maximum principle, u may not achieve m at an interior point, hence m = min∂ Ω u ≥ 0 and u > 0 in Ω.

A.3

Harnack’s inequality

Another useful consequence of the mean-value formulae is the following inquality. Proposition A.3.1 Let Ω ⊂ RN , N ≥ 1 denote a domain. Take a point x 0 ∈ Ω and r > 0 such that B(x 0 , 4r) ⊂ Ω. Assume that u is nonnegative and harmonic in Ω. Then, sup u ≤ 3N inf u. B(x 0 ,r)

B(x 0 ,r)

Proof. Take any two points y, z ∈ B(x 0 , r) and apply the mean-value theorem (Proposition A.1.2). Then, ˆ ˆ 1 1 u( y) = u(x) d x ≤ u(x) d x = B B( y,r) B B(z,3r) r r ˆ 1 N =3 u(x) d x = 3N u(z). B B(z,3r) 3r

209

Stable solutions of elliptic PDEs The desired estimate follows. Harnack’s inequality can be extended to a much wider class of equations (see, for example, [126]). For our purposes, the following generalization will suffice. Proposition A.3.2 Let Ω ⊂ RN , N ≥ 1, denote a domain. Assume that V (x) is a bounded function on Ω and u ∈ C 2 (Ω) is a nonnegative solution to −∆u = V (x)u

in Ω.

Take a point x 0 ∈ Ω and r > 0 such that B(x 0 , 6r) ⊂ Ω. Then, there exists a constant C = C(N , kV k L ∞ (Ω) , r) such that sup u ≤ C inf u. B(x 0 ,r)

B(x 0 ,r)

p Proof. Let λ = kV k L ∞ (Ω) . It suffices to prove the proposition for r < π/(10 λ). Take any two points y, z ∈ B(x 0 , r). By Exercise A.1.5, we have u dx ≤ C

u( y) ≤ C

u(x) d x. B(z,3r)

B( y,r)

p Consider the function v(t, x) = cos( λt)u(x) defined for (t, x) ∈ R×Ω. Then, v is superharmonic, v ≥ 0 in (−5r, 5r) × Ω, and so u(z) = v(0, z) ≥ ≥

1

v(t, x) d x d t B((0,z),5r) ˆ 4r

|B((0, z), 5r)|

ˆ p cos( λt) d t

−4r

u(x) d x ≥ c B(z,3r)

u(x) d x. B(z,3r)

Collecting the above two inequalities, the proposition follows.

A.4

The boundary-point lemma

We have just seen that a superharmonic function u must achieve its minimum on the boundary of the domain. In particular, at any such point of minimum, u must be nonincreasing in the direction of the exterior normal to the boundary. The boundary point lemma states that in fact more can be said, provided ∂ Ω is sufficiently smooth: u must be strictly decreasing at such boundary points. 210

Appendix A. Maximum principles Definition A.4.1 Let N ≥ 1. A domain Ω ⊂ RN satisfies an interior sphere condition at a point x 0 ∈ ∂ Ω, if there exists a ball B = B( y0 , r0 ) ⊂ Ω such that x 0 ∈ ∂ Ω ∩ ∂ B. Definition A.4.2 Let N ≥ 1. A vector v ∈ RN is an interior vector to a ball B if there exists t 0 > 0 and x 0 ∈ ∂ B such that x 0 + t v ∈ B for all t in the interval (0, t 0 ). Proposition A.4.1 ([225]) Let N ≥ 1 and let Ω ⊂ RN denote a domain satisfying an interior sphere condition at some point x 0 ∈ ∂ Ω. Then, given any function u ∈ C 2 (Ω) ∩ C(Ω) satisfying  in Ω,  −∆u ≥ 0 u>0 in Ω, (A.13)  u(x 0 ) = 0, there holds

u(x 0 + t v)

> 0, (A.14) t where v is any interior vector to B at the point x 0 (and where B is an interior sphere tangent to ∂ Ω at x 0 ). lim inf + t→0

Proof. We follow [188]. First, note that the function v(x) = |x|−λ , λ > N − 2, satisfies −∆v = λ(−λ + N − 2) |x|−λ−2 ≤ 0 in RN \ {0}. Next, observe that u ≥ c > 0 on ∂ B( y0 , r0 /2). Now consider −λ v1 (x) = c1 x − y0 − r0−λ , x ∈ A = B( y0 , r0 ) \ B( y0 , r0 /2), where c1 > 0 is chosen so small that v1 ≤ c ≤ u on ∂ B( y0 , r0 /2). Since, v1 = 0 on ∂ B( y0 , r0 ), we conclude that v1 ≤ u on ∂ A. By the maximum principle, since −∆(u − v1 ) ≥ 0 in A, we obtain that u ≥ v1 in A. Equation (A.14) holds for v1 by direct inspection and so (A.14) also holds for u. Corollary A.4.1 Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain, and let n denote the exterior normal unit vector to ∂ Ω. Then, given any function u ∈ C 2 (Ω) ∩ C 1 (Ω) solving −∆u ≥ 0 in Ω, (A.15) u=0 on ∂ Ω, 211

Stable solutions of elliptic PDEs there holds at every x 0 ∈ ∂ Ω, −

∂u ∂n

(x 0 ) > 0.

(A.16)

In particular, there exists a constant c > 0, depending on u and Ω only, such that u(x) ≥ c dΩ (x)

∀x ∈ Ω,

(A.17)

where dΩ (x) = dist(x, ∂ Ω) is the distance to the boundary. Proof. For (A.16), we simply note that if B denotes an interior sphere tangent to ∂ Ω at x 0 , then −n(x 0 ) is an interior vector to B at x 0 . We then apply (A.14): −

∂u ∂n

(x 0 ) = lim inf +

u(x 0 − t n(x 0 ))

t→0

t

> 0.

Since ∂ Ω is compact, it follows that − ∂∂ un ≥ c > 0 on ∂ Ω. Letting c > 0 smaller if necessary, the inequality remains valid on some given neighborhood ω ⊂ Ω of ∂ Ω. Now, every x ∈ Ω sufficiently close to the boundary has a unique projection x 0 ∈ ∂ Ω. Let ω ⊂ Ω denote a neighborhood of ∂ Ω containing all such points. It clearly suffices to establish (A.17) in ω. Given x ∈ ω, the segment [x, x 0 ] lies in ω and by the fundamental theorem of calculus, u(x) = u(x) − u(x 0 ) = ˆ 1 ˆ 1 d = u(x 0 + t(x − x 0 )) d t = ∇u(x 0 + t(x − x 0 )) · (x − x 0 ) d t = 0 dt 0 ˆ 1 ∂u = − (x 0 + t(x − x 0 )) d t x − x 0 ≥ c dΩ (x). ∂n 0 (A.17) follows. We have seen in (A.17), that superharmonic functions vanishing on ∂ Ω, are bounded below by a constant c > 0 times the distance to the boundary. The constant c can be further quantified, as the following proposition shows. Proposition A.4.2 ([163, 226]) Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain and let dΩ (x) = dist (x, ∂ Ω) denote the distance to the boundary of Ω. Assume that f ≥ 0 belongs to L ∞ (Ω) and let u denote the solution to −∆u = f in Ω, (A.18) u=0 on ∂ Ω. 212

Appendix A. Maximum principles Then,

ˆ u(x) ≥ c

Ω

f dΩ (x) d x

dΩ (x),

for all x ∈ Ω,

(A.19)

where c > 0 is a constant depending on Ω only. Proof. We follow [27]. Step 1. For any compact set K ⊂ Ω, we show that ˆ ∀x ∈ K, u(x) ≥ c f dΩ (x) d x,

(A.20)

Ω

where c > 0 depends only on K and Ω. To prove (A.20), let ρ = dist (K, ∂ Ω)/2 and cover K by m balls of radius ρ: K ⊂ Bρ (x 1 ) ∪ · · · ∪ Bρ (x m ) ⊂ Ω. For i = 1, . . . , m, let ζi denote the solution to ¨ −∆ζi = χBρ (x i ) in Ω, u=0

on ∂ Ω,

where χA denotes the characteristic function of A. The boundary point lemma (Corollary A.4.1) implies that there exists a constant c > 0 such that ζi (x) ≥ c dΩ (x)

∀x ∈ Ω

∀i = 1, . . . , m.

Take x ∈ K and a ball Bρ (x i ) containing x. Then, Bρ (x i ) ⊂ B2ρ (x) ⊂ Ω and since −∆u ≥ 0 in Ω, we conclude that

ˆ

ˆ u(x) ≥

B2ρ (x)

u dx = c

B2ρ (x)

=c

u dx ≥ c

ˆ Ω

Bρ (x i )

u dx =

u(−∆ζi ) d x = c

ˆ Ω

ˆ f ζi d x ≥ c

Ω

f dΩ d x.

´ Step 2. Fix a smooth compact set K ⊂ Ω. By (A.20), u ≥ c Ω f dΩ d x in K, so that it suffices to prove (A.19) for x ∈ Ω \ K. Let w be the solution to  in Ω \ K,  −∆w = 0 w=0 on ∂ Ω,  w=1 on ∂ K. 213

Stable solutions of elliptic PDEs The boundary point lemma (Corollary A.4.1) gives again w(x) ≥ c dΩ (x) ∀x ∈ Ω \ K. ´ Since u is superharmonic and u ≥ c Ω f dΩ d x on ∂ K, the maximum principle implies that ˆ u(x) ≥ c

Ω

ˆ

f dΩ d x

w(x) ≥ c

Ω

f dΩ d x

dΩ (x)

∀x ∈ Ω \ K.

This completes the proof.

A.5

Elliptic operators

In this section and the next, we shall discuss the generalization of the maximum principle to more general elliptic operators. To this end, observe that the conclusion of Proposition A.2.2 can be divided in two parts: • (Weak comparison principle) If u is superharmonic in Ω and u ≥ 0 on ∂ Ω, then u ≥ 0 in Ω. • (Strong comparison principle) If u is superharmonic in Ω and u ≥ 0 in all of Ω, then in fact u > 0 in Ω, unless u ≡ 0. This two-step procedure will be used in what follows. We consider elliptic operators of the form −Lu = −A(x).Hu + B(x) · ∇u + V (x)u N N X X ∂u ∂ 2u + bi (x) + V (x)u, ai j (x) =− ∂ x i ∂ x j i=1 ∂ xi i, j=1

(A.21)

where ai, j , bi , V ∈ C(Ω) for all i, j = 1, . . . , N . In addition, we assume that V ≥0

in Ω.

(A.22)

Definition A.5.1 An operator of the form (A.21) is said to be elliptic at a point x ∈ Ω if the matrix A(x) = [ai j (x)] is positive; that is, if λ(x) and Λ(x) denote respectively the minimum and maximum eigenvalues of A(x), then 0 < λ(x) |ξ|2 ≤ (A(x).ξ) · ξ ≤ Λ(x) |ξ|2 ,

for all ξ ∈ RN \ {0}.

The operator is uniformly elliptic if Λ/λ is bounded in Ω. 214

(A.23)

Appendix A. Maximum principles Proposition A.5.1 Let Ω denote a bounded domain of RN , N ≥ 1, let L denote an elliptic operator in Ω such that (A.22) holds. Let u ∈ C 2 (Ω) denote a function satisfying −Lu ≥ 0 in Ω, u≥0 on ∂ Ω. Then, u ≥ 0 in Ω. Proof. We follow [126]. Assume first that −Lu > 0 in Ω. Assume by contradiction that u(x 0 ) < 0, for some x 0 ∈ Ω. We may always assume that u achieves its minimum at x 0 . Then, u(x 0 ) ≤ 0, ∇u(x 0 ) = 0 and Hu(x 0 ) is a nonnegative matrix. Since L is elliptic, A(x 0 ) is positive, hence −Lu(x 0 ) = −A(x 0 )Hu(x 0 ) + V (x 0 )u(x 0 ) ≤ 0, contradicting −Lu > 0 in Ω. Now take a subdomain Ω0 ⊂⊂ Ω. Since ai, j , bi , V are continuous and since L is elliptic, the quotients b1 /λ and |V | /λ are bounded by some constant b0 on Ω0 . Then, since a11 ≥ λ, there exists a constant γ sufficiently large such that L eγx 1 = γ2 a11 − γb1 − V eγx 1 ≥ λ γ2 − γb0 − b0 eγx 1 > 0 in Ω0 . Take a constant c > 0 so large that eγx 1 ≤ c in Ω. Then, for any " > 0, −L(u + "(c − eγx 1 )) > 0 in Ω0 and u + "(c − eγx 1 ) ≥ 0 on ∂ Ω0 . Hence, u + "(c − eγx 1 ) ≥ 0 in Ω0 . This being true for all " > 0, Ω0 ⊂⊂ Ω, the proposition follows. To prove the strong maximum principle for general elliptic operators, we use the following boundary point lemma. Lemma A.5.1 ([129, 169]) Let Ω denote a smoothly bounded domain of RN , N ≥ 1. Assume that L is a uniformly elliptic operator such that (A.22) holds. Let u ∈ C 2 (Ω) satisfy −Lu ≥ 0 in Ω. Assume that for some x 0 ∈ ∂ Ω, u(x 0 ) ≤ 0 and u(x 0 ) < u(x) for all x ∈ Ω. Then, the outer normal derivative of u at x 0 satisfies ∂u

(A.24) (x 0 ) > 0. ∂n Exercise A.5.1 Adapt the proof of Proposition A.4.1 to establish (A.24). −

Corollary A.5.1 ([129, 169]) Let Ω denote any domain of RN , N ≥ 1. Assume that L is a uniformly elliptic operator. Let u ∈ C 2 (Ω) satisfy −Lu ≥ 0 in Ω, u≥0 in Ω. Then, u > 0 in Ω, unless u ≡ 0. 215

Stable solutions of elliptic PDEs Proof. Assume by contradiction that u vanishes at some point in Ω, while the set Ω+ = {x ∈ Ω : u(x) > 0} is nonempty. We first note that ∂ Ω+ ∩ Ω 6= ;. Otherwise, writing Ω0 = {x ∈ Ω : u(x) = 0}, we would have Ω = Ω+ t Ω0 = Ω+ t Ω0 , contradicting the fact that Ω is connected. So, there exists a point x 1 ∈ ∂ Ω+ ∩ Ω. Let d = d(x 1 , ∂ Ω) > 0 and take a ∈ Ω+ such that |a − x 1 | < d/3. In particular, d(a, ∂ Ω) ≥ 2d/3. Now set R = d(a, Ω0 ) > 0, so that B(a, R) ⊂ Ω+ . Since R ≤ |a − x 1 | < d/3, we also have B(a, R) ⊂ Ω. Then, u(x 0 ) = 0 for some x 0 ∈ ∂ B(a, R) (hence x 0 is a point of minimum of u), while u > 0 in B(a, R). Applying Lemma A.5.1, we deduce that ∇u(x 0 ) 6= 0. This contradicts the fact that x 0 is an interior point of minimum of u.

A.6

The Laplace operator with a potential

In this section, we work with elliptic operators of the form −Lu = −∆u − V (x)u, where V ∈ C 0,α (Ω). This time, we do not make any assumption on the sign of V . The validity of the maximum principle for −L is very much related to its spectrum. Definition A.6.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let V ∈ C(Ω). Assume that there exists λ ∈ R and ϕ ∈ C 2 (Ω), ϕ 6≡ 0 such that −∆ϕ − V (x)ϕ = λϕ in Ω, (A.25) ϕ=0 on ∂ Ω. Then, λ is called an eigenvalue of −L = −∆ − V (x) (with Dirichlet boundary conditions) and ϕ an eigenvector associated to λ. Definition A.6.2 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let V ∈ C(Ω). The principal eigenvalue of the operator −L = −∆ − V (x) is denoted by λ1 = λ1 (−∆ − V (x); Ω) and defined by ˆ ˆ 2 2 ∇ϕ d x − V (x)ϕ d x . (A.26) λ1 = inf ϕ∈Cc1 (Ω), kϕk L 2 (Ω) =1

Ω

Ω

We shall soon verify that λ1 is indeed an eigenvalue of −L. We start by proving that the maximum principle for −L holds if λ1 > 0. 216

Appendix A. Maximum principles Proposition A.6.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let V ∈ C 0,α (Ω). Assume that the principal eigenvalue of the operator −L = −∆ − V (x) satisfies λ1 (−∆ − V (x); Ω) > 0. If u ∈ C 2 (Ω) satisfies

−∆u − V (x)u ≥ 0 u≥0

in Ω, on ∂ Ω,

(A.27)

then, u ≥ 0 in Ω. Proof. Let u denote a solution to (A.27). Clearly, u− ∈ H01 (Ω) and so we may multiply (A.27) by u− and integrate by parts.

ˆ 0≤

ˆ −

Ω

∇u∇u d x −

V (x)u u− d x Ω ˆ ˆ 2 2 − − ∇u d x − V (x) u =− dx . Ω

Ω

Since λ1 (−∆ − V (x); Ω) > 0 is given by (A.26), we deduce that u− ≡ 0. We now return to the characterization of the principal eigenvalue λ1 . It turns out that λ1 is an eigenvalue of −L. More precisely, λ1 is the smallest eigenvalue of −L. Theorem A.6.1 Let Ω denote a smoothly bounded domain of RN , N ≥ 1 and let V ∈ C 1 (Ω). Then, −L = −∆ − V (x) has a smallest eigenvalue, called the principal eigenvalue of −L and denoted by λ1 = λ1 (−∆ − V (x); Ω). Furthermore, λ1 is characterized by either of the following statements. 1. λ1 is given by (A.26). 2. λ1 is a simple eigenvalue and there exists an eigenvector ϕ1 associated to λ1 such that ϕ1 > 0 in Ω. Furthermore, if ϕ > 0 is an eigenvector associated to an eigenvalue λ, then in fact λ = λ1 . 3. λ1 is the supremum of all λ ∈ R such that there exists v ∈ C 2 (Ω) such that v > 0 in Ω and −∆v − V (x)v ≥ λv in Ω. (A.28) 217

Stable solutions of elliptic PDEs Proof. Let µ1 = infϕ∈Cc1 (Ω),kϕk L2 (Ω) =1 Q(ϕ), where

ˆ ˆ 2 Q(ϕ) = ∇ϕ d x − V (x)ϕ 2 d x. Ω

(A.29)

Ω

Since V is bounded, Q(ϕ) ≥ −kV k L ∞ (Ω) whenever kϕk L 2 (Ω) = 1. We deduce that µ1 > −∞. Let (ϕn ) denote a minimizing sequence for µ1 , that is, µ1 ≤ Q(ϕn ) ≤ µ1 +

1 n

and kϕn k L 2 (Ω) = 1. Then, (ϕn ) is bounded in H01 (Ω), since ˆ ˆ 1 ∇ϕ 2 d x = Q(ϕ ) + V (x)ϕ 2 d x ≤ µ + + kV k ∞ ≤ C. n n 1 L (Ω) n n Ω Ω In particular, a subsequence ϕkn converges weakly in H01 (Ω) and strongly in L 2 (Ω) (see Theorem IX.16 in [25]) to a function ϕ1 ∈ H01 (Ω) such that kϕ1 k L 2 (Ω) = 1 and Q(ϕ1 ) ≤ µ1 . We also deduce that Q(ϕ1 ) = µ1 = minϕ∈H01 (Ω),kϕk L2 (Ω) =1 Q(ϕ), since Cc1 (Ω) is dense in H01 (Ω). We claim that ϕ1 is a weak solution to (A.25) with λ = µ1 , that is, ˆ ˆ ˆ B(ϕ1 , ϕ) := ∇ϕ1 ∇ϕ d x − V (x)ϕ1 ϕ d x − µ1 ϕ1 ϕ d x = 0, (A.30) Ω

Ω

Ω

for all ϕ ∈ H01 (Ω). Let Q 2 (ϕ, ϕ) := B(ϕ, ϕ). Then, for any t ∈ R and ϕ ∈ H01 (Ω), 0 ≤ Q 2 (ϕ1 + tϕ) = B(ϕ1 + tϕ, ϕ1 + tϕ) = Q 2 (ϕ1 ) + t 2Q 2 (ϕ) + 2t B(ϕ1 , ϕ) = t 2Q 2 (ϕ) + 2t B(ϕ1 , ϕ). So that,

−2t B(ϕ1 , ϕ) ≤ t 2Q 2 (ϕ).

Dividing by |t| and letting t → 0± , it follows that B(ϕ1 , ϕ) = 0, that is, (A.30) holds. Using elliptic regularity theory (see Appendix B.1), we deduce that ϕ1 ∈ C 2 (Ω) is an eigenvector of (A.25) associated to the eigenvalue λ = µ1 . Let λ ∈ R denote another eigenvalue with eigenvector ϕ normalized by kϕk L 2 (Ω) = 1. Multiply (A.25) by ϕ and integrate. Then, λ = Q(ϕ) ≥ µ1 , since µ1 minimizes Q among all functions ϕ ∈ H01 (Ω) such that kϕk L 2 (Ω) = 1. Hence, µ1 is the smallest eigenvalue of −L = −∆ − V (x) and µ1 =: λ1 is given by (A.26). 218

Appendix A. Maximum principles Now let ϕ1 denote an eigenvector associated to λ1 , normalized by kϕ1 k L 2 (Ω) = 1. We claim that ϕ1 is of constant sign throughout Ω. Indeed, with Q given by (A.29), there holds ˆ ˆ 2 + − + 2 λ1 = Q(ϕ1 ) = Q(ϕ1 ) + Q(ϕ1 ) ≥ λ1 ϕ1 d x + λ1 ϕ1− d x = λ1 . Ω

Ω

In particular, Q(ϕ1± ) = λ1 kϕ1± k2L 2 (Ω) . At least one of the functions ϕ1+ , ϕ1− cannot be identical to zero. Say ϕ1+ 6≡ 0. Then, ϕ1+ /kϕ1+ k L 2 (Ω) is a minimizer for (1.10). Working as previously, we deduce that ϕ1+ is itself an eigenvector associated to λ1 . By the strong maximum principle, we deduce that ϕ1+ > 0 in Ω, hence ϕ1 = ϕ1+ > 0. So, the sign of any eigenvector associated to λ1 is ˜1 and let constant. Now take two eigenvectors ϕ1 , ϕ ´ ˜1 d x ϕ t = ´Ω . (A.31) ϕ d x 1 Ω ˜1 = tϕ1 , that is, λ1 is simple. If this were not the case, then ψ := Then, ϕ ˜1 − ´tϕ1 would be an eigenvector, hence it would be of constant sign. But ϕ then Ω ψ d x = 0 by (A.31), a contradiction. Next, we show that if ϕ > 0 is an eigenvector associated to λ, then λ = λ1 . Let ϕ1 > 0 denote an eigenvector associated to λ1 , multiply (A.25) by ϕ1 , and integrate. We obtain ˆ ˆ ˆ ∇ϕ∇ϕ1 − V (x)ϕϕ1 d x = λ1 ϕϕ1 d x λ ϕϕ1 d x = Ω

Ω

Ω

and thus, λ = λ1 . This proves Point 2 of the theorem. Let µ1 denote the supremum of all λ ∈ R such that for some v ∈ C 2 (Ω), v > 0 in Ω, (A.28) holds. Fix λ < µ1 . Take ϕ ∈ Cc1 (Ω) and multiply (A.28) by ϕ 2 /v: ˆ ˆ ϕ2 2 λ ϕ d x ≤ (−∆v − V (x)v) dx v Ω Ω ˆ ϕ2 2 ≤ ∇v · ∇ − V (x)ϕ dx v Ω ˆ 2 ˆ ˆ ϕ ϕ 2 ≤− |∇v| d x + 2 ∇v · ∇ϕ d x − V (x)ϕ 2 d x 2 Ω v Ω ˆ Ω v ˆ 2 ≤ ∇ϕ d x − V (x)ϕ 2 d x, Ω

Ω

219

Stable solutions of elliptic PDEs where we used Young’s inequality in the last inequality. By (A.26), we deduce that λ ≤ λ1 , hence µ1 ≤ λ1 . Now, by Point 2 of the theorem, for λ = λ1 , there exists v = ϕ1 > 0 satisfying (A.28). So, λ1 ≤ µ1 , hence λ1 = µ1 and Point 3 of the theorem is proven.

A.7

Thin domains and unbounded domains

In this section, we collect two useful versions of the maximum principle. The first result applies to bounded domains of small measure. Proposition A.7.1 ([18]) Let Ω denote a bounded open set of RN , N ≥ 1, V (x) ∈ L p (Ω), p > N /2. There exists " > 0 such that if |Ω| < ", then for any function u ∈ C 2 (Ω) ∩ C 1 (Ω) satisfying −∆u + V (x)u ≥ 0 in Ω, u≥0 on ∂ Ω, we have u ≥ 0 in Ω. Proof. Multiply the equation by u− and integrate by parts. We obtain ˆ ˆ ∇u− 2 d x + V (x) u− 2 d x ≤ 0. Ω

Ω

By Sobolev and Hölder’s inequalities, there exists a constant CN > 0 such that ˆ CN

Ω

2N /(N −2) u− dx

N −2 N

ˆ ˆ 2 2 − ≤ ∇u d x ≤ − V (x) u− d x Ω

≤ kV (x)k L p (Ω) hence (CN − kV (x)k L p (Ω) |Ω|

2p−N 2p

)

ˆ Ω

ˆ Ω

Ω

2N /(N −2) u− dx

2N /(N −2) u− dx 2p−N

N −2 N

|Ω|

2p−N 2p

,

N −2 N

≤ 0.

If |Ω| > 0 is sufficiently small, CN −kV (x)k L p (Ω) |Ω| 2p > 0 and we deduce that u− ≡ 0, that is, u ≥ 0 in Ω. Our next result applies in any domain, provided there exists a suitable barrier function. 220

Appendix A. Maximum principles Proposition A.7.2 Let Ω denote an open set of RN , N ≥ 1, V (x) ∈ C(Ω). Assume that there exists a function w ∈ C 2 (Ω) such that ¨ −∆w − V (x)w ≥ 0 in Ω (A.32) w>0 in Ω. Assume that u ∈ C 2 (Ω) satisfies −∆u − V (x)u ≥ 0 u≥0

in Ω on ∂ Ω,

(A.33)

and if Ω is unbounded, also assume that lim

|x|→+∞, x∈Ω

u(x) w(x)

= 0.

(A.34)

Then, u ≥ 0 in Ω. Proof. Assume by contradiction that ω = {x ∈ Ω : u(x) < 0} is nonempty. Then, σ = wu solves − ∇ · (w 2 ∇σ) = w(−∆u) − u(−∆w) = w(−∆u − V (x)u) − u(−∆w − V (x)w) ≥ 0

in ω.

In addition, σ ≥ 0 on ∂ ω and lim|x|→+∞ σ(x) = 0. So, given " > 0, there exists R0 > 0 such that for every R > R0 , ¨ −∇ · (w 2 ∇σ) ≥ 0 in ω ∩ BR , (A.35) σ ≥ −" on ∂ (ω ∩ BR ). By the maximum principle (Proposition A.5.1), we deduce that σ ≥ −" in ω ∩ BR for every R > R0 . Letting R → +∞ and then " → 0, we deduce that σ ≥ 0 in ω. This is a contradiction with the definition of ω.

A.8

Nonlinear comparison principle

Proposition A.8.1 Let Ω denote a bounded domain of RN , N ≥ 1, f : R → R a locally Lipschitz function and u, u ∈ C 2 (Ω) a sub- and a supersolution to (1.3). Then, if u ≤ u in Ω, either u ≡ u or u < u. 221

Stable solutions of elliptic PDEs Proof. Let a = kuk∞ + kuk∞ and let K denote the Lipschitz constant of f on [−a, a]. Then, f (u) − f (u) ≥ −K(u − u). In particular, w := u − u solves

−∆w + K w = f (u) − f (u) + K w ≥ 0 w≥0

in Ω, on ∂ Ω.

By the strong maximum principle (Corollary A.5.1), w ≡ 0 or w > 0 in Ω.

A.9

L 1 theory for the Laplace operator

A.9.1

Linear theory and weak comparison principle

Lemma A.9.1 ([29]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Let dΩ denote the distance to the boundary of Ω, as defined in (3.4). Given f ∈ L 1 (Ω, dΩ (x)d x), there exists a unique solution u ∈ L 1 (Ω) of −∆u = f in Ω, u=0 on ∂ Ω, in the sense that

ˆ Ω

ˆ u(−∆ϕ) d x =

Ω

f ϕ d x,

for every ϕ ∈ C02 (Ω). In addition, there exists a constant C = C(Ω, N ) > 0 such that kuk L 1 (Ω) ≤ Ck f k L 1 (Ω,dΩ (x)d x) (A.36) and the following comparison principle holds: f ≥ 0 a.e.

=⇒

u ≥ 0 a.e.

Proof. Step 1. Existence of a solution By splitting f into its positive and negative parts f = f + − f − , it suffices to consider the case f ≥ 0 a.e. Given n ∈ N, let f n = min( f , n). Since f n is bounded, there exists a unique un solving −∆un = f n in Ω, (A.37) un = 0 on ∂ Ω. 222

Appendix A. Maximum principles Note that by the standard maximum principle, the sequence (un ) is nondecreasing. Now take ψ ∈ Cc∞ (Ω) and let ϕ ∈ C02 (Ω) denote the solution to

−∆ϕ = ψ ϕ=0

in Ω, on ∂ Ω.

(A.38)

Multiplying the above equation by un and integrating, we obtain ˆ ˆ un ψ d x = f n ϕ d x. Ω

Ω

It follows that ˆ 0 u ψ d x ≤ Ck f k 1 n n L (Ω,dΩ (x)d x) kϕkC 1 (Ω) ≤ C k f k L 1 (Ω,dΩ (x)d x) kψk L ∞ (Ω) . Ω

Taking the supremum over all functions ψ such that kψk L ∞ (Ω) ≤ 1, we deduce that kun k L 1 (Ω) ≤ C 0 k f k L 1 (Ω,dΩ (x)d x) . By monotone convergence, the sequence (un ) tends to u ∈ L 1 (Ω) solving the equation in the weak sense and satisfying (A.36). Step 2. Comparison principle and uniqueness Clearly, uniqueness is a direct consequence of the comparison principle. So, it suffices to prove the latter. If ψ ≥ 0, we have, as in Step 1, ˆ ˆ f ϕ d x ≥ 0. uψ d x = Ω

Ω

This being true for every ψ ≥ 0, we deduce that u ≥ 0 a.e. Lemma A.9.1 can be extended to the setting of measures, as follows.

Corollary A.9.1 Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Given a Radon measure µ such that dΩ ∈ L 1 (Ω, d|µ|), there exists a unique solution u ∈ L 1 (Ω) of −∆u = µ in Ω, u=0 on ∂ Ω, in the sense that

ˆ Ω

ˆ u(−∆ϕ) d x =

for every ϕ ∈ C02 (Ω). In addition, 223

Ω

ϕ dµ,

Stable solutions of elliptic PDEs (i) For every p ∈ [1, NN−1 ), there exists a constant C = C(Ω, N , p) > 0 such that kuk L p (Ω) ≤ CkdΩ µkM (Ω) , (A.39) where k · kM (Ω) denotes that total variation of a measure. (ii) If a sequence (dΩ µn ) is bounded in M (Ω), then the corresponding sequence of solutions (un ) is relatively compact in L p (Ω) for all p ∈ [1, NN−1 ). (iii) The following comparison principle holds: µ ≥ 0 as a measure

=⇒

u ≥ 0 a.e.

Proof. The comparison principle and uniqueness are proved exactly as in Step 2 of the previous lemma. For the existence of a solution, choose a sequence ( f n ) in L 1 (Ω, dΩ (x)d x) such that dΩ f n * dΩ µ in M (Ω). Let un ∈ L 1 (Ω) denote the solution to (A.37) and ϕ the solution to (A.38) for a given ψ ∈ Cc∞ (Ω). Multiplying (A.38) by un and integrating, we obtain ˆ ˆ un ψ d x = f n ϕ d x. Ω

Ω

It follows that ˆ 0 u ψ d x ≤ Ck f k 1 n n L (Ω,dΩ (x)d x) kϕkC 1 (Ω) ≤ C kdΩ µkM (Ω) kψk L q (Ω) , Ω

where q > N /2 (so that kϕkC 1 (Ω) ≤ Ckψk L q , by elliptic regularity). Letting p denote the conjugate exponent of q and taking the supremum over all functions ψ such that kψk L q (Ω) ≤ 1, we deduce that kun k L p (Ω) ≤ C 0 kdΩ µkM (Ω) . In particular, a subsequence of (un ) converges weakly in L p (Ω) to a solution u of the equation satisfying (A.39). It remains to be proven the compactness result (ii). Take a sequence of measures (µn ) such that kdΩ µn kM (Ω) ≤ C and a smooth domain ω ⊂⊂ Ω. Let vn ∈ L 1 (ω) denote the solution to ¨ −∆vn = µn ω in ω, vn = 0

on ∂ ω.

224

Appendix A. Maximum principles We claim that (vn ) is bounded in W 1,p (ω) for every p < NN−1 . To see this, take any ψ ∈ Cc1 (ω) and let ϕ ∈ C02 (ω) denote the solution to

−∆ϕ = ψ ϕ=0

in ω, on ∂ ω.

By Theorem B.4.1, kϕk L ∞ (ω) ≤ CkψkW −1,p0 (ω) . So, ˆ ˆ v ψ d x = ϕdµ ≤ kϕk ∞ kµ k n n L (ω) n M (ω) ≤ Ckϕk L ∞ (ω) ≤ CkψkW −1,p0 (ω) . ω

ω

This being true for all ψ, the claim follows. In addition to this, since un − vn is harmonic in ω, we also have for ω0 ⊂⊂ ω, kun − vn kC 1 (ω0 ) ≤ Cω0 kun − vn k L 1 (ω) ≤ C 0 kµn dΩ kM (Ω) ≤ C 00 . So, the sequence (un ) is bounded in W 1,p (ω0 ) for any p < NN−1 and any ω0 ⊂⊂ Ω. By a standard diagonalization argument, there exists a subsequence ukn → u a.e. in Ω. Since (ukn ) is bounded in L p (Ω) for p < NN−1 , we conclude using Egorov’s theorem. Exercise A.9.1 Assume that µ ∈ M (Ω), that is, µ is integrable up to the boundary. Prove that for every p ∈ [1, NN−2 ), there exists a constant C = C(Ω, N , p) such that kuk L p (Ω) ≤ CkµkM (Ω) .

A.9.2

The boundary-point lemma

The refined boundary point lemma (see Proposition A.4.2) also holds in the general L 1 setting. This is the content of the next corollary. Corollary A.9.2 Let N ≥ 1, let Ω ⊂ RN denote a smoothly bounded domain and let dΩ given by (3.4). Assume that µ ≥ 0 is a Radon measure such that dΩ is µ-integrable and let u ∈ L 1 (Ω) denote the solution to −∆u = µ in Ω, (A.40) u=0 on ∂ Ω. 225

Stable solutions of elliptic PDEs Then,

ˆ u(x) ≥ c

Ω

dΩ dµ dΩ (x),

for a.e. x ∈ Ω,

(A.41)

where c > 0 is a constant depending on Ω only. Proof. Take a sequence of functions f n ∈ Cc∞ (Ω) such that dΩ f n * dΩ µ in M (Ω). Let un denote the associated solution to (A.37). By (A.39), un * u in L p (Ω) for p ∈ (1, NN−1 ). By Lemma A.4.2, we also have for x ∈ Ω, ˆ un (x) ≥ c f n dΩ d y dΩ (x). Ω

Take any ψ ∈ Cc∞ (Ω) such that ψ ≥ 0 and integrate. Then, ˆ ˆ un (x) − c f n dΩ d y dΩ (x) ψ d x ≥ 0. Ω

Ω

Passing to the (weak) limit, it follows that ˆ ˆ u(x) − c dΩ dµ dΩ (x) ψ d x ≥ 0. Ω

Ω

The above inequality being true for all ψ ≥ 0, (A.41) follows.

A.9.3

Sub- and supersolutions in the L 1 setting

We establish the following extension of the method of sub- and supersolutions. Theorem A.9.1 ([159]) Let N ≥ 1 and Ω ⊂ RN denote a smoothly bounded domain. Let dΩ denote the distance to the boundary of Ω, as defined in (3.4). Let f ∈ C 1 (R). Assume that there exists u, u ∈ L 1 (Ω) a weak sub- and supersolution of (1.3), that is, f (u)dΩ , f (u)dΩ ∈ L 1 (Ω) and ˆ ˆ − u∆ϕ d x ≤ f (u)ϕ d x, for all ϕ ∈ C02 (Ω), ϕ ≥ 0 Ω

Ω

and the reverse inequality holds for u. Assume that u ≤ u a.e. and f (v)dΩ ∈ L 1 (Ω),

for every v ∈ L 1 (Ω) such that u ≤ v ≤ u a.e.

(A.42)

Then, there exists the minimal solution u ∈ L 1 (Ω) of (1.3) relative to u, that is, u solves (1.3) and u ≤ u ≤ v a.e., for every weak supersolution v such that v ≥ u a.e. 226

Appendix A. Maximum principles Exercise A.9.2 Using the monotone iteration scheme (see Exercise 1.2.1), prove Theorem A.9.1 under the additional assumption that f is nondecreasing. Note that in this case, (A.42) is automatically satisfied. To prove the theorem, we first establish a series of intermediate results. Lemma A.9.2 Let (w n ) denote a sequence of functions in L 1 (Ω) and let (En ) a sequence of measurable subsets of Ω such that ˆ |En | → 0 and |w n | d x ≥ 1 ∀n ≥ 1. (A.43) En

Then, there exists a subsequence (w nk ) and a sequence of disjoint measurable sets (Fk ) such that ˆ 1 |w nk | d x ≥ Fk ⊂ Enk and ∀k ≥ 1. (A.44) 2 Fk Proof. Set n1 = 1, n2 = 2, A1 = E1 , and A2 = E2 . By induction, we construct an increasing sequence of integers (nk ) and measurable sets (Ak ) as follows. Assume that k ≥ 3, n1 , . . . , nk−1 and A1 , . . . , Ak−1 are such that A j ⊂ E j and ˆ 1 1 |w n j | d x ≤ − k−1 ∀ j = 1, . . . , k − 2. 2 2 A j+1 ∪···∪Ak−1 Since |En | → 0, then for nk > nk−1 sufficiently large we have ˆ 1 |w n j | d x ≤ k ∀ j = 1, . . . , k − 1. 2 Enk Let Ak = Enk . Then, ˆ A j+1 ∪···∪Ak

Now set

|w n j | d x ≤

1 2

−

1 2k

∀ j = 1, . . . , k − 1.

F k = A k \ ∪∞ A. i=k+1 i

Then the sets Fk are disjoint and ˆ ˆ |w nk | d x = lim Fk

i→+∞

1 |w nk | d x ≥ . 2 Ak \(Ak+1 ∪···∪Ai ) 227

Stable solutions of elliptic PDEs Proposition A.9.1 Let g : Ω × R → R be a Carathéodory function and u, u ∈ L 1 (Ω) be such that u ≤ u a.e. Assume that g(·, v)dΩ ∈ L 1 (Ω) for every v ∈ L 1 (Ω) such that u ≤ v ≤ u a.e. Then, the set ¦ © B = g(·, v) ∈ L 1 (Ω, dΩ d x) : v ∈ L 1 (Ω) and u ≤ v ≤ u a.e.

(A.45)

(A.46)

is bounded and equi-integrable in L 1 (Ω, dΩ d x). Proof. Recall that a set B ⊂ L 1 (Ω; dΩ d x) is equi-integrable if for every " > 0, there exists δ > 0 such that ˆ E ⊂ Ω and |E| < δ =⇒ |g|dΩ d x < " ∀ g ∈ B. E

Since Ω is bounded, it suffices to show that B is equi-integrable. Assume by contradiction that B is not equi-integrable. Then, there exists " > 0, a sequence (un ) in L 1 (Ω) with u ≤ un ≤ u a.e., and a sequence of measurable sets (En ) such that ˆ g(x, un )dΩ d x ≥ " ∀n ≥ 1. |En | → 0 and En

Applying Lemma A.9.2 with w n = g(·, un )dΩ /", we find a subsequence (unk ) and a sequence of disjoint measurable sets (Fk ) such that ˆ " |g(x, unk )|dΩ d x ≥ ∀k ≥ 1. (A.47) 2 Fk Set

¨ v(x) =

unk (x) if x ∈ Fk for some k ≥ 1, u(x) otherwise.

Then, u ≤ v ≤ u a.e. Hence, v ∈ L 1 (Ω). Moreover,

ˆ Ω

|g(x, v)|dΩ d x ≥

+∞ ˆ X k=1

Fk

|g(x, unk )|dΩ d x = +∞.

This contradicts (A.46). Therefore B is equi-integrable in L 1 (Ω; dΩ d x). 228

Appendix A. Maximum principles Proposition A.9.2 Let g : Ω × R → R be a Carathéodory function such that g(·, v)dΩ ∈ L 1 (Ω)

for every v ∈ L 1 (Ω).

(A.48)

Then, the Nemytskii operator ¨ G:

L 1 (Ω) → L 1 (Ω; dΩ d x) v 7→ g(·, v)

is continuous. Proof. Assume that vn → v in L 1 (Ω). Extract a sequence (vnk ) such that vnk → v a.e. and |vnk | ≤ V a.e., for some V ∈ L 1 (Ω). In particular, g(·, vnk ) → g(·, v)

a.e.

Moreover, by Proposition A.9.1 (applied with u = −V , U = V ), the sequence (g(·, vnk )) is equi-integrable in L 1 (Ω; dΩ d x). By Egorov’s theorem, g(·, vnk ) → g(·, v)

in L 1 (Ω; dΩ d x).

Since the limit is independent of the subsequence (vnk ), we deduce that G(vn ) → G(v) in L 1 (Ω; dΩ d x). Proof of Theorem A.9.1. Step 1. There exists a solution u of (1.3) such that u ≤ u ≤ u a.e. For t ≥ 0, x ∈ Ω, set   f (u(x)) if t < u(x), f (t) if u(x) ≤ t ≤ u(x), g(x, t) =  f (u(x)) if t > u(x). Then g is a Carathéodory function and by (A.42), g(·, v)dΩ ∈ L 1 (Ω)

for every v ∈ L 1 (Ω).

Consider the operators ¨ G:

L 1 (Ω) → L 1 (Ω; dΩ d x), v 7→ g(·, v), 229

Stable solutions of elliptic PDEs and

¨ K:

L 1 (Ω; dΩ d x) → L 1 (Ω) h 7→ w,

where w is the unique solution to −∆w = h w=0

in Ω, on ∂ Ω.

Proposition A.9.2 shows that G is continuous, while Corollary A.9.1 implies that K is compact. Hence, K G : L 1 (Ω) → L 1 (Ω) is compact. By Proposition A.9.1, G(L 1 (Ω)) is bounded in L 1 (Ω; dΩ d x). So, kK(G(v))k L 1 (Ω) ≤ C1 kG(v)dΩ k L 1 (Ω) ≤ C

∀v ∈ L 1 (Ω).

By Schauder’s fixed-point theorem, K G has a fixed point u ∈ L 1 (Ω). In other words, u solves −∆u = g(x, u) in Ω, u=0 on ∂ Ω. We claim that u≤u≤u

a.e.

(A.49)

This implies in particular that u solves (1.3). To prove (A.49), we note that g(·, u) = g(·, u) a.e. on the set [u ≥ u]. Thus, applying Kato’s inequality (see (3.19)) to w = u − u, we get ˆ ˆ + − w ∆ϕ d x ≤ (g(x, u) − g(x, u))ϕ d x = 0 ∀ϕ ∈ C02 (Ω), ϕ ≥ 0. Ω

[u≥u]

By the maximum principle (see Corollary A.9.1), w + ≤ 0 a.e. This implies that u ≤ u a.e. The inequality u ≥ u is obtained similarly. So, we have obtained a solution u of (1.3) such that u ≤ u ≤ u. Step 2. There exists the smallest solution u in the interval [u, u]. Claim. If u1 , u2 are two solutions of (1.3) such that u ≤ u1 , u2 ≤ u a.e., then there exists a solution w such that u ≤ w ≤ min(u1 , u2 ) ≤ u

a.e.

(A.50)

Indeed, by Kato’s inequality, min(u1 , u2 ) = u2 − (u1 − u2 )− is a supersolution to (1.3). We then apply Step 1 to find the desired solution w. Now set ˆ A = inf u d x : u ≤ u ≤ u a.e. and u is a solution to (1.3) . Ω

230

Appendix A. Maximum principles It follows from the above claim that one can find a nonincreasing sequence of solutions (w n ) such that ˆ u ≤ w n ≤ u and w n d x → A. (A.51) Ω

By monotone convergence, there exists u ∈ L 1 (Ω) such that w n → u a.e., ˆ ˆ u ≤ u ≤ u and w n d x → u d x = A. Ω

Ω

By Proposition A.9.1, ( f (w n )) is equi-integrable and bounded in L 1 (Ω; dΩ d x), hence f (w n ) → f (u) in L 1 (Ω; dΩ d x). So, u is a solution to (1.3) and

ˆ Ω

u d x = A.

According to the claim, u is the smallest solution to (1.3) lying in the interval [u, u]. Step 3. u ≤ v a.e., for every supersolution v such that v ≥ u a.e. Using Kato’s inequality, w = min(u, v) = u − (v − u)− is a supersolution ˜ lying in the interval to (1.3) and u ≤ w. By Step 1, there exists a solution u ˜ ≤ w ≤ u. Since u is the smallest solution in the [u, w]. By definition of w, u ˜ lies in that interval, we also have u ≤ u ˜. Hence, interval [u, u] and since u ˜ ≤ w = min(u, v) and so u ≤ v. u=u

231

Appendix B Regularity theory for elliptic operators The following two sections follow [11, 126, 222].

B.1

Harmonic functions

In this section, we establish basic elliptic regularity estimates for harmonic functions and solve the Dirichlet problem.

B.1.1

Interior regularity

1 Proposition B.1.1 Let Ω be an open set of RN , N ≥ 1. Let u ∈ L loc (Ω) be a ∞ N harmonic function. Then, u ∈ C (Ω), and given any k ∈ N , there exists a constant CN ,k , depending on N and k only, such that

|D k u(x 0 )| ≤

CN ,k r |k|

kuk L ∞ (B(x 0 ,r)) ,

(B.1)

for every ball B(x 0 , r) ⊂ Ω. Proof. Let u ∈ L l1oc (Ω) be a harmonic function. By Exercise A.1.3, u ∈ C 2 (Ω). We establish (B.1) by induction on |k|. The case |k| = 0 is a straightforward consequence of the mean-value formula (A.10). Now differentiate the meanvalue formula (A.10) with respect to x i . Then, v = ∂ u/∂ x i also satisfies the 233

Stable solutions of elliptic PDEs mean-value formula. By Exercise A.1.3, v is harmonic. In particular, u ∈ C 3 (Ω). Iterating this argument, we deduce that u ∈ C ∞ (Ω). In addition, ∂u ∂u (x ) = d y ∂ x 0 ∂ x B(x 0 ,r/2) i i N ˆ 2 1 un dσ = i |B | r ∂ B(x 0 ,r/2)

1

≤

C r

kuk L ∞ (∂ B(x 0 ,r/2)) .

(B.2)

This implies (B.1) for |k| = 1. Now take n ≥ 1 and assume that (B.1.1) holds for all |k| ≤ n. Take k0 ∈ NN such that |k0 | = n + 1. Then, 0

Dk u =

∂ ∂ xi

D k u,

for some i ∈ {1, . . . , N } and k ∈ NN such that |k| = n. By (B.2) applied to v = D k u, we have C 0 |D k u(x 0 )| ≤ kD k uk L ∞ (∂ B(x 0 ,r/2)) . (B.3) r Now, if x ∈ ∂ B(x 0 , r/2), then B(x, r/2) ⊂ B(x 0 , r) ⊂ Ω. Using the induction hypothesis, we deduce that kD k uk L ∞ (∂ B(x 0 ,r/2)) ≤

C r |k|

kuk L ∞ (B(x 0 ,r)) .

This together with (B.3) yields the desired result. N Exercise B.1.1 Let u be a harmonic function in Ω ⊂ R , N ≥ 1. Prove that for every ball B(x 0 , r) ⊂ Ω and every k ∈ NN , |D k u(x 0 )| ≤

CN ,k r N +|k|

kuk L 1 (B(x 0 ,r)) ,

where CN ,0 = 1/|B1 | and for |k| ≥ 1, CN ,k ≤

|k| 2N +1 N |k| |B1 |

Deduce that u is analytic. 234

.

Appendix B. Regularity theory for elliptic operators

B.1.2

Solving the Dirichlet problem on the unit ball

In this section, we show that given any function g ∈ C(∂ B), there exists a unique function u ∈ C 2 (B) ∩ C(B) solving −∆u = 0 in B, (B.4) u= g on ∂ B. Suppose for the moment that u ∈ C 2 (B) is a solution to (B.4). Fix a point x ∈ B, multiply by Γ( y − x), where Γ is the fundamental solution of the Laplace operator, given by (A.6) and (A.7), and integrate over B \ B(x, "), " > 0. Then, ˆ 0= −∆u( y)Γ( y − x) d y = B\B(x,") ˆ ∂u ∂Γ − ( y)Γ( y − x) + u( y) ( y − x) dσ( y)− ∂n ∂n ∂B ˆ ∂u ∂Γ − ( y)Γ( y − x) + u( y) ( y − x) dσ( y). ∂n ∂n ∂ B(x,") Letting " → 0, it is not hard to deduce that ˆ ∂u ∂Γ − ( y)Γ( y − x) + u( y) ( y − x) dσ( y). u(x) = ∂n ∂n ∂B More generally, if h ∈ C 2 (B) is harmonic and G(x, y) = Γ( y − x) − h( y), the same computation leads to ˆ ∂u ∂G u(x) = − ( y)G(x, y) + u( y) (x, y) dσ( y). ∂n ∂ ny ∂B If h can be chosen so that G(x, y) = 0 for all y ∈ ∂ B, then the above equation simplifies to ˆ ∂G u(x) = g( y) (x, y) dσ( y). (B.5) ∂ ny ∂B Note that (B.5) provides a representation formula for u in terms of its boundary value g. G is called the Green’s function for the Laplace operator on the ball, and h the associated corrector function. It remains to construct such a corrector function. To this end, we shall use the following simple geometric identity. 235

Stable solutions of elliptic PDEs Lemma B.1.1 Take two nonzero vectors x, y ∈ RN , N ≥ 1. Then, x y = . − | y|x − |x| y |x| | y| Proof. Simply square both sides and expand using the inner product. Recall that we are looking for a harmonic function h( y) = h x ( y) that agrees with Γ(x − y) for y ∈ ∂ B. Let x ∗ = x/|x|2

(B.6)

be the image of x under the inversion in the unit sphere ∂ B. In particular, since x ∈ B, x ∗ ∈ RN \ B and so, given any constant k ∈ R, hk ( y) = Γ(k( y − x ∗ )) is harmonic in B. By the symmetry lemma (Lemma B.1.1), we have for all y ∈ ∂ B, x − |x| y = |x| · | y − x ∗ |. | y − x| = |x| Hence, choosing k = |x|, we see that h( y) = Γ(|x|( y − x ∗ )) is the desired corrector function. Plugging this in the definition of the Green’s function G = G(x, y)Γ( y − x) − h( y), we obtain after simplification P(x, y) =

∂G ∂ ny

(x, y) =

1 − |x|2 |x − y|N

,

for x ∈ B, y ∈ ∂ B.

(B.7)

P is called the Poisson kernel for the ball. Theorem B.1.1 Let g ∈ C(∂ B). There exists a unique solution u ∈ C 2 (B)∩C(B) to (B.4), given by  u(x) =



ˆ ∂B

P(x, y)g( y) dσ( y) if x ∈ B,

(B.8)

g(x) if x ∈ ∂ B,

 where P is the Poisson kernel given by (B.7).

Proof. The uniqueness of u follows from the maximum principle. To prove that u is indeed a solution, we proceed in steps. Step 1. Let y ∈ ∂ B. Then, P(·, y) is harmonic in RN \ { y}. 236

Appendix B. Regularity theory for elliptic operators By construction, y 7→ G(x, y) = Γ(x, y) − h( y) is a harmonic function of y in RN \ {x}. In addition, using the symmetry lemma (Lemma B.1.1), G(x, y) = Γ(x − y) − Γ(|x| · | y − x ∗ |) = G( y, x), for all x 6= y. Hence, x 7→ G(x, y) is harmonic in RN \ { y}, and so must be P(·, y) = ∂∂nG (·, y). y

Step 2. The Poisson kernel verifies (a) P(x, y) > 0, for all x ∈ B, y ∈ ∂ B, ´ (b) ∂ B P(x, y) dσ( y) = 1, for all x ∈ B, and ˆ (c) for every y0 ∈ ∂ B and every δ > 0, lim x→ y0

P(x, y) dσ( y) = 0.

| y− y0 |>δ

Properties (a) and (c) follow directly from (B.7), while (b) is a consequence of (B.5) and (B.7) , applied to u ≡ 1. Step 3. The function u given by (B.8) is harmonic in B. By Step 1, P(·, y) is harmonic in B. We may then safely differentiate under the integral sign to conclude that u is harmonic in B. Step 4. u is continuous on B. Fix y0 ∈ ∂ B and " > 0. Choose δ > 0 so small that |g( y) − g( y0 )| < ", for | y − y0 | < δ. Using Step 2, we deduce that ˆ |u(x) − u( y0 )| = (g( y) − g( y0 ))P(x, y) dσ( y) ∂B ˆ ˆ |g( y)−g( y0 )|P(x, y) dσ( y) |g( y)−g( y0 )|P(x, y) dσ( y)+ ≤ | y− y0 |>δ | y− y0 |≤δ ˆ ≤ " + 2kgk L ∞ (∂ B) P(x, y) dσ( y). | y− y0 |>δ

The last term above is less than " for x sufficiently close to y0 , hence, u is continuous at y0 .

B.1.3

Solving the Dirichlet problem on smooth domains

In this section, we describe the Perron method for solving the Dirichlet problem in a smoothly bounded domain Ω ⊂ RN , that is, we prove the following theorem. 237

Stable solutions of elliptic PDEs Theorem B.1.2 Assume that Ω ⊂ RN is a smoothly bounded domain and let g ∈ C(∂ Ω). There exists a unique solution u ∈ C 2 (Ω) ∩ C(Ω) to −∆u = 0 in Ω, (B.9) u= g on ∂ Ω. To prepare the proof for the above theorem, we shall use the following basic properties of subharmonic functions. Proposition B.1.2 If u1 and u2 are subharmonic functions on Ω, then so is max(u1 , u2 ). Proof. This simply follows from the fact that a function is subharmonic if and only if it satisfies the mean-value inequality (see Exercise A.1.3). Proposition B.1.3 Let u be subharmonic in Ω and take a ball B ⊂ Ω. Let w be the subharmonic lift of u, that is, take v ∈ C 2 (B) ∩ C(B) the solution to −∆v = 0 in B, v=u on ∂ B. given by Theorem B.1.1 and set w=v w=u

in B, in Ω \ B.

Then, w is subharmonic, and u ≤ w. Proof. Apply the maximum principle in B. It follows that u ≤ w in Ω. Now fix a point x ∈ Ω. If x ∈ B, then w is harmonic in any ball B(x, r) ⊂ B. In particular, w(x) =

w dσ. ∂ B(x,r)

If x 6∈ B, then, w(x) = u(x). Since u is subharmonic and u ≤ w, we have for all the balls B(x, r) ⊂ Ω, w(x) = u(x) ≤

∂ B(x,r)

u dσ ≤

w dσ. ∂ B(x,r)

So, w satisfies the mean-value inequality for every x ∈ Ω and for every sufficiently small ball B(x, r), hence w is subharmonic in Ω. 238

Appendix B. Regularity theory for elliptic operators Proof of Theorem B.1.2. Take g ∈ C(∂ Ω), and let m = min∂ Ω g, M = max∂ Ω g. We write v ∈ S g whenever v ∈ C(Ω) is subharmonic in Ω and v ≤ g on ∂ Ω. Note that S g is nonempty, since v ≡ m ∈ S g . We want to prove that the function u defined for x ∈ Ω by u(x) = sup{v(x) : v ∈ S g } (B.10) is the solution to (B.9). Note that u is well defined and that m ≤ u ≤ M on Ω. Step 1. u is harmonic on Ω. Fix a point x 0 ∈ Ω and a ball B ⊂ Ω centered at x 0 . Choose a sequence (vk ) in S g , such that vk (x 0 ) → u(x 0 ). Replace vk by the fonction w k obtained as the subharmonic lift of max{m, v1 , . . . , vk } in B. Then, w k ∈ S g , w k is harmonic in B, and (w k ) is a nondecreasing sequence. In particular, w k (x 0 ) → u(x 0 ). Also, (w k ) is uniformly bounded. By Proposition B.1.1, (w k ) converges uniformly on compact subsets of B to a smooth function w. Passing to the limit in the mean-value inequality, we deduce that w is harmonic in B. It remains to be proven that u = w in B. Clearly, w ≤ u in B. To prove the reverse inequality, take v ∈ S g , and let zk be the subharmonic lift of max{w k , v} in B. Since u(x 0 ) = w(x 0 ) and zk ∈ S g , we have zk (x 0 ) ≤ w(x 0 ) for all k. In addition, zk is harmonic in B and max{w k , v} ≤ zk in B by the maximum principle. Thus, w(x 0 ) ≥ zk (x 0 ) =

zk d x ≥ B

max{w k , v} d x. B

Applying the mean-value equality on the left-hand side and letting k → +∞ on the right-hand side, we obtain w dx ≥ B

max(w, v) d x. B

It follows that v ≤ w in B. This being true for all v ∈ S g , we deduce that u ≤ w in B, as desired. Step 2. Given any x 0 ∈ ∂ Ω, there exists a barrier function ζ at x 0 , that is, ζ ∈ C(Ω) is superharmonic in Ω, ζ > 0 in Ω \ {x 0 }, and ζ(x 0 ) = 0. Since Ω is bounded and smooth, there exists an exterior ball B = B( y0 , r) ⊂ N R \ Ω such that ∂ Ω ∩ ∂ B = {x 0 }. Define ζ in RN \ { y0 } by ζ(x) = Γ(x − y0 ) − Γ(r), where Γ is the fundamental solution to the Laplace equation given by (A.6) and (A.7). Then, ζ is the desired barrier. 239

Stable solutions of elliptic PDEs Step 3. End of proof. Let ζ be the barrier at x 0 ∈ ∂ Ω, constructed in Step 2. Let " > 0. By continuity of g on ∂ Ω, if r > 0 is sufficiently small, g(x 0 ) − " < g < g(x 0 ) + " in ∂ Ω ∩ B(x 0 , r). Since ζ is positive and continuous on the compact set Ω \ B(x 0 , r), there exists a positive constant C > 0 such that g(x 0 ) − " − Cζ < g < g(x 0 ) + " + Cζ,

on ∂ Ω \ B(x 0 , r).

(B.11)

Note that g(x 0 ) − " − Cζ ∈ S g , hence g(x 0 ) − " − Cζ ≤ u, where u is given by (B.10). Now, for any v ∈ S g , v ≤ g on ∂ Ω, and therefore v − Cζ < g(x 0 ) + " on ∂ Ω, by (B.11). By the maximum principle, v − Cζ < g(x 0 ) + " on Ω. This being true for all v ∈ S g , we obtain at last g(x 0 ) − " − Cζ ≤ u ≤ g(x 0 ) + " + Cζ,

on Ω.

Since ζ is continuous at x 0 , ζ(x 0 ) = 0, and " > 0 is arbitrary, we deduce that lim x→x 0 u(x) = g(x 0 ), as desired.

B.2 B.2.1

Schauder estimates Poisson’s equation on the unit ball

We consider next the Poisson equation −∆u = f

in B,

(B.12)

posed on the unit ball B ⊂ RN , N ≥ 1. By the fundamental theorem of calculus, in dimension N = 1, every solution to −u00 = f , with f ∈ C(−1, 1), belongs to the space C 2 (−1, 1). Unfortunately, such a regularity result fails in higher dimensions. Example B.2.1 ([126]) Let N ≥ 2 and let B be the unit ball of RN . There exists a function f ∈ C(B) such that no function u ∈ C 2 (B) solves (B.12). Proof. Take a harmonic homogeneous polynomial of degree two, for example, P(x) = x 1 x 2 , and a cutoff function η ∈ Cc2 (RN ), such that η ≡ 1 in B1 and η ≡ 0 in RN \ B2 . Now let − f (x) =

+∞ X 1 k=1

k

∆(ηP)(2k x), 240

for x ∈ B.

Appendix B. Regularity theory for elliptic operators Since P is harmonic, ∆(ηP) = P∆η + 2∇η · ∇P. So, ∆(ηP)(2k x) is supported in B2−k+1 \ B2−k . It follows that f (0) = 0 and for x ∈ B \ {0}, 1

− f (x) =

kx

∆(ηP)(2k x x),

where k = k x is the unique integer such that 2−k < |x| ≤ 2−k+1 . Hence, f ∈ C(B). Now, let u(x) =

+∞ X 1 k=1

k4k

(ηP)(2k x),

for x ∈ B.

Then, u ∈ C 1 (B) ∩ C 2 (B \ {0}), −∆u = f in B \ {0}, and ∂ 2u ∂ x1∂ x2

+∞ X 1 ∂ 2 (ηP)

(x) =

k=1

k ∂ x1∂ x2

(2 x) = k

+∞ X 1 k=1

∂ 2P

k

η

∂ x1∂ x2

(2k x) + v(x),

for some v ∈ C(B). Now, for P(x) = x 1 x 2 , +∞ X 1 k=1

k

η

∂ 2P

∂ x1∂ x2

(2 x) ≥ k

kX x −1 k=1

1 k

→ +∞,

as |x| → 0.

˜ ∈ C 2 (B) solves So, u 6∈ C 2 (B). Assume by contradiction that some function u ˜ is harmonic in B \ {0}, and in fact, also in B, −∆˜ u = f in B. Then, z = u − u by Exercise A.1.3. This is not possible, by Proposition B.1.1. Example B.2.1 shows that some quantitative information on the modulus of continuity of f is needed in order to gain C 2 regularity of solutions. This is what we discuss next. Definition B.2.1 Let Ω be an open set of RN , N ≥ 1. We say that f is Dini continuous in Ω and write f ∈ CDini (Ω) if

ˆ

d

ω(r)

0

r

d r < +∞,

where d is the diameter of Ω, and where ω(r) =

sup

| f (x) − f ( y)|,

{x, y∈Ω : |x− y|
241

for r ∈ (0, d).

Stable solutions of elliptic PDEs The following theorem gives an interior a priori estimate on the modulus of continuity of D2 u, for any u ∈ C 2 solving the Poisson equation (Equation (B.12)), with Dini continuous right-hand side f . Theorem B.2.1 ([33]) Let B denote a ball of radius 1 in RN , N ≥ 1. Let f ∈ CDini (B). Assume that u ∈ C 2 (B) solves (B.12). Then, there exists a constant CN depending on N only, such that for all x, y ∈ B1/2 ,

ˆ

|D2 u(x) − D2 u( y)| ≤ CN

d

d sup |u| + B1

ω(r)

0

r

ˆ

1

dr + d d

ω(r) r2

d r , (B.13)

where d = |x − y|. Proof. We follow [222]. It suffices to establish (B.13) for x = 0 and | y| < 1/8. For k ≥ 1, let Bk = Bρ k (0), where ρ = 1/2. Let uk be the solution to

−∆uk = f (0) uk = u

in Bk , on ∂ Bk .

(B.14)

kuk − uk L ∞ (Bk ) ≤ Cρ 2k ω(ρ k ).

(B.15)

We claim that Indeed, let ζk (x) = be the solution to

1 2k ρ − |x|2 2N

−∆ζ = 1 ζ=0

in Bk , on ∂ Bk .

Since v = uk − u solves

−∆v = f (0) − f v=0

in Bk , on ∂ Bk ,

it follows from the maximum principle that −ω(ρ k )ζk ≤ v ≤ ω(ρ k )ζk . Equation (B.15) follows. We also deduce that kuk − uk+1 k L ∞ (Bk+1 ) ≤ Cρ 2k ω(ρ k ). 242

(B.16)

Appendix B. Regularity theory for elliptic operators Since uk+1 − uk is harmonic, it follows from (B.1) that k∇(uk+1 − uk )k L ∞ (Bk+2 ) ≤ Cρ k ω(ρ k ), kD2 (uk+1 − uk )k L ∞ (Bk+2 ) ≤ Cω(ρ k ).

(B.17)

Let vk denote the difference of uk and the quadratic part of u at 0, that is, 1 2 vk (x) = uk (x) − u(0) + ∇u(0) · x + D u(0).x · x . 2 Since u ∈ C 2 (B), we deduce from (B.15) that vk = o(ρ 2k ) in Bk . By definition of uk , we also have that vk is harmonic in Bk . Hence, by (B.1), ∇u(0) = lim ∇uk (0),

D2 u(0) = lim D2 uk (0).

k→+∞

(B.18)

k→+∞

Fix a point y ∈ B1/8 (0). We have |D2 u( y) − D2 u(0)| ≤ |D2 u( y) − D2 uk ( y)| + |D2 uk ( y) − D2 uk (0)|+ + |D2 uk (0) − D2 u(0)| =: I1 + I2 + I3 . Then, by (B.17) and (B.18), I3 ≤

+∞ X

˜ j ( y) − D u ˜ j+1 ( y)| ≤ C |D u 2

2

+∞ X j=k

j=k

ˆ ω(ρ ) ≤ C j

0

ρ k−1

ω(r) r

d r.

(B.19)

˜ j be the Next, we estimate I1 . For j ≥ k + 1, let B j ( y) = B( y, ρ j ) and let u solution to ¨ −∆˜ u j = f ( y) in B j ( y), ˜j = u u

on ∂ B j ( y).

Then, ˜k+1 ( y) − D2 uk ( y)| ˜k+1 ( y)| + |D2 u I1 ≤ |D2 u( y) − D2 u +∞ X ˜ j ( y) − D2 u ˜ j+1 ( y)| + |D2 u ˜k+1 ( y) − D2 uk ( y)| ≤ |D2 u j=k+1

ˆ ≤C 0

ρk

ω(r) r

˜k+1 ( y) − D2 uk ( y)|. (B.20) d r + |D2 u

243

Stable solutions of elliptic PDEs Let k ≥ 1 be the unique integer such that ρ k+3 ≤ | y| < ρ k+2 .

(B.21)

Then, Bk+1 ( y) = B( y, ρ k+1 ) ⊂ B(0, ρ k ) and the function ˜k+1 − uk − ( f ( y) − f (0))ζk w := u is harmonic in Bk+1 ( y). Hence, |D2 w( y)| ≤ In addition,

C ρ

kwk L ∞ (Bk+1 ( y)) ≤ Cω(ρ k ).

2k

|( f ( y) − f (0))D2 ζk ( y)| ≤ Cω(ρ k ),

so that

˜k+1 ( y) − D2 uk ( y)| ≤ Cω(ρ k ). |D2 u

Using this in (B.20), we deduce that

ˆ

ρ k−1

I1 ≤ C

ω(r) r

0

(B.22)

d r.

We estimate at last I2 . For j ≤ k, let h j = u j+1 − u j . It follows from (B.1) and (B.16) that |D2 h j ( y) − D2 h j (0)| ≤ kD3 h j k L ∞ (Bk+2 ) | y| ≤ kD3 h j k L ∞ (B j+2 ) | y| ≤ C

| y| ρj

ω(ρ j ).

Hence, I2 ≤ |D2 uk−1 ( y) − D2 uk−1 (0)| + |D2 hk−1 ( y) − D2 hk−1 (0)| k−1 X ≤ |D2 u1 ( y) − D2 u1 (0)| + |D2 h j ( y) − D2 h j (0)| j=1 k−1 X

ku1 k L ∞ (B1 ) +

≤ C| y|

≤ C| y| kuk L ∞ (B) +

ˆ

! ρ − j ω(ρ j )

j=1 1

| y|

ω(r) r2 244

dr

.

(B.23)

Appendix B. Regularity theory for elliptic operators Collecting (B.19), (B.22), and (B.23), we obtain

ˆ

ρ k−1

|D2 u( y) − D2 u(0)| ≤ C

ω(r) r

0

ˆ d r + | y|kuk L ∞ (B) + | y|

1

ω(r)

| y|

r2

Recalling (B.21), (B.13) follows.

! dr

.

Corollary B.2.1 Let B1 be a ball of radius 1 in RN , N ≥ 1, and α ∈ (0, 1). Let f ∈ C α (B). Assume that u ∈ C 2 (B) solves (B.12). Then, there exists a constant CN depending on N only, such that 1 k f kC α (B) . (B.24) kukC 2,α (B1/2 ) ≤ CN kuk L ∞ (B1 ) + α(1 − α) If f is Lipschitz, then for all x, y ∈ B1/2 , |D2 u(x) − D2 u( y)| ≤ CN d kuk L ∞ (B1 ) + k f kC 0,1 (B1 ) | ln d| ,

(B.25)

where d = |x − y|. The previous corollary is an immediate consequence of Theorem B.2.1 and the following interpolation inequality (with " = 1), the proof of which can be found in [126]. Lemma B.2.1 Let α ∈ (0, 1],N ≥ 1 and Ω a smoothly bounded domain of RN . For every " > 0, there exists a constant C = C(N , Ω, ") such that for every u ∈ C 2,α (Ω), kukC 2 (Ω) ≤ Ckuk L ∞ (Ω) + "[D2 u]α , where

¨ [D u]α = sup 2

|D2 u(x) − D2 u( y)| |x − y|α

« : x, y ∈ Ω, x 6= y .

Boundary estimates can be obtained similarly to Theorem B.2.1. Theorem B.2.2 Let B + = {x ∈ B : x N > 0} be a half-ball of radius 1 in RN , N ≥ 1. Let f ∈ CDini (B + ). Assume that u ∈ C 2 (B) ∩ C(B + ) solves

in B + , on [x N = 0].

−∆u = f u=0 245

(B.26)

Stable solutions of elliptic PDEs + Then, there exists a constant C depending on N only, such that for all x, y ∈ B1/2 , α + (B.13) holds. In addition, if f ∈ C (B ) for some α ∈ (0, 1), then 1 (B.27) k f kC α (B+ ) . kukC 2,α (B+ ) ≤ C kuk L ∞ (B+ ) + 1/2 α(1 − α)

and if f is Lipshitz continuous, then |D2 u(x) − D2 u( y)| ≤ C d kuk L ∞ (B+ ) + k f kC 0,1 (B+ ) | ln d| ,

(B.28)

where d = |x − y|. Proof. The proof is the same as that for Theorem B.2.1, provided we replace Bk by Bk ∩ [x N > 0] and note that if w is a harmonic function in B + such that w = 0 on [x N = 0], then w is harmonic in B after its odd extension in the x N variable (see Exercise A.1.4). Thanks to the a priori estimates, we may now solve Poisson’s equation on a ball. Theorem B.2.3 Let B be a ball of radius 1 in RN , N ≥ 1. Let f ∈ C α (B), α ∈ (0, 1). There exists a unique solution u ∈ C 2,α (B) to

−∆u = f u=0

in B, on ∂ B.

(B.29)

Furthermore, there exists a constant C depending on N only, such that kukC 2,α (B) ≤ Ck f kC α (B) .

(B.30)

Proof. By a standard density argument, it suffices to consider the case where f ∈ Cc∞ (RN ). Let v = Γ∗ f , where Γ is the fundamental solution of the Laplace operator, given by (A.6) and (A.7). Then, v ∈ C ∞ (RN ) and −∆v = f in RN . Hence, any solution to (B.29) can be written as u = v + w, where w solves the Dirichlet problem (B.4) with boundary data g = − v|∂ B . By Theorem B.1.1, such a solution is unique and belongs to C 2 (B) ∩ C(B). Hence, the same holds true for the solution u to (B.29). It remains to be proven that u ∈ C 2,α (B). By Corollary B.2.1, for every x 0 ∈ B, there exists r > 0 and a constant C, which may depend on r such that kukC 2,α (B(x 0 ,r)) ≤ C kuk L ∞ (B) + k f kC α (B) ≤ C 0 k f kC α (B) , (B.31) 246

Appendix B. Regularity theory for elliptic operators 1 where the last inequality is obtained by comparing u and ±k f k L ∞ (B) 2N (1 − 2 |x| ). Now take a point x 0 ∈ ∂ B. Without loss of generality, we may assume that 0 ∈ ∂ B is the antipodal point of x 0 . Note that the sphere inversion x ∗ given by (B.6) maps B onto a half-space H, ∂ B onto ∂ H, and x 0 to x 0∗ 6= 0. Let u∗ denote the Kelvin transform of u, that is, for y ∈ H, let

u∗ (x) = | y|2−N u( y ∗ ). Then, u∗ ∈ C 2 (H) ∩ C(H) solves ¨ −∆u∗ = | y|−2−N f ( y ∗ ) u∗ = 0

in H, on ∂ H.

Using Theorem B.2.2 and the smoothness of the mapping x 7→ x ∗ near x 0 , we deduce that for some r > 0, there exists a constant C such that kukC 2,α (B(x 0 ,r)∩Ω) ≤ Ck f kC α (B) .

(B.32)

Since B is compact, it can be covered by finitely many balls where either (B.31) or (B.32) holds. Equation (B.30) follows.

B.2.2

A priori estimates for C 2,α solutions

In this section, we want to generalize Theorem B.2.3 to uniformly elliptic operators defined on a smoothly bounded domain. As a first step, we prove a priori estimates for C 2,α solutions. Theorem B.2.4 ([198]) Let N ≥ 1, let Ω denote a smoothly bounded domain of RN and given α ∈ (0, 1), let f ∈ C α (Ω). Let L denote a uniformly elliptic operator, that is, (A.21) and (A.23) hold. In addition, assume that the coefficients of L are such that A, B, V ∈ C α (Ω). Assume that u ∈ C 2,α (Ω) solves −Lu = f in Ω, (B.33) u=0 on ∂ Ω. Then, there exists a constant C depending only on N , Ω, α, the ellipticity constant of L, and the norms kAkC α (Ω) , kBkC α (Ω) , kV kC α (Ω) , such that kukC 2,α (Ω) ≤ C kuk L ∞ (Ω) + k f kC α (Ω) . 247

(B.34)

Stable solutions of elliptic PDEs Remark B.2.1 The above theorem does not say that there indeed exists a solution u to (B.33) belonging to the class C 2,α (Ω). This will be proven later on under the extra requirement that V ≥ 0 in Ω. Proof. Changing coordinates if necessary, we may assume that 0 ∈ Ω and ai j (0) = δi j . Let u ∈ C 2,α (Ω) be a solution to (B.33). Then, −∆u = f +

N X

(ai j (x) − ai j (0))∂i j u +

i, j=1

N X

! bi ∂i u + V (x)u

=: R

in Ω.

i=1

Applying Corollary B.2.1 (to v(x) = u(r x)), there exists constants C and C 0 such that for any r > 0 small, 1 kuk L ∞ (Br ) + kRkC α (Br ) kukC 2,α (Br/2 ) ≤ C r 2+α 1 0 ≤C kuk L ∞ (Br ) + k f kC α (Br ) + kukC 2 (Br ) + 2+α r sup kai j (·) − ai j (0)k L ∞ (Br ) kukC 2,α (Br )

.

(B.35)

i, j=1...N

Since ai j is continuous, C 0 sup kai j (·) − ai j (0)k L ∞ (Br ) < 1/4,

(B.36)

i, j=1...N

for r > 0 sufficiently small. Fix such a r. By Lemma B.2.1, there exists a constant C such that 1 C 0 kukC 2 (Br ) ≤ Ckuk L ∞ (Br ) + kukC 2,α (Br ) . 4

(B.37)

Plugging (B.36) and (B.37) in (B.35), we obtain 1 kukC 2,α (Br/2 ) ≤ C kuk L ∞ (Ω) + k f kC α (Ω) + kukC 2,α (Ω) . 2

(B.38)

The above estimate applies to any sufficiently small ball B(x 0 , r) ⊂ Ω. Now take a point x 0 ∈ ∂ Ω. Since Ω has a smooth boundary, there exists r > 0 and a smooth coordinate chart y = φ(x) such that φ(Ω ∩ B(x 0 , r)) = B + 248

Appendix B. Regularity theory for elliptic operators and φ(∂ Ω ∩ B(x 0 , r)) = B + ∩ [x N = 0]. Writing Equation (B.33) in the new coordinates, we obtain ¨ −L 0 u = f in B + , (B.39) u=0 on B + ∩ [x N = 0], 2

PN

where L 0 u =

i, j=1

ai0 j ( y) ∂ ∂y ∂uy + i

j

PN i=1

bi0 ( y) ∂∂ yu + V u is a uniformly elliptic i

operator, which coefficients depend on L and φ. We may further assume that ai0 j (0) = δi j , so that (B.39) can be rewritten as   −∆u = f +

N X

(ai0 j ( y) − ai0 j (0))∂i j u +

i, j=1



u=0

N X

! bi0 ∂i u + V u

=: R0

in B + ,

i=1

on B + ∩ [x N = 0].

Applying Theorem B.2.2, we deduce that kukC 2,α (B+ ) ≤ C kuk L ∞ (B+ ) + kR0 kC α (B+ ) . 1/2

By localization and interpolation, we obtain as previously that 1 kukC 2,α (B+0 ) ≤ C kuk L ∞ (Ω) + k f kC α (Ω) + kukC 2,α (Ω) , r 2 for some r 0 > 0. Going back to the original variables, it follows that for some r > 0, 1 kukC 2,α (Ω∩B(x 0 ,r)) ≤ C kuk L ∞ (Ω) + k f kC α (Ω) + kukC 2,α (Ω) . 2

(B.40)

Since Ω is compact, it can be covered by finitely many balls where either (B.38) or (B.40) holds, hence kukC 2,α (Ω) ≤ C kuk L ∞ (Ω) + k f kC α (Ω) , as desired.

B.2.3

Existence of C 2,α solutions

We now generalize Theorem B.2.3 to the case of a uniformly elliptic equation posed on the ball. 249

Stable solutions of elliptic PDEs Theorem B.2.5 Let B be a ball of radius 1 in RN , N ≥ 1. Let f ∈ C α (B), g ∈ C 2,α (∂ B), α ∈ (0, 1). Let L be a uniformly elliptic operator, that is, (A.21) and (A.23) hold. In addition, assume that the coefficients of L are such that A, B, V ∈ C α (B), and V ≥ 0 in B. Then, there exists a unique solution u ∈ C 2,α (B) to −Lu = f in B, (B.41) u= g on ∂ B. Furthermore, there exists a constant C depending on N , α, the ellipticity constant of L, and the norms kAkC α (B) , kBkC α (B) , kV kC α (B) only, such that kukC 2,α (B) ≤ C k f kC α (B) + kgkC 2,α (∂ B) . (B.42) To establish Theorem B.2.5, we shall use the following method of continuity. Proposition B.2.1 Let X , Y be two Banach spaces and L0 , L1 two bounded linear operators from X to Y . For each t ∈ [0, 1], set L t = (1 − t)L0 + t L1 and suppose that there exists a constant C such that for all u ∈ X , t ∈ [0, 1].

kukX ≤ CkL t ukY , If L0 is invertible, then so is L1 .

Proof. Suppose that Ls is invertible for some s ∈ [0, 1]. For t ∈ [0, 1] and f ∈ Y , the equation L t u = f can be rewritten as Ls u = f + (Ls − L t )u = f + (t − s)(L0 − L1 )u, that is,

u = Ls−1 f + (t − s)Ls−1 (L0 − L1 )u =: Tu.

The mapping T is a contraction in X if |t − s| < δ :=

1 C(kL0 k + kL1 k)

,

and hence Ls is invertible for all t ∈ [0, 1] such that |t −s| < δ. By dividing the interval [0, 1] into subintervals of length less than δ, we see that the mapping L t is invertible for all t ∈ [0, 1] provided L0 is. Proof of Theorem B.2.5. First, we note that we can assume that g = 0. Indeed, take η ∈ Cc∞ (0, 2) such that η(1) = 1 and let G(x) = η(|x|)g(x/|x|), 250

Appendix B. Regularity theory for elliptic operators for x ∈ B. Then, G ∈ C 2,α (B), kGkC 2,α (B) ≤ CkgkC 2,α (∂ B) , and G|∂ B = g. Then, u = v + G, where v vanishes on ∂ B and solves −∆u = f + ∆G in B. Now let X be the Banach space of functions u ∈ C 2,α (B) such that u = 0 on ∂ B. Also let Y = C α (B). For t ∈ [0, 1], let X →Y Lt : u 7→ (1 − t)∆u + t Lu. Given f ∈ Y , we want to find a solution to −L1 u = f .

(B.43)

By Proposition B.2.1 and Corollary B.2.3, it suffices to show that there exists a constant C such that for all u ∈ X and t ∈ [0, 1], kukC 2,α (B) ≤ CkL t ukC α (B) .

(B.44)

Noting that the ellipticity constant and C α norms of the coefficients of L t are uniformly controlled independently of t ∈ [0, 1], it follows from Theorem B.2.4 that kukC 2,α (B) ≤ C kuk L ∞ (B) + kL t ukC α (B) . (B.45) For γ > 0 sufficiently large, the function ζ(x) = eγ − eγx 1 is positive in B and satisfies −Lζ > 1 in B. Comparing u with ±kL t uk L ∞ (B) ζ, we deduce that kuk L ∞ (B) ≤ CkL t uk L ∞ (B) . (B.44) follows. We turn at last to the case of a uniformly elliptic equation posed on a smoothly bounded domain. Theorem B.2.6 Let Ω be a smoothly bounded domain in RN , N ≥ 1. Let f ∈ C α (Ω), α ∈ (0, 1). Let L be a uniformly elliptic operator, that is, (A.21) and (A.23) hold. In addition, assume that the coefficients of L are such that A, B, V ∈ C α (Ω), and V ≥ 0 in Ω. Then, there exists a unique solution u ∈ C 2,α (Ω) to −Lu = f in Ω, (B.46) u=0 on ∂ Ω. Furthermore, there exists a constant C depending on N , α, Ω, the ellipticity constant of L, and the norms kAkC α (Ω) , kBkC α (Ω) , kV kC α (Ω) only, such that kukC 2,α (Ω) ≤ Ck f kC α (Ω) . Proof. Thanks to the continuity method (Proposition B.2.1) and the a priori estimate of Theorem B.2.4, it suffices to prove that for L = ∆, f ∈ Cc∞ (Ω), 251

Stable solutions of elliptic PDEs there exists a solution u ∈ C 2,α (Ω) to (B.46). Writing u = Γ ∗ f + v, where Γ is the fundamental solution to the Laplace operator, and where v is the solution to the Dirichlet problem with boundary data g = −Γ ∗ f ∂ Ω given by Perron’s method (Theorem B.1.2), we obtain a solution u ∈ C 2 (Ω) ∩ C(Ω). 2,α By Corollary B.2.1, u ∈ Cloc (Ω). It remains to be proven that u is C 2,α up to the boundary. Take a point x 0 ∈ ∂ Ω. Since Ω has a smooth boundary, there exists a neighborhood V of x 0 and a coordinate chart y = φ(x), such that (0, . . . , 0, 1) = φ(x 0 ), φ(Ω ∩ V ) = B, and φ(∂ Ω ∩ V ) = { y ∈ ∂ B : yN > 0}. In the new coordinates, we obtain for some uniformly elliptic operator L 0 , −L 0 u = f in B, u=0 on ∂ B ∩ [ yN > 0]. Take g n ∈ C 2,α (∂ B), such that g n converges uniformly to u|∂ B on ∂ B and g n vanishes on ∂ B ∩ [ yN > 1/2]. Let un ∈ C 2,α (B) be the solution to −L 0 un = f in B, un = g n on ∂ B, given by Theorem B.2.5. Working with the proof of Theorem B.2.4, we obtain kun kC 2,α (B∩[ yN >3/4]) ≤ C kun k L ∞ (B) + k f k L ∞ (B) . Passing to the limit as n → +∞, we deduce that u is C 2,α in a neighborhood of x 0 , as desired.

B.3

Calderon-Zygmund estimates

We state here without proof the following extension of Theorem B.2.4 to the setting of Sobolev spaces. For a proof in the case p = 2, see, for example, [25]. Theorem B.3.1 ([1, 48]) Let N ≥ 1, let Ω denote a smoothly bounded domain of RN and given p ∈ (1, +∞), let f ∈ L p (Ω). Let L denote a uniformly elliptic operator, that is, (A.21) and (A.23) hold. In addition, assume that the coefficients of L are such that A ∈ C(Ω), B, V ∈ L ∞ (Ω), and V ≥ 0 a.e. in Ω. Then, there exists a unique solution u ∈ W 2,p (Ω) of (B.33) in the sense that 1,p −Lu = f a.e. in Ω and u ∈ W0 (Ω) ∩ W 2,p (Ω). Furthermore, there exists a constant C = C(Ω, L, N , p) such that kukW 2,p (Ω) ≤ Ck f k L p (Ω) .

(B.47)

Remark B.3.1 As follows from Example B.2.1 (and its proof), the above theorem is false for p = +∞. It is also false for p = 1 (work by contradiction with f = ρn , a standard mollifier converging to the Dirac mass δ0 ). 252

Appendix B. Regularity theory for elliptic operators

B.4

Moser iteration

In this section, we give two useful elliptic regularity results giving uniform a priori estimates via the Moser iteration method. Theorem B.4.1 ([206]) Let N ≥ 2, let Ω denote a bounded domain of RN and given p > N , let f ∈ W −1,p (Ω). Then, there exists a unique solution u ∈ H01 (Ω) ∩ L ∞ (Ω) of −∆u = f in Ω, (B.48) u=0 on ∂ Ω. Furthermore, there exists a constant C = C(Ω, n, p) such that kuk L ∞ (Ω) ≤ Ck f kW −1,p (Ω) .

(B.49)

Proof. Since p > N , W −1,p (Ω) ⊂ H −1 (Ω). It follows from the Lax-Milgram lemma that (B.33) is uniquely solvable in H01 (Ω). It remains to prove (B.49). By an obvious scaling argument, we may assume that k f kW −1,p (Ω) = 1. Since f ∈ W −1,pP (Ω) and Ω is bounded, there exists f i ∈ L p (Ω), i = 1, . . . , N , such N ∂f that f = i=1 ∂ xi and maxi=1,...,N k f i k L p (Ω) = k f kW −1,p (Ω) = 1 (see Proposition i IX.20 in [25]). Given k ≥ 0, consider the function v = G(u) ∈ H01 (Ω), where G is the Lipschitz function defined for t ∈ R by 

 t + k if t ≤ −k 0 if −k < t < k G(t) =  t − k if t ≥ k. Multiply (B.48) by v and integrate. Then,

ˆ

ˆ |∇v| d x = 2

Ω

Ω

∇u · ∇v d x = −

N ˆ X j=1

Ak

fj

∂v ∂ xj

d x,

where Ak is the set [|u| ≥ k]. Applying the Cauchy-Schwarz inequality, we deduce that ˆ N ˆ X 2 |∇v| d x ≤ f j2 d x. Ω

j=1

253

Ak

Stable solutions of elliptic PDEs Using Sobolev’s inequality on the one hand, and Hölder’s inequality on the other hand, it follows that ˆ Ω

2/2∗

2∗

≤ Ck f k2W −1,p (Ω) |Ak |1−2/p ≤ C|Ak |1−2/p .

|v| d x

In other words,

ˆ

2/2∗

∗

(|u| − k)2 d x

≤ C|Ak |1−2/p .

Ak

Now take h > k ≥ 0. In particular, Ah ⊂ Ak . Since |u| ≥ h in Ah, we deduce that 2/2∗

(h − k) |Ah| 2

ˆ

2/2∗

2∗

(|u| − k) d x

≤

≤ C|Ak |1−2/p ,

Ak

that is,

1

|Ah| ≤ C

1/2−1/p

2∗

(h − k)

|Ak | 1/2−1/N .

(B.50)

For s ∈ N, d > 0, let hs = d − 2ds , θs = |Ahs |, α = 2∗ , β = (B.50) applied with k = hs , h = hs+1 yields s+1 α 2 θsβ . θs+1 ≤ C d β−1

Fix at last d α = Cθ0

1/2−1/p 1/2−1/N

> 1. Then

αβ

2 β−1 . We claim that α s − β−1

θs ≤ θ0 2

for all s ∈ N.

,

(B.51)

Equation (B.51) is obviously satisfied for s = 0. Assume that it holds for some s ∈ N. Then, by (B.50), θs+1 ≤ C

2s+1 d

α

αβ

s β − θ0 2 β−1

β

=C

2α θ0 dα

αs − β−1

2

β

=C

2α θ0

β−1 Cθ0 2

αβ β−1

αs − β−1

2

α − β−1 (s+1)

= θ0 2

. (B.52)

We have just proved (B.51) by induction. Let at last s → +∞ in (B.51): |Ad | = 0 and (B.49) follows. 254

Appendix B. Regularity theory for elliptic operators Theorem B.4.2 ([201], [212]) Let N ≥ 1, let Ω denote a domain of RN and assume that B(x 0 , 2R) ⊂⊂ Ω. Assume that u ∈ C 2 (Ω) satisfies −∆u − V (x)u ≤ 0 in Ω, (B.53) u>0 in Ω, N

where V ∈ L l2−" (Ω) for some " > 0. Then, for every t > 1, oc kuk L ∞ (B(x 0 ,R)) ≤ CST R−N /t kuk L t (B(x 0 ,2R)) , where CS T is a constant depending only on N , t and R" kV k

(B.54) N

L 2−" (B(x 0 ,2R))

.

Proof. Translating and scaling space if necessary, we may assume that x 0 = 0 and R = 1. Take j ≥ t/2 > 1/2, ϕ ∈ Cc2 (B2 ) and multiply (B.53) with u2 j−1 ϕ 2 . Then, ˆ ˆ 2 j−1 2 ∇u · ∇ u ϕ dx ≤ V (x)u2 j ϕ 2 d x. RN

RN

Expand the left-hand side: ˆ ˆ 2 2 2 j−2 (2 j − 1) u |∇u| ϕ d x ≤ RN

ˆ V (x)u ϕ d x − 2j

RN

2

RN

u2 j−1 ∇u∇ϕ 2 d x.

Integrate by parts the last term in the above: ˆ ˆ ˆ 2 1 2j − 1 2 j 2 j 2 ∇u ϕ d x ≤ V (x)u ϕ d x + u2 j ∆ϕ 2 d x. 2 N N N j 2j R R R 2 2 Since ∇(u j ϕ) = ∇u j ϕ 2 + ∇ϕ · ∇(u2 j ϕ), it follows that 2j − 1 j2

ˆ RN

∇(u j ϕ) 2 d x ≤

ˆ V (x)u ϕ d x + 2j

RN

2

+

1

ˆ

2 j RN ˆ 2j − 1 j2

u2 j ∆ϕ 2 d x

RN

∇ϕ · ∇(u2 j ϕ) d x.

Integrate by parts the last term in the above and multiply by j 2 /(2 j −1). Then, ˆ ∇(u j ϕ) 2 d x ≤ RN ˆ ˆ 2j 2 2j 2 C j |V (x)|u ϕ d x + u (|∆ϕ | + |ϕ∆ϕ|) d x , RN

RN

255

Stable solutions of elliptic PDEs for some constant C depending on t only. Apply Sobolev’s inequality to the left-hand side and Hölder’s inequality to the first term on the right-hand side. Then, ku ϕk j

2 2N

L N −2 (RN )

≤C

N jkV k 2−" ku j ϕk2−" ku j ϕk"L 2 (RN ) + 2N L (B2 ) L N −2 (RN ) ˆ 2j 2 u (|∆ϕ | + |ϕ∆ϕ|) d x RN 0 j " j 2−" ≤ C jku ϕk 2N ku ϕk L 2 (RN ) + B ,

L N −2 (RN )

where C 0 = C 0 (t, N , kV k If B ≥ jku j ϕk2−" 2N

L N −2 (RN )

) and where B =

N L 2−"

(B2 ) j " ku ϕk L 2 (RN ) ,

2N

L N −2 (RN )

L N −2 (RN )

ku j ϕk2

L

RN

u2 j (|∆ϕ 2 | + |ϕ∆ϕ|) d x.

then

ku j ϕk2 while if B < jku j ϕk2−" 2N

´

≤ 2C 0 B,

(B.55)

ku j ϕk"L 2 (RN ) , then

2N N −2 (RN )

≤ 2C 0 jku j ϕk2−" 2N L

hence ku j ϕk2

L

N −2 (RN )

2

2N N −2 (RN )

ku j ϕk"L 2 (RN ) ,

≤ (C j) " ku j ϕk2L 2 (RN ) .

(B.56)

From (B.55) and (B.56), we deduce that in all cases, ˆ 2 j 2 j 2 u2 j (|∆ϕ 2 | + |ϕ∆ϕ|) d x. ku ϕk 2N N ≤ (C j) " ku ϕk L 2 (RN ) + C L N −2 (R )

RN

Given 1 < ρ 0 < ρ < 2, we now choose our function ϕ such that 0 ≤ ϕ ≤ 1, C C ϕ = 1 in Bρ0 , ϕ = 0 in RN \ Bρ , |∇ϕ| ≤ ρ−ρ 0 , and |∆ϕ| ≤ (ρ−ρ 0 )2 . It follows that ku j k2

L

2N N −2 (B 0 ) ρ

≤ (C j)2/" ku j k2L 2 (B ) + ρ

C

ku j k2L 2 (B ) , ρ (ρ − ρ 0 )2 C 2/" = (C j) + ku j k2L 2 (B ) . (B.57) ρ (ρ − ρ 0 )2 256

Appendix B. Regularity theory for elliptic operators Setting k = 2 j, (B.57) can be rephrased as kuk

L

k NN −2

(Bρ0 )

≤

(C k)

2/"

+

C

1/k kuk L k (Bρ ) .

(ρ − ρ 0 )2

(B.58)

At last, choose sequences kl and (ρl ), which are defined by kl = t

l

N

for l ∈ N

,

N −2

(B.59)

and ρl = 1 + 2−l ,

for l ∈ N.

(B.60)

Apply (B.57) with k = kl , ρ = ρl , ρ 0 = ρl+1 . It follows that kuk L kl+1 (Bρ

) l+1

1/kl l kuk L kl (Bρ ) ≤ C˜ "kl kuk L kl (Bρ ) . ≤ C 2l/" + C4l l

l

Iterating the above, we obtain 1

kuk L kl (Bρ ) ≤ C˜ "

Pl−1

m m=1 km

l

kuk L t (B2 ) .

Passing to the limit as l → +∞ yields the desired inequality.

B.5

The inverse-square potential

In this section, we study elliptic regularity for an equation of the form  c  −∆φ − φ = g in Ω, |x − ξ|2 (B.61)  φ = h on ∂ Ω, where c ∈ R, ξ ∈ Ω and f , g are given, say, smooth, functions. Note that the potential V (x) = c/|x − ξ|2 6∈ L p (Ω) for any p ≥ N /2, so standard elliptic regularity results, for example, Theorem B.4.2, cannot be applied to the operator L = −∆ − V (x). In fact, elliptic regularity fails for this operator: φ can be unbounded even if f , g are smooth. Still, regularity results can be recovered, provided the data f , g satisfy certain orthogonality relations. This is what we describe in this section. 257

Stable solutions of elliptic PDEs

B.5.1

The kernel of L = −∆ − |x|c 2

In Section C.4, we study the Laplace-Beltrami operator −∆S N −1 on the sphere S N −1 and derive the following properties: the eigenvalues of −∆S N −1 are given by λk = k(N + k − 2),

k ≥ 0,

and there exists an orthonormal basis of L 2 (S N −1 ) formed with eigenvectors {ϕk,l : k ≥ 0, l = 1, . . . , mk }, where mk denotes the multiplicity of λk and ϕk,l , l = 1, . . . , mk the eigenfunctions associated to λk . We choose the first functions to be 1/2 xl N 1 , ϕ1,l = ´ = x l , l = 1, . . . , N . ϕ0,1 = |S N −1 | |S N −1 |1/2 ( S N −1 x l2 )1/2 Now, we seek solutions of −∆w −

c |x|2

w=0

in RN \ {0}

(B.62)

of the form w(x) = f (r)ϕk,l (ω), where r = |x| and ω = x/r for x ∈ RN \ {0}. By Lemma C.4.2, this is equivalent to asking that f solves the following ordinary differential equation: f 00 +

N −1 r

f0+

c − λk r2

f = 0,

for r > 0.

(B.63)

Equation (B.63) is of Euler type and it admits a basis of solutions of the form ± f (r) = r −αk , where −α± are the roots of the associated characteristic equak tion, that is, r N − 2 2 N − 2 α± = ± − c + λk . k 2 2 Note that α± may have a nonzero imaginary part only for finitely many k’s. If k k0 is the first integer k such that α± ∈ R then k ≤ . . . < α− < α− k0 +1 k0

N −2 2

≤ α+ < α+ < ..., k0 +1 k0

whereas, if k < k0 , we denote the imaginary part of α+ by k È N −2 2 bk = c − − λk . 2 258

Appendix B. Regularity theory for elliptic operators For k ≥ 0, l = 1, . . . , mk , we have just found a family of real-valued solutions 1 2 of (B.62), denoted by w 1 = w k,l , w2 = w k,l and defined on RN \ {0} by  x N −2 2 1 −α+ if ( 2 ) − c + λk > 0: w (x) = |x| k ϕk,l ,   |x|     x  2 −α−  w (x) = |x| k ϕk,l ,    |x|    x  N −2 2 1 − N 2−2   , w (x) = |x| log |x| ϕk,l  if ( 2 ) − c + λk = 0: |x|  x N −2  2 −  2 ϕ w (x) = |x| ,  k,l  |x|     x  N −2 2 1 − N 2−2  if ( 2 ) − c + λk < 0: sin(bk log |x|)ϕk,l w (x) = |x| ,   |x|     x  2 − N 2−2 cos(bk log |x|)ϕk,l . w (x) = |x| |x| (B.64) Each of the functions Wk,l defined by ( if ( N 2−2 )2 − c + λk > 0: Wk,l (x) = w 1 (x) − w 2 (x), if ( N 2−2 )2 − c + λk ≤ 0:

Wk,l (x) = w 1 (x),

(B.65)

then solves   −∆Wk,l − 

B.5.2

c |x|2

Wk,l = 0 in B \ {0}, Wk,l = 0 on ∂ B.

Functional setting

From the previous analysis, one can expect that solutions of an equation of c the form −∆u − |x−ξ| 2 u = f behave near ξ like a (possibly negative) power of |x − ξ|. It is therefore convenient to work in the functional setting (see [12, 44, 186]) described below. Given Ω a smooth bounded domain of RN , ξ ∈ Ω, k ∈ N, α ∈ (0, 1), r ∈ (0, dist(x, ∂ Ω)/2), and u ∈ Clk,α (B \ {ξ}) we define: oc k k k X |∇ u(x) − ∇ u( y)| |u|k,α,r,ξ = sup r j |∇ j u(x)| + r k+α sup . |x − y|α r≤|x−ξ|≤2r j=0 r≤|x−ξ|,| y−ξ|≤2r 259

Stable solutions of elliptic PDEs Let d = dist(ξ, ∂ Ω) and for any ν ∈ R let kukk,α,ν,ξ;Ω = kukC k,α (Ω\Bd/2 (ξ)) + sup r −ν |u|k,α,r,ξ . 0
Define the space k,α k,α Cν,ξ (Ω) = { u ∈ Cloc (Ω \ {ξ}) : kukk,α,ν,ξ;Ω < ∞ }. k,α One can easily check that Cν,ξ (Ω) is a Banach space. It embeds continuously in the space of bounded functions whenever ν ≥ 0. From here on, given h ∈ C(∂ Ω) and g ∈ C(Ω \ {ξ}), we shall say that a k,α function φ ∈ Cν,ξ (Ω) (k ≥ 2) solves (B.61) whenever the boundary condition c φ|∂ Ω = h holds and −∆φ(x) − |x−ξ| 2 φ(x) = g(x) for all x ∈ Ω \ {ξ}.

B.5.3

The case ξ = 0

In this section, we investigate (B.61) in the case Ω = B and ξ = 0. First observe that, by Hardy’s inequality (Proposition C.1.1), in the case (N −2)2 c < H N = 4 , N ≥ 3, the operator L = −∆ − |x|c 2 is coercive in H01 (B).

In particular, given g ∈ H −1 (B), h = 0, (B.61) is uniquely solvable in H01 (B). However, even if g ∈ C ∞ (B), the solution φ need not be regular at the origin. 1 −α− 2 0 Check that for g = 1 and h = 0, φ = 2N +c |x| − |x| . In other words, 1 −1 although the operator L : H0 (B) → H (B) is coercive, thus invertible, L need not be invertible if acting on a space of bounded functions, for example, L : 0,α 2,α (B) → C−2,0 (B). This situation is similar to Fredholm’s alternative: in C0,0 order to solve (B.61) in a given function space, one must request that the data is orthogonal to the kernel of the adjoint of the operator. Now, L is formally self-adjoint and it has a nontrivial kernel: the functions Wk,l given in (B.65). In the next lemma, we present the orthogonality relations under which the 2,α equation is solvable in Cν,0 (B) spaces. Lemma B.5.1 Let N ≥ 3. Let c, ν ∈ R and assume that ∃k1 ≥ k0 such that

< ν < −α− . − α− k1 k1 +1

0,α Let g ∈ Cν−2,0 (B) and h ∈ C 2,α (∂ B) and consider  c  −∆φ − φ = g in B, |x|2  φ = h on ∂ B.

260

(B.66)

(B.67)

Appendix B. Regularity theory for elliptic operators 2,α Then (B.67) has a solution in Cν,0 (B) if and only if

ˆ

ˆ gWk,l d x = B

h ∂B

∂ Wk,l ∂n

dσ,

∀k = 0, . . . , k1 , ∀l = 1, . . . , mk .

(B.68)

2,α Under this condition the solution φ ∈ Cν,0 (B) to (B.67) is unique and it satisfies

kφk2,α,ν,0;B ≤ C( kgk0,α,ν−2,0;B + khkC 2,α (∂ B) )

(B.69)

where C is independent of g and h. Remark B.5.1 Under the hypotheses of Lemma B.5.1 we have ν > −α− ≥− k1

N −2 2

,

(B.70)

where the last inequality follows from the discussion in Section B.5.1. This implies that the integrals in the left-hand side of (B.68) are finite. Remark B.5.2 By taking k1 sufficiently large, one can choose ν ≥ 0 in the previous lemma. In particular, the corresponding solution φ is bounded. Corollary B.5.1 Assume that (B.66), (B.67), and (B.68) hold. In addition, assume that ν ≥ 0. If |x|2 g is continuous at the origin, then so is φ. Proof of Lemma B.5.1. Write φ as φ(x) =

mk ∞ X X

φk,l (r)ϕk,l (ω),

x = rω, 0 < r < 1, ω ∈ S

N −1

.

k=0 l=1

Then φ solves −∆φ − |x|c 2 φ = g in B \ {0} if and only if φk,l satisfies the ordinary differential equation 00 φk,l +

N −1 r

0 φk,l +

c − λk r2

φk,l = −g k,l

for all k ≥ 0 and l = 1, . . . , mk , where ˆ g k,l (r) = g(rω)ϕk,l (ω) dσ, S

N −1

261

0 < r < 1,

0 < r < 1.

(B.71)

Stable solutions of elliptic PDEs 2,α Note that if φ ∈ Cν,0 (B) then there exists a constant C > 0 independent of r such that

|φk,l (r)| ≤ C r ν .

(B.72)

Furthermore, φ = h on ∂ B if and only if φk,l (1) = hk,l for all k, l, where ˆ hk,l = h(ω)ϕk,l (ω) dσ. N −1

S

Step 1. Clearly, sup0≤t≤1 t 2−ν |g k,l (t)| < ∞ and observe that (B.68) still holds when g is replaced by g k,l ϕk,l and h by hk,l ϕk,l . We claim that there is a unique φk,l that satisfies (B.71), (B.72), and φk,l (1) = hk,l .

(B.73)

We also have |φk,l (r)| ≤ Ck r ν

sup t 2−ν |g k,l (t)| ,

0≤t≤1

0 < r < 1.

(B.74)

g k,l (s) ds,

(B.75)

Case k = 0, . . . , k1 . A solution to (B.71) is given by: • if α± 6∈ R k,l φk,l (r) = • if α+ = α− = k,l k,l

1

ˆ

r

s

b

s N −2 r

0

sin bk log

s r

N −2 : 2

ˆ φk,l (r) =

r

s

s N −2 2

r

0

• if α± ∈ R, α± 6= k,l k,l φk,l (r) =

2

N −2 : 2

1 α+ − α− k k

ˆ

r

s 0

log

s α+k r

s

−

r

g k,l (s) ds, and

s α−k r

g k,l (s) ds.

(B.76)

(B.77)

In each case, (B.74) holds and (B.73) follows from (B.68). Concerning uniqueness, suppose that φk,l satisfies (B.71) with g k,l = 0 and (B.73) with hk,l = 0. Then φk,l is a linear combination of the functions w 1 , w 2 defined in (B.64). By (B.66), (B.70), and (B.74), φk,l has to be zero. 262

Appendix B. Regularity theory for elliptic operators Case k ≥ k1 + 1. Observe that (B.71) is equivalent to −∆φ˜k,l +

λk − c |x|2

φ˜k,l = g˜k,l

in B \ {0},

where φ˜k,l (x) = φk,l (|x|) and g˜k,l (x) = g k,l (|x|). Since α± ∈ R we must have k N −2 2 λk −c ≥ −( 2 ) . By Hardy’s inequality (Proposition C.1.1) and Lax-Milgram’s lemma, the equation  λ −c  −∆φ˜k,l + k φ˜k,l = g˜k,l in B |x|2 (B.78)  ˜ φk,l = hk,l on ∂ B, has a unique solution φ˜k,l ∈ H, where H is the completion of C0∞ (B) with the norm ˆ λk − c 2 2 ϕ , kϕkH = |∇ϕ|2 + |x|2 B see [215]. To show (B.74), observe that for some constant C depending only on N , λk , and ν, Ak,l (r) = r ν C

sup t 2−ν |˜ g k,l (t)| + |hk,l |

0
is a supersolution to (B.78) and −Ak,l is a subsolution. To see this, we emphasize that the condition −α− > ν > −(N −2)/2 implies ν 2 +(N −2)ν+c−λk < 0. k It follows that |φ˜k,l (x)| ≤ Ak,l (|x|) for 0 < |x| ≤ 1. To show that φ˜k,l is uniquely determined, we simply observe that any solution w of (B.78) such that |w(x)| ≤ C|x|ν must belong to H (where uniqueness holds). Indeed, by scaling, it can be checked that |∇w(x)| ≤ C|x|ν−1 (see Claim B.5.1 and (B.82) later) and this together with (B.70) implies w ∈ H 1 (B), which is contained in H. The computations above also yield the necessity of condition (B.68). In2,α deed, assuming a solution φ ∈ Cν,0 (B) exists, since φk,l satisfies the ODE (B.71) we see that for k = 0, . . . , k1 the difference between φk,l and one of the particular solutions (B.75), (B.76), or (B.77) can be written in the form + − we have ck,l = dk,l = 0 ck,l r −αk + dk,l r −αk . Since |φk,l (r)| ≤ C r ν and ν > −α− k1 and this implies (B.68). 263

Stable solutions of elliptic PDEs Step 2. Define for m ≥ 1 ( ) m X X Gm = g = g k,l (r)ϕk,l (ω) : |x|2−ν g(x) ∈ L ∞ (B) k=0

l

and ( Hm =

h=

m X X k=0

) hk,l ϕk,l (ω) : hk,l ∈ R

.

l

Let gP (B.68) hold. Write g m (x) = m ∈ Gm , hm ∈ Hm be such P Pthat m m g (r)ϕ (ω) and h (σ) = h ϕ (ω). Let φk,l be the unique k,l m l k,l k=0 k=0 k,l k,l solution to (B.71), Pm P (B.72), and (B.73) associated to g k,l , hk,l , and define φm (x) = k=0 l φk,l (r)ϕk,l (ω). We claim that there exists C independent of m such that ν 2−ν |φm (x)| ≤ C|x| sup | y| |g m ( y)| + sup |hm | , 0 < |x| < 1. (B.79) ∂B

B

By the previous step, (B.79) holds for some constant C, which may depend on m. In particular, choosing m = k1 , we obtain a bound on the first components φk,l , k = 0 . . . k1 . Hence, it suffices to prove (B.79) in the case g k,l ≡ 0 and hk,l = 0, k = 0, . . . , k1 . Working as in [186] (the argument already appeared in unpublished notes of Pacard), we argue by contradiction assuming that kφm |x|−ν k L ∞ (B) ≥ Cm (kg m |x|2−ν k L ∞ (B) + khm k L ∞ (∂ B) ), where Cm → ∞ as m → ∞. Replacing φm by φm /kφm |x|−ν k L ∞ (B) if necessary, we may assume that kφm |x|−ν k L ∞ (B) = 1, kg m |x|2−ν k L ∞ (B) + khm k L ∞ (∂ B) → 0

as m → ∞.

(B.80)

Let x m ∈ B \ {0} be such that |φm (x m )||x m |−ν ∈ [ 21 , 1]. Let us show that x m → 0 as m → ∞. Otherwise, up to a subsequence x m → x 0 6= 0. By standard elliptic regularity, up to another subsequence, φm → φ uniformly on compact sets of B \ {0} and hence  c  −∆φ − φ = 0 in B \ {0}, |x|2  φ = 0 on ∂ B. 264

Appendix B. Regularity theory for elliptic operators Moreover, φ satisfies |φ(x 0 )||x 0 |−ν ∈ [ 21 , 1] and |φ(x)| ≤ |x|ν in B. Writing X X φ(x) = φk,l (r)ϕk,l (ω), k≥k1 +1

l

we see that φk,l solves (B.63). The growth restriction |φk,l (r)| ≤ C r ν and the explicit functions w 1 , w 2 given by (B.64) rule out the cases α± 6∈ R, α− = α+ k k k − and force φk,l = ak,l r −αk . But φk,l (1) = 0 so we deduce φk,l ≡ 0 and hence φ ≡ 0, contradicting |φ(x 0 )||x 0 |−ν 6= 0. The above argument shows that x m → 0. Define rm = |x m | and vm (x) = rm−ν φm (rm x),

x ∈ B1/rm .

Then |vm (x)| ≤ |x|ν in B1/rm , |vm ( r m )| ∈ [ 12 , 1] and x

m

−∆vm (x) −

c |x|2

vm (x) = rm2−ν g(rm x) in B1/rm .

But rm2−ν g(rm x) ≤ kg m ( y)| y|2−ν k L ∞ (B) |x|ν−2 → 0, by (B.80). Passing to a subsequence, we have that N

xm rm

as m → ∞ → x 0 with |x 0 | = 1,

vm → v uniformly on compact sets of R \ {0} and v satisfies −∆v −

c |x|2

v=0

in RN \ {0}.

Furthermore, |v(x)| ≤ |x|ν in RN \ {0} and |v(x 0 )| 6= 0. Write v(x) =

∞ X X k=0

vk,l (r)ϕk,l (σ).

l

Then |vk,l (r)| ≤ Ck r ν for r > 0. But vk,l has to be a linear combination of the functions w 1 , w 2 given in (B.64), and none of these is bounded by C r ν for all r > 0. Thus v ≡ 0 yielding a contradiction. Step 3. Fix an integer d ≥ 3(N − 2)/2 + 1. Suppose now that g ∈ C ∞ (B \ {0}) and |∇i g(x)| ≤ C|x|ν−2−i for 0 < |x| < 1 and for i = 0, . . . , d. Let h ∈ C ∞ (∂ B) 2,α such that (B.68) holds. We will show that there exists φ ∈ Cν,0 (B) solving (B.67) and satisfying the estimate kφ|x|−ν k L ∞ (B) ≤ C kg|x|2−ν k L ∞ (B) + khk L ∞ (∂ B) . (B.81) 265

Stable solutions of elliptic PDEs To prove this, define for m ∈ N g m (x) =

m X X k=0

g k,l (r)ϕk,l (σ) and hm (σ) =

l

m X X k=0

hk,l ϕk,l (σ).

l

We have X l

X ˆ |g k,l (r)| =

S

l

≤

N −1

X 1 g(rσ)ϕk,l (σ) dσ = λ l

C mk r 2d

≤ Cr

sup |∇2d g(x)| d λk |x|=r ν−2 −2d+2(N −2)

k

ˆ

S

N −1

g(rσ)∆ϕk,l (σ) dσ

kϕk,l k L ∞ (S N −1 )

,

k

where we used integration by parts d times to obtain the inequality and the facts: λk ∼ k2 as k → ∞, |ϕk,l | ≤ C k N −2 in S N −1 and mk ≤ C k N −2 , where mk is the multiplicity of λk , see Section C.4. It follows that g m (x)|x|2−ν converges uniformly in B to g(x)|x|2−ν and hence kg m |x|2−ν k L ∞ (B) → kg|x|2−ν k L ∞ (B) as m → ∞. Similarly, hm converges uniformly to h on ∂ B and thus limm→∞ khm k L ∞ (∂ B) = khk L ∞ (∂ B) . Now g m ∈ Gm and hm ∈ Hm verify the orthogonality conditions (B.68). By the previous step, the associated solution φm satisfies kφm |x|−ν k L ∞ (B) ≤ C kg m |x|2−ν k L ∞ (B) + khm k L ∞ (∂ B) . Using elliptic regularity, up to a subsequence, φm → φ uniformly in B \ {0}, for some φ satisfying the equations −∆φ − |x|c 2 φ = g in B \ {0}, φ = h on ∂ B and the estimate (B.81). Claim B.5.1 φ is a solution to the equation in the whole ball B. To see this, it suffices to prove that |∇φ(x)| ≤ C|x|ν−1

for x ∈ B1/2 .

(B.82)

Recall that ν − 1 > − N2 . This implies that φ ∈ H 1 (B) and thus solves the equation in B. Let x 0 ∈ B1/2 , d = |x 0 | and for x ∈ B3/4 , v(x) = φ(x 0 + d x). Then, −∆v −

c d2 |x 0 + d x|

2

v = d 2 g(x 0 + d x) in B3/4 . 266

Appendix B. Regularity theory for elliptic operators 2

d Observing that 0 ≤ |x c+d ≤ 16c, it follows by elliptic regularity that for some x|2 0 constants C independent of d, |∇v(0)| ≤ C kd 2 g(x 0 + d x)k L ∞ (B3/4 ) + kvk L ∞ (B3/4 ) ≤ C d ν kg|x|2−ν k L ∞ (B) + kφ|x|−ν k L ∞ (B) ≤ C|x 0 |ν kg|x|2−ν k L ∞ (B) + khk L ∞ (∂ B) ,

where we used (B.81) in the last inequality. Hence, |∇φ(x 0 )| ≤ C 0 |x 0 |ν−1 , which is the desired result. 0,α Step 4. We assume now that g ∈ Cν−2,0 (B) and h ∈ C 2,α (∂ B) satisfy (B.68). For " > 0 let h" be the convolution product of h with a standard mollifier on the sphere ∂ B. Let ρ" be a standard mollifier in RN and define g" (x) = |x|ν−2 ρ" (x) ∗ (g|x|2−ν ), where g is first extended by zero outside B. Since g(x)|x|2−ν ∈ L ∞ (B), we have g" ∈ C ∞ (B \ {0}) and

|∇i g" (x)| ≤ C(i, ")|x|ν−2−i . Moreover, g" → g a.e. in B, h" → h a.e. as " → 0 on ∂ B and kg" |x|2−ν k L ∞ (B) ≤ kg|x|2−ν k L ∞ (B)

and

kh" k L ∞ (∂ B) ≤ khk L ∞ (∂ B) .

From this and (B.68), we deduce that for all k = 0, . . . , k1 and l = 1, . . . , mk , ˆ ˆ ∂ Wk,l dσ → 0 as " → 0. g" Wk,l d x − h" ∂n B ∂B Let (") ak,l

=´

∂ B Wk,l

ˆ

ˆ

1 ∂ Wk,l ∂n

dσ

and ˜h" = h" +

B

g" Wk,l d x −

k1 X mk X

∂B

h"

∂ Wk,l ∂n

dσ

(")

ak,l Wk,l .

k=0 l=1 2,α Then g" , ˜h" satisfy the orthogonality conditions (B.68). Let φ" ∈ Cν,0 (B) denote the solution to (B.67) with data g" , ˜h" . We have kφ" |x|−ν k L ∞ (B) ≤ C kg" |x|2−ν k L ∞ (B) + k˜h" k L ∞ (∂ B) ≤ C kg|x|2−ν k L ∞ (B) + khk L ∞ (∂ B) .

267

Stable solutions of elliptic PDEs As in the previous step, from here we deduce that φ = lim"→0 φ" is a solution to (B.67) with data g, h. In addition, (B.81) holds. Finally, the estimate (B.69) is obtained by scaling, working as in Claim B.5.1. Proof of Corollary B.5.1. Let (αn ) denote an arbitrary sequence of real numbers converging to zero, g˜ (x) = |x|2 g(x) and φn (x) = φ(αn x) for x ∈ B1/αn (0). Then φn solves −∆φn −

c 2

|x|

φn =

g˜ (αn x)

in B1/αn (0).

|x|2

Also, (φn ) is uniformly bounded so that up to a subsequence, it converges in the topology of C 1,α (RN \ {0}) to a bounded solution Φ of −∆Φ −

g˜ (0) Φ = |x|2 |x|2 c

in RN \ {0}.

Now, Φ + g˜ (0)/c is bounded and solves (B.62), so it must be identically zero. It follows that the whole sequence (φn ) converges to −˜ g (0)/c. Now let (x n ) denote an arbitrary sequence of points in RN converging to 0 and αn = |x n |. x Then, φ(x n ) = φn ( |x n | ) and up to a subsequence, φ(x n ) → −˜ g (0)/c. Again, n since the limit of such a subsequence is unique, the whole sequence converges.

B.5.4

The case ξ 6= 0

The purpose of this section is to extend Lemma B.5.1 to a general bounded, smooth domain Ω of RN , N ≥ 3 and general ξ ∈ Ω, by redefining the functions Wk,l , which appear in (B.68). For this we restrict ourselves to the values of c in the range 0
(N − 2)2 4

,

which guarantees α− < N 2−2 < α+ . 0 0 0,α Take g ∈ Cν−2,0 (Ω) ∩ H −1 (Ω) and h ∈ C 2,α (∂ Ω). Hardy’s inequality and the condition c <

(N −2)2 4

ensure that equation  c  −∆φ − φ = g in Ω, |x − ξ|2  φ = h on ∂ Ω, 268

(B.83)

Appendix B. Regularity theory for elliptic operators has a unique solution φ ∈ H 1 (Ω). We define Wk,l,ξ , which will play the same role as in (B.68), to be smooth functions in Ω \ {ξ} satisfying  c −∆W − Wk,l,ξ = 0 in Ω \ {ξ}, k,l,ξ  |x − ξ|2   Wk,l,ξ = 0 on ∂ Ω, (B.84)   x −ξ  + ) x ∼ ξ. Wk,l,ξ (x) ∼ |x − ξ|−αk ϕk,l ( |x − ξ| Indeed, let −α+ k

Wk,l,ξ (x) = |x − ξ|

ϕk,l

x −ξ

|x − ξ|

− ψk,l,ξ (x),

(B.85)

where ψk,l,ξ ∈ H 1 (Ω) is the unique solution to  c −∆ψ − ψk,l,ξ = 0 in Ω, k,l,ξ  |x − ξ|2 x −ξ +  ψk,l,ξ = |x − ξ|−αk ϕk,l ( ) on ∂ Ω. |x − ξ| −

−

Observe that for C > 0 large enough, C|x − ξ|−α0 and −C|x − ξ|−α0 are respectively a super- and a subsolution of the above equation, hence by the maximum principle (which is valid in virtue of Hardy’s inequality and the re− (N −2)2 striction c < 4 ), |ψ| ≤ C|x − ξ|−α0 and Wk,l,ξ satisfies (B.84). Remark B.5.3 If Ω = B1 (0) and ξ = 0, our definition is consistent with (B.65), since −

ψk,l,ξ = |x|−αk ϕk,l (x/|x|) and +

−

Wk,l,ξ = (|x|−αk − |x|−αk )ϕk,l (x/|x|) .

(B.86)

Theorem B.5.1 Let N ≥ 3 and 0 < c < (N − 2)2 /4. Assume that ∃k1 ≥ k0

such that

− α− > ν > −α− . k1 +1 k1

(B.87)

0,α Let Ω a smooth bounded domain of RN , ξ ∈ Ω, g ∈ Cν−2,ξ (Ω) ∩ H −1 (Ω) and h ∈ C 2,α (∂ Ω). If ˆ ˆ ∂ Wk,l,ξ dσ, ∀k = 0, . . . , k1 , ∀l = 1, . . . , mk (B.88) gWk,l,ξ d x = h ∂n Ω ∂Ω

269

Stable solutions of elliptic PDEs 2,α then there exists a unique φ ∈ Cν,ξ (Ω) ∩ H 1 (Ω) solution to  c  −∆φ − φ = g in Ω |x − ξ|2  φ = h on ∂ Ω,

(B.89)

and it satisfies kφk2,α,ν,0 ≤ C( kgk0,α,ν−2,ξ;Ω + khkC 2,α (∂ Ω) )

(B.90)

where C is independent of g and h. Proof of Theorem B.5.1. By translating the domain, we consider from here on ξ = 0. By Lemma B.5.1, a straightforward scaling argument implies that Theorem B.5.1 holds when Ω = BR (0) and ξ = 0. In this case, Wk,l,0 takes the form −α+k −α−k ! |x| x |x| f . (B.91) − ϕk,l Wk,l (x) = R R |x| This is obtained by scaling the functions in (B.86) and is the same as in definition (B.85) except for a multiplicative constant. Take R > 0 small, so that BR (0) ⊂ Ω. Then the unique solution φ ∈ H 1 (Ω) of (B.89) satisfies (B.90) if ˆ ˆ fk,l ∂W f g Wk,l d x = φ dσ, ∀k = 0, . . . , k1 , ∀l = 1, . . . , mk , (B.92) ∂n BR ∂ BR fk,l is defined in (B.91). Since W fk,l satisfies where W fk,l − −∆W

c |x|2

fk,l = 0 W

in RN \ {0},

multiplying this equation by φ and integrating in Ω \ BR , we obtain ! ˆ ˆ ˆ fk,l fk,l ∂W ∂W ∂φ fk,l fk,l d x. (B.93) φ−W dσ − φ dσ = gW ∂n ∂n ∂ BR ∂ n Ω\BR ∂Ω Adding (B.92) and (B.93) we see that (B.92) is equivalent to ! ˆ ˆ fk,l ∂W ∂φ fk,l fk,l d x. φ−W dσ = gW ∂n ∂n ∂Ω Ω 270

(B.94)

Appendix B. Regularity theory for elliptic operators ˜ k,l ∈ H 1 (Ω) be the solution to Let ψ  ˜ k,l −  −∆ψ

c |x|2

˜ k,l = 0 ψ

in Ω,

˜ k,l = W fk,l on ∂ Ω. ψ



Multiplying this equation by φ and integrating by parts yields ˆ ˜ ˆ ∂ ψk,l ∂φ ˜ ˜ k,l d x. φ− ψk,l dσ = gψ ∂ n ∂ n ∂Ω Ω Subtracting this equation from (B.94) we obtain that (B.92) is equivalent to

ˆ ∂Ω

˜ k,l ) fk,l − ψ ∂ (W ∂n

ˆ φ dσ =

Ω

˜ k,l ) d x. fk,l − ψ g(W

˜ k,l is the same as Wk,l,0 as defined in fk,l − ψ Up to multiplicative constant W (B.85).

271

Appendix C Geometric tools In this chapter, we gather the main geometric results used in the book.

C.1

Functional inequalities

We begin our discussion with three fundamental inequalities that are deeply related to the geometry of RN .

C.1.1

The isoperimetric inequality

Theorem C.1.1 Let N ≥ 2. Among all sets Ω ⊂ RN having a smooth boundary of given finite (N − 1) Hausdorff measure, the ball has the largest N -dimensional volume, that is, |Ω|

N −1 N

≤ C |∂ Ω| ,

(C.1)

N −1 with C = B1 N / ∂ B1 . Equality holds if and only if Ω is the ball of radius 1 N −1 |∂ Ω| r = ∂B . | 1| Proof. We follow [189]. For x ∈ Ω, consider the sets Ai (x) = {z : z j = x j for j < i and zi ≤ x i } and Bi (x) = {z : z j = x j for j < i}, 273

Stable solutions of elliptic PDEs with the convention that B1 = Ω. Consider the map yΩ : x → y defined by ´ A (x)∩Ω dzi . . . dzN yi = ´ i . Bi (x)∩Ω dzi . . . dzN Since Ai ⊂ Bi , we have 0 ≤ yi ≤ 1, that is, yΩ maps Ω into the cube [0, 1]N . Furthermore, yΩ is a triangular map, that is, for each i, yi is a function of x 1 , . . . , x i only. Also, each partial derivative ∂ yi /∂ x i is nonnegative and equal to ´ Bi+1 (x)∩Ω dzi+1 . . . dzN  ´ if 1 ≤ i < N ,   ∂ yi dz . . . dz i N Bi (x)∩Ω (x) =  ∂ xi 1   ´ if i = N , BN (x)∩Ω dzN for every x ∈ Ω. Thus, the Jacobian of yΩ is equal to J(x) =

N Y ∂ yi i=1

∂ xi

=

1 |Ω|

.

In the particular case where Ω = B is the unit ball, the map yB is invertible. Its inverse z : (0, 1)N → B is still triangular, with Jacobian equal to |B|. Now, consider the map F = z ◦ yΩ : Ω → B. By construction, F has nonnegative partial derivatives ∂ Fi /∂ x i and its Jacobian is given by |B| , J F (x) = |Ω| for all x ∈ Ω. Thus, the divergence ∇ · F satisfies 1 N

∇·F ≥

1/N JF

=

|B|

1/N

|Ω|

.

Furthermore, since |F | ≤ 1, N

|B| |Ω|

ˆ

1/N |Ω| ≤

Ω

ˆ ∇· F dx =

and equality holds only if Ω is a ball.

∂Ω

F · n dσ ≤ |∂ Ω|,

274

Appendix C. Geometric tools

C.1.2

The Sobolev inequality

The celebrated Sobolev inequality reads as follows. Theorem C.1.2 Let N ≥ 2 and p ∈ [1, N ). Then, there exists a constant C = C(N , p) > 0 such that for all ϕ ∈ Cc1 (RN ),

kϕk L p∗ (RN ) ≤ Ck∇ϕk L p (RN ) , where

1 p

∗

=

1

−

p

1 N

(C.2)

.

Proof. Step 1. We begin by proving the inequality when p = 1. Let ϕ ∈ Cc∞ (RN ). We may always assume that ϕ ≥ 0. Letting χA be the characteristic function of the set A, we have ˆ +∞

ϕ(x) = 0

χ[ϕ(x)>t] ds.

So,

ˆ kϕk

L

N N −1 (RN )

ˆ

+∞

≤ 0

kχ[ϕ(·)>t] k

L

N N −1 (RN )

ds =

+∞

N −1 [x : ϕ(x) > t] N ds.

0

By Sard’s theorem (see Theorem C.3.1), for almost every t, the level set Ω t = [x : ϕ(x) > t] has a smooth boundary of finite perimeter. The isoperimetric inequality then implies that N −1 [x : ϕ(x) > t] N ≤ C [x : ϕ(x) = t] , hence

ˆ kϕk

L

N N −1 (RN )

ˆ

+∞

[x : ϕ(x) = t] d t = C

≤C 0

RN

∇ϕ d x,

where we used the coarea formula, see Equation (C.23). Step 2. It remains to prove the case 1 1 so that ψ ∈ Cc1 (RN ) and we may apply the Sobolev inequality (C.2) with p = 1 to get ˆ 1/1∗ ˆ 1/1∗ ˆ p∗ 1∗ ϕ d x ∇ψ d x. = ψ dx ≤C (C.3) RN

RN

RN

275

Stable solutions of elliptic PDEs t−1 Now, ∇ψ = t ϕ ∇ϕ . By Hölder’s inequality, it follows that

ˆ RN

∇ψ d x ≤ t

ˆ RN

∇ϕ p d x

1/p ˆ RN

(t−1)p0 ϕ dx

1/p0 .

Observe that by the definition of p∗ , 1

−

1∗

1 p∗

=

1 1

1

−

p

=

1 p0

.

(C.4)

In particular, (t − 1)p = 0

p∗

1 1 0 ∗ −1 p = p − ∗ p0 = p∗ . ∗ ∗ 1 1 p

Hence,

ˆ RN

∇ψ d x ≤ t

ˆ RN

∇ϕ p d x

1/p ˆ RN

p∗ ϕ d x

1/p0 .

Plugging this into (C.3), we obtain ˆ RN

p∗ ϕ d x

ˆ

1/1∗ −1/p0 ≤C

RN

∇ϕ p d x

1/p .

By (C.4), 1/1∗ − 1/p0 = 1/p∗ . The desired inequality follows.

C.1.3

The Hardy inequality

Proposition C.1.1 For N ≥ 3, let H N =

ˆ HN

RN

ϕ2 2

|x|

ˆ dx ≤

RN

(N −2)2 . 4

∇ϕ 2 d x,

Then, for all ϕ ∈ Cc1 (RN ).

Furthermore, the constant H N is sharp. Proof. Note that the inequality is invariant under rotation: replacing ϕ by ϕ(T x) where T is an orthogonal transformation leaves the inequality unchanged. Similarly, the inequality is preserved when replacing ϕ by 276

Appendix C. Geometric tools N −2

R 2 ϕ(Rx). It is therefore natural to look for a minimizer that shares both the rotation invariance and the scaling property, that is, N −2 2

ϕ0 = |x|−

.

Unfortunately, as can be seen by direct inspection, ϕ0 makes both integrals in the inequality infinite. So we use the change of unknown ϕ = ϕ0 ψ, where we assume for the time being that ψ ∈ Cc1 (RN \ {0}). Then, ϕ ∈ Cc1 (RN \ {0}) and

ˆ

ˆ |∇ϕ| d x = 2

RN

ˆ |∇(ϕ0 ψ)| d x =

|ψ∇ϕ0 + ϕ0 ∇ψ|2 d x ˆ ˆ ˆ 2 2 = ψ |∇ϕ0 | d x + 2 ϕ0 ψ∇ϕ0 ∇ψ d x + ϕ02 |∇ψ|2 d x RN RN RN ˆ ˆ ϕ02 (N − 2)2 1 2 = dx + ψ ∇(ψ2 ) · ∇(ϕ02 ) d x+ 2 N N 4 |x| 2 R R ˆ + ϕ02 |∇ψ|2 d x N R ˆ ˆ (N − 2)2 1 ϕ2 ≥ dx − ψ2 ∆(ϕ02 ) d x 2 N N 4 2 R R |x| 2 ˆ 2 (N − 2) ϕ ≥ d x. 2 4 RN |x| 2

RN

RN

In the last inequality, we used the fact that ϕ02 = |x|2−N is a constant multiple of the fundamental solution of the Laplace equation, hence it is harmonic on the support of ψ. We have just established Hardy’s inequality for test functions ϕ ∈ Cc1 (RN \ {0}). Now take a cutoff function η ∈ C 1 (R), such that η(r) ≡ 0 for r ≤ 1 and η ≡ 1 for r ≥ 2. Given, ϕ ∈ Cc1 (RN ), let ηn (x) = η(n |x|) and ϕn = ηn ϕ. Then, Hardy’s inequality holds for ϕn , that is,

ˆ HN

RN

ϕn2 2

|x|

ˆ dx ≤ 277

RN

∇ϕ 2 d x n

(C.5)

Stable solutions of elliptic PDEs and ˆ

ˆ ∇ϕ (x) 2 d x = η ∇ϕ + nϕ∇η(n |x|) 2 d x n n RN RN ˆ ˆ 2 2 2 2 2 = ηn ∇ϕ d x + 2nϕηn ∇ϕ∇η(n |x|) + n ϕ ∇η (n |x|) d x RN RN ˆ 2 =: η2n ∇ϕ d x + En . (C.6) RN

We control the error term by

ˆ

En ≤ 2nkϕk∞ kηk∞ k∇ηk∞ + n2 kϕk2∞ k∇ηk2∞

[x : n|x|≤2]

d x ≤ C n2−N .

Using this together with (C.6) in (C.5), we obtain ˆ ˆ 2 ϕ2 2 η2n ∇ϕ d x + C n2−N . HN ηn 2 d x ≤ |x| RN RN We may then easily pass to the limit by monotone convergence. It remains to prove that the constant H N is optimal. Take a larger constant H > H N and test the equation with ϕn = ϕ0 ψn = |x|−(N −2)/2 ψn , where this time ψn ∈ Cc1 (RN \ {0}) is given by ψn = ηn (x)(1 − η(|x|)). Working as previously, we obtain ˆ ˆ ˆ ˆ 2 ϕn2 ϕn2 2 ∇ϕ 2 d x − H dx d x = (H − H) d x + ϕ ∇ψ N n n 0 2 2 N N N N R |x| R R R |x| ˆ ˆ 2 ψn 2 d x + n2 |x|2−N ∇η (n |x|) d x + C ≤ (H N − H) N RN RN |x| ˆ ˆ 2 ψn y 2−N ∇η 2 d y + C = (H N − H) d x + N RN |x| RN ˆ 1 ≤ (H N − H) d x + C. N [x : 1/n≤|x|≤1] |x| The right-hand side converges to −∞ as n → +∞, which implies the optimality of H N .

C.2

Submanifolds of RN

Here is a quick review of some basic facts in differential geometry. See, for example, [170, 171] for a thorough introduction to the subject. 278

Appendix C. Geometric tools

C.2.1

Metric tensor, tangential gradient

Definition C.2.1 Let N ≥ m ≥ 1. A set M ⊂ RN is an m-dimensional C 2 submanifold of RN , if given any point x 0 ∈ M , there exists open sets D ⊂ Rm , Ω ⊂ RN and a C 2 map D→Ω x: t = (t 1 , . . . , t m ) 7→ x(t) = (x 1 (t), . . . , x N (t)) such that

x 0 ∈ Ω ∩ M = x(D),

and such that the vectors ∂∂ tx (t), i = 1, . . . , m are linearly independent for each i t ∈ D. The mapping x is called a representation of M at x 0 and the vector space spanned by ∂∂ tx (t 0 ), i = 1, . . . , m, where x(t 0 ) = x 0 , is called the tangent space i Tx 0 M to M at x 0 . Remark C.2.1 As follows from the implicit function theorem, if a ∈ R is a regular value of a function u ∈ C 2 (RN ), then the level set L a = { y ∈ RN : u( y) = a} is an N − 1 submanifold of RN . Definition C.2.2 Let N ≥ m ≥ 1. Given an m-dimensional C 2 submanifold M ⊂ RN , the first fundamental form or metric tensor of M is the matrix-valued function (g i j )i, j=1,...,m defined for t ∈ D by N ∂ x k (t) ∂ x k (t) ∂ x(t) ∂ x(t) X · = . g i j (t) = ∂ ti ∂ tj ∂ ti ∂ tj k=1

Remark C.2.2 It can be easily checked that given x 0 = x(t 0 ) ∈ M , the relation X g(v, w) = g i j (t 0 )vi w j , i, j=1,...m

Pm Pm where v = i=1 vi ∂∂ tx (t 0 ), w = i=1 w i ∂∂ tx (t 0 ) are two arbitrary vectors belongi i ing to Tx 0 M defines an inner product on Tx 0 M . Furthermore, g is independent of the choice of representation x. Using a standard abuse of notations, we shall also write g = det (g i j ), while (g i j ) = (g i j )−1 denotes the inverse matrix of (g i j ). Thanks to the metric g, one can easily compute the orthogonal projection (with respect to the standard inner product) of a vector in RN on the tangent space Tx 0 M . 279

Stable solutions of elliptic PDEs Proposition C.2.1 Let N ≥ m ≥ 1. Given an m-dimensional C 2 submanifold ˜ 0 ) = (˜ M ⊂ RN and a point x 0 = x(t 0 ) ∈ M , the matrix G(x g i j (x 0 ))i, j=1,...,N defined by m X ∂ xj ∂ xi ij g˜ (x 0 ) = (t 0 ) (t 0 ) g rs (t 0 ) ∂ tr ∂ ts r,s=1 represents the orthogonal projection on Tx 0 M in the canonical basis of RN , that is, ¨ v for all v ∈ Tx 0 M , ˜ 0 ).v = G(x 0 for all v ∈ (Tx 0 M )⊥ . In particular, for all x ∈ M , X g˜ ii (x) = m, i=1,...,N

0≤

N X

g˜ i j (x)vi v j ≤ |v|2 ,

for all v = (v1 , . . . , vN ) ∈ RN ,

i, j=1

g˜ i j (x) = g˜ ji (x),

for all i, j = 1, . . . , N .

Proof. For any v = (v1 , . . . , vN ) ∈ RN , N X j=1

g˜ i j v j =

m X N X r,s=1 j=1

g rs

∂ xi ∂ x j ∂ t r ∂ ts

vj =

m X

g rs

r,s=1

∂ xi ∂ x ∂ t r ∂ ts

· v.

So, (˜ g i j ) leaves invariant all the vectors ∂∂ tx , . . . , ∂∂tx , while it vanishes on the 1 m orthogonal complement of those vectors. Using the projection matrix, the tangential gradient is simply defined as the projection of the gradient on the tangent space. Proposition C.2.2 Let N ≥ m ≥ 1. Given an m-dimensional C 2 submanifold M ⊂ RN , a point x 0 = x(t 0 ) ∈ M , an open set Ω ⊂ RN such that x 0 ∈ Ω, and a function ϕ ∈ C 1 (Ω, R), the tangential gradient of ϕ at x 0 is defined as the orthogonal projection of ∇ϕ on Tx 0 M . We have ˜ 0 ).∇ϕ(x 0 ) = ∇ T ϕ(x 0 ) := G(x

m X

m X

r=1

s=1

280

g rs (t 0 )

∂ϕ ∂ ts

! (t 0 )

∂x ∂ tr

(t 0 ).

Appendix C. Geometric tools Proof. Let ei , i = 1, . . . , N , denote the canonical basis of RN . We compute ˜ 0 ).∇ϕ(x 0 ) = G(x

N X

g˜ i j

i, j=1

=

∂ϕ

N X m X

g rs

i, j=1 r,s=1

= =

N X m X

g rs

i=1 r,s=1 m X ∂ϕ rs

g

r,s=1

ei

∂ xj

∂ xi ∂ x j ∂ ϕ ∂ t r ∂ ts ∂ x j

∂ xi ∂ ϕ ∂ t r ∂ ts ∂x

∂ ts ∂ t r

ei

ei

.

C.2.2

Surface area of a submanifold

Let x : D ⊂ Rm → Ω denote a representation of an m-dimensional C 2 submanifold M ⊂ RN . Given a smoothly bounded subdomain ω ⊂⊂ D, the image of ω on M has its m-dimensional area equal to ˆ ˆ p dσ = g d t1 . . . d t m. x(ω)

ω

Using a standard partition of unity, one can use the above formula to compute the integral over M of any function, which is, say, continuous in a neighborhood of M . The following elementary property of the area element will be useful in the sequel. Lemma C.2.1 Let 1 ≤ m ≤ N . Let M denote an m-dimensional C 2 submanifold p of the Euclidean space RN and let dσ = g d t 1 . . . d t m denote its volume element. Also let ωm denote the Lebesgue measure of the unit ball in Rm . Then, for all x 0 ∈ M , σ(Sρ (x 0 )) lim+ = ωm , ρ→0 ρm where

Sρ (x 0 ) = {x ∈ M : x − x 0 ≤ ρ}. 281

Stable solutions of elliptic PDEs Proof. Let x be a representation of M at x 0 = x(t 0 ). By Taylor’s formula, m X ∂x x(t) = x 0 + (t − t 0 )i (t 0 ) + o(|t − t 0 |). (C.7) ∂ ti i=1 Without loss of generality, we may assume that the canonical basis of Rm is orthonormal for the innner product g(t 0 ), that is, there exists λ1 , . . . , λm > 0 such that g i j (t 0 ) = λi δi j . Using this and (C.7), we obtain |x(t) − x 0 | = 2

m X

(t − t 0 )i

i, j=1

=

m X

∂x ∂ ti

(t 0 )(t − t 0 ) j

∂x ∂ tj

(t 0 ) + o(|t − t 0 |2 )

g i j (t 0 )(t − t 0 )i (t − t 0 ) j + o(|t − t 0 |2 )

(C.8) (C.9)

i, j=1

=

m X

λi (t − t 0 )2i + o(|t − t 0 |2 ).

(C.10)

i=1

Given " > 0, we deduce that for ρ sufficiently small ( ) m X λi (t − t 0 )2i ≤ (1 − ")ρ 2 ⊂ x −1 (Sρ (x 0 )) t ∈ Rm : i=1

( t ∈ Rm :

⊂

m X

) λi (t − t 0 )2i ≤ (1 + ")ρ 2 .

i=1

It follows that lim+

ρ→0

σ(Sρ (x 0 )) ρm

´ = lim+ ρ→0

= lim+

{t :

Pm

2 2 i=1 λi (t−t 0 )i ≤ρ }

´ {s :

Pm

2 2 i=1 si ≤ρ }

p

g d t1 . . . d t m

ρm ds1 . . . dsm

ρm

ρ→0

= ωm , where we used the change of variable si =

C.2.3

p

λi (t − t 0 )i , i = 1, . . . , m.

Curvature, Laplace-Beltrami operator

Tangent vector and curvature vector of a curve A regular curve on a submanifold M is a C 1 mapping x : I = (α, β) → M , such that |x 0 (τ)| > 0, for all τ ∈ I. Let t 1 (τ), . . . , t m (τ) denote the coordinates of 282

Appendix C. Geometric tools x(τ) in a given representation of M . Then, |x 0 (τ)|2 =

m X

g i j t i0 (τ)t 0j (τ)

i, j=1

and the length of the curve x(τ) is given by ˆ β L= |x 0 (τ)| dτ. α

´τ

Now set s(τ) = α |x 0 (τ)| dτ. Then, since the curve is regular, the mapping s : (α, β) → (0, L) is invertible. s is called the arc-length parameter and the mapping ¨ (0, L) → RN s 7→ x(τ(s)) where τ(s) is the inverse map of s(τ), the parametrization of the curve by arc length. The unit tangent vector to the curve is given by T=

dx ds

=

x 0 (τ) s0 (τ)

and the curvature vector of the curve is given by N=

dT ds

=

d2 x ds2

.

Note that since |T |2 = 1, T · ddsT = 0, that is, the unit tangent vector and the curvature vector are orthogonal. Normal curvature, second fundamental form Given a submanifold M , a point x 0 ∈ M , the orthogonal complement of the tangent space (with respect to the standard inner product in RN ) Nx 0 (M ) = (Tx 0 M )⊥ is called the normal space of M at x 0 . Given a regular curve x(s) parametrized by arc-length, its unit tangent vector and curvature vector can be computed in the coordinates of a representation as follows: dx ds

=

m X d ti ∂ x i=1

ds ∂ t i

283

Stable solutions of elliptic PDEs and

d2 x ds2

=

m X d2 ti ∂ x i=1

ds2 ∂ t i

m X d ti d t j ∂ 2 x

+

i, j=1

ds ds ∂ t i ∂ t j

.

Hence, given a normal vector N ∈ Nx 0 M , 2 m X d ti d t j ∂ x d2 x ·N . ·N = 2 ds ∂ ti∂ t j ds ds i, j=1 The right-hand side of the above expression can be seen as a quadratic form acting on the vector T = ddsx . This quadratic form is called the second fundamental form of M with respect to the normal N and it is represented in the basis ( ∂∂ tx ) of Tx 0 M by the matrix i

Bi j = Bi j (N ) =

∂ 2x ∂ ti∂ t j

· N.

(C.11)

The matrix Bi j can also be expressed as Bi j = − where τi = ∂N ∂ τi

· τj =

∂x . ∂ ti

∂N ∂ τi

· τj,

(C.12)

Indeed,

N X ∂ Nk ∂ x k k=1

N N X X ∂ xk ∂ Nk ∂ x l ∂ x k ∂N ∂x (∇Nk · τi ) = = = · , ∂ τi ∂ t j ∂ tj ∂ xl ∂ ti ∂ t j ∂ ti ∂ t j k=1 k,l=1

while, since N · τ j = 0, 0=

∂ ∂ ti

(N · τ j ) =

∂N

·

∂x

∂ ti ∂ t j

+N ·

∂ 2x ∂ t j∂ ti

=

∂N

·

∂x

∂ ti ∂ t j

+ Bi j .

Equation (C.12) follows. Letting T = d x/ds, the second fundamental form calculated at T , that is, the quantity d2 x k(N , T ) = 2 · N ds is called the normal curvature to M in the direction of T , with respect to N . Take an orthonormal basis of Tx 0 M and let (Bi j ) be the matrix representing 284

Appendix C. Geometric tools the second fundamental form in this basis. Then the eigenvalues ki = ki (N ), i = 1, . . . m of (Bi j ) are called the principal curvatures of M with respect to the normal N . Their arithmetic mean H(N ) =

k1 (N ) + · · · + km (N ) m

is the mean curvature of M with respect to N . Since the quantity H(N ) is linear in N , there exists a unique vector H ∈ Nx 0 M such that H(N ) = H · N . H is called the mean curvature vector of M .

Laplace-Beltrami operator and mean curvature vector Proposition C.2.3 Let x : D ⊂ Rm → Ω denote a representation of an mdimensional C 2 submanifold M ⊂ RN . Let ϕ ∈ C 2 (Ω). The Laplace-Beltrami operator acting on ϕ is defined by

ˆ

ˆ (−∆ M ϕ)ψ dσ :=

∇ T ϕ · ∇ T ψ dσ,

M

for all ψ ∈ Cc1 (Ω).

M

In coordinates, given a point x 0 = x(t 0 ) ∈ Ω ∩ M , ∆ M ϕ(x 0 ) = p

m X ∂

1 g(t 0 )

i=1

p

∂ ti

g(t)

m X j=1

! ij g (t) (t) ∂ tj ∂ϕ

. t=t 0

Proof. By definition of the tangential gradient, ∇T ϕ · ∇T ψ =

m X

g

r,s=1

=

m X

∂ϕ ∂ x rs ∂ ts ∂ t r rs

g g

µν

g rν

r,s,µ,ν=1

285

! ·

m X µ,ν=1

∂ϕ ∂ψ ∂ ts ∂ tµ

=

g

∂ψ ∂ x µν

!

∂ tµ ∂ tν

m X r,s=1

g rs

∂ϕ∂ψ ∂ ts ∂ t r

.

Stable solutions of elliptic PDEs Hence, ˆ ∇ T ϕ · ∇ T ψ dσ = M

m ˆ X r,s=1

g rs D

ˆ

=−

ψ D

ˆ

∂ ϕ ∂ ψp ∂ ts ∂ t r

m X ∂ r=1

∂ tr

g d t1 . . . d t m

m p X rs ∂ ϕ g g ∂ ts s=1

m 1 X ∂ =− ψp g r=1 ∂ t r M

! d t1 . . . d t m

m p X rs ∂ ϕ g g ∂ ts s=1

! dσ,

as claimed.

Proposition C.2.4 Let x : D ⊂ Rm → Ω denote a representation of an mdimensional C 2 submanifold M ⊂ RN . Then, the mean curvature vector of M satisfies H = ∆ M x. Proof. We begin by proving that ∆ M x belongs to the normal space Nx 0 M , at any point x = x 0 ∈ M . It suffices to show that ∆ M x · ∂∂ tx = 0, for k = 1, . . . , m. k Now, ! X ∂ ∂x ∂x p X ij ∂ x p = · g∆ M x · g g ∂ tk ∂ ti ∂ tj ∂ tk i j ! X ∂ Xp X ∂ x ∂ 2x p X ij ∂ x ∂ x ij = · g g − g g ∂ ti ∂ t j ∂ tk ∂ t j ∂ ti∂ tk i i j j ! X ∂ 1 X p X i j ∂ gi j p X ij = g g g jk − g g ∂ ti 2 i ∂ tk i j j X ∂ pg 1 p X i j ∂ gi j = δik − g g . ∂ ti 2 ∂ tk i ij Now, using the letter G to denote the matrix (g i j ), p X p d gi j ∂ g 1 dg 1 dG g −1 = p = p g tr G = gij , ∂ ti 2 g d ti 2 g dt 2 ij dt and so ∆x · ∂∂ tx = 0. k

286

Appendix C. Geometric tools It remains to prove that ∆ M x = H. To see this, simply note that 1 X ∂ p i j ∂ x X i j ∂ 2 x ∆M x = p gg + g g i j ∂ ti ∂ tj ∂ ti∂ t j ij and so, for any normal vector N ∈ Tx 0 M , ∆M x · N =

X ij

gij

∂ 2x ∂ ti∂ t j

· N = H · N.

As an immediate consequence of Proposition C.2.4, we obtain the following useful formula. Lemma C.2.2 Let 1 ≤ m ≤ N . Let x : D ⊂ Rm → Ω denote a representation of an m-dimensional C 2 submanifold M ⊂ RN . For all ϕ ∈ Cc1 (Ω), there holds

ˆ ∇ T ϕ + Hϕ dσ = 0. M

C.2.4

The Sobolev inequality on submanifolds

This section is dedicated to the proof of the following theorem. Theorem C.2.1 ([5, 153]) Let 1 ≤ m ≤ N . Let M denote an m-dimensional C 2 submanifold of the Euclidean space RN and U an open set containing M . For all p ∈ [1, m), there exists a constant C = C(m, p) > 0 such that for all ϕ ∈ Cc1 (U), ˆ

p∗ ϕ dσ M

ˆ

1/p∗

∇ ϕ p + Hϕ p dσ T

≤C

1/p ,

(C.13)

M

where 1/p∗ = 1/p − 1/m, where H is the mean curvature vector of M and where ∇ T ϕ is the projection of the gradient of ϕ on the tangent space of M . Remark C.2.3 • Note that the constant C appearing in (C.13) is independent of the given manifold M . The geometry of M enters only through its mean curvature vector H. 287

Stable solutions of elliptic PDEs • The inequality is local: using a standard partition of unity, it suffices to prove it on a neighborhood of a point x 0 ∈ M . At the expense of making global assumptions on the manifold M (for example, if M is compact or if the following three assumptions hold: M is complete, M has a nonnegative Ricci curvature, and M has maximal volume growth), one can derive the standard (global) Sobolev inequality, that is, ˆ

p∗ ϕ dσ

ˆ

1/p∗

∇ ϕ p dσ T

≤C

1/p

M

M

holds for all ϕ ∈ Cc1 (M ), with C = C(m, p, M ). See [189]. The proof of Theorem C.2.1 uses the following two lemmata. Lemma C.2.3 Suppose λ ∈ C 1 (R) is a nondecreasing function such that λ(t) ≤ 0 for t ≤ 0. Let ϕ ∈ Cc1 (U), ϕ ≥ 0. For x 0 ∈ M , define ϕ x 0 , ψ x 0 ∈ C 1 (0, +∞) by ˆ ϕ x 0 (ρ) = ϕ(x)λ(ρ − r) dσ(x) M

ˆ

and ψ x 0 (ρ) =

|∇ T ϕ(x) + |H(x)|ϕ(x) λ(ρ − r) dσ(x), M

where r = |x − x 0 |. Then, ϕ x 0 (ρ) ψ x 0 (ρ) d , ≤ − dρ ρm ρm

for all ρ > 0.

Proof. Using Lemma C.2.2 with (x − x 0 )i λ(ρ − r)ϕ in place of ϕ and summing over i = 1, . . . , N , we have   ˆ X ˆ N N X δi [(x − x 0 )i λ(ρ − r)ϕ] dσ = − ϕλ(ρ − r)  (x − x 0 )i H i  dσ, M i=1

M

i=1

where δi , H i are the components of ∇ T , H in the canonical base of RN . Now, δi [(x−x 0 )i λ(ρ−r)ϕ] = g˜ λ(ρ−r)ϕ−rλ (ρ−r)ϕ 0

ii

N X (x − x 0 )i (x − x 0 ) j j=1

+ λ(ρ − r)(x − x 0 )i δi ϕ, 288

r

r

g˜ i j

for all i = 1, . . . , N ,

Appendix C. Geometric tools and hence, using Proposition C.2.1 together with N X (x − x 0 )i 2 r

i=1

we obtain

ˆ

mϕ x 0 (ρ) −

,

ˆ rλ (ρ − r)ϕ dσ ≤

rλ(ρ − r)[|∇ T ϕ| + |H|ϕ] dσ.

0

M

M

Since λ(ρ − r) = 0 when r ≥ ρ, so that rλ(ρ − r) ≤ ρλ(ρ − r) and

rλ0 (ρ − r) ≤ ρλ0 (ρ − r),

we obtain mϕ x 0 (ρ) − ρϕ 0x 0 (ρ) ≤ ρψ x 0 (ρ). This last inequality can be rewritten in the form ψ x 0 (ρ) ϕ x 0 (ρ) d ≤ , − m dρ ρ ρm as requested.

Lemma C.2.4 Let ϕ be as in Lemma C.2.3 and let x 0 ∈ M such that ϕ(x 0 ) ≥ 1. Define ϕ x 0 , ψ x 0 on (0, +∞) by

ˆ ϕ x 0 (ρ) =

Sρ (x 0 )

ϕ dσ,

ˆ ψ x 0 (ρ) =

Sρ (x 0 )

[|∇ T ϕ| + |H|ϕ] dσ,

where Sρ (x 0 ) = {x ∈ M : x − x 0 ≤ ρ}. 1/m ´ Then, there exists ρ such that 0 < ρ < 2 ω−1 ϕ dσ and M m ϕ x 0 (4ρ) ≤ 4

m

ˆ ω−1 m

ϕ dσ M

1/m

ψ x 0 (ρ),

where ωm denotes the Lebesgue measure of the unit ball in Rm . 289

Stable solutions of elliptic PDEs Proof. Let ϕ x 0 , ψ x 0 be as in Lemma C.2.3, so that −

d

ϕ x 0 (ρ) ρm

dρ

Let ρ0 = 2

ψ x 0 (ρ)

≤

ρm

ˆ ω−1 m

ϕ dσ

1/m

.

(C.14)

> 0.

M

Assume that t ∈ (0, ρ0 ) and integrate (C.14) over the interval (t, ρ0 ). This yields ˆ ρ0 −m −m t ϕ x 0 (t) ≤ ρ0 ϕ x 0 (ρ0 ) + ρ −m ψ x 0 (ρ) dρ ˆ t ρ0 (C.15) −m −m ≤ ρ0 ϕ x 0 (ρ0 ) + ρ ψ x 0 (ρ) dρ. 0

Now let " ∈ (0, t) and suppose that the function λ appearing in the definition of ϕ x 0 , ψ x 0 is chosen so that λ(t) = 1 for all t ≥ ". It follows from (C.15) that

ˆ t

−m

ϕ x 0 (t − ") ≤

ρ0−m ϕ x 0 (ρ0 ) +

0

ρ0

ρ −m ψ x 0 (ρ) dρ.

Since t < ρ0 and " ∈ (0, t) are arbitrary, this gives ˆ ρ0 −m −m ρ −m ψ x 0 (ρ) dρ. sup t ϕ x 0 (t) ≤ ρ0 ϕ x 0 (ρ0 ) + t∈(0,ρ0 )

(C.16)

0

Assume that, contrary to the statement of the lemma, ψ x 0 (ρ) 2.4−m ρ0−1 ϕ x 0 (4ρ), for all ρ ∈ (0, ρ0 ). Then,

ˆ 0

ρ0

ˆ ρ

−m

ψ x 0 (ρ) dρ ≤ 2.4

−m

ρ0−1

ˆ

0

ρ0

<

ρ −m ϕ x 0 (4ρ) dρ

4ρ0

1 = ρ0−1 t −m ϕ x 0 (t) d t 2 0 ˆ ρ ˆ +∞ 0 1 −1 −m −m t ϕ x 0 (t) d t ≤ ρ0 t ϕ x 0 (t) d t + 2 0 ρ0 ˆ 1 −1 1 −m 1−m ≤ ρ0 ρ0 sup t ϕ x 0 (t) + ρ ϕ dσ , 2 m−1 0 t∈(0,ρ0 ) M 290

Appendix C. Geometric tools where we used the fact that ϕ x 0 (t) ≤ from (C.16) that 1

sup t 2 t∈(0,ρ0 )

−m

ϕ x 0 (t) ≤

´

ρ0−m

M

ϕ dσ for all t > 0. Hence, it follows ˆ

1

1+

2(m − 1)

ϕ dσ, M

so that, using the definition of ρ, sup t

−m

t∈(0,ρ0 )

ϕ x 0 (t) ≤ 2

1−m

1

ωm 1 +

2(m − 1)

< ωm .

Using Lemma C.2.1 and the assumption ϕ(x 0 ) ≥ 1, we obtain a contradiction. Proof of Theorem C.2.1. Step 1. We begin by proving that the case p ∈ (1, m) can be derived from the case p = 1. Assume that (C.13) holds in the latter case, take ϕ ∈ Cc1 (U) and t−1 let ψ = ϕ ϕ, where t = p∗ /1∗ , for some p ∈ (1, m). In particular, t > 1 so that ψ ∈ Cc1 (U) and we may apply (C.13) to get ˆ

p∗ ϕ dσ

1/1∗

=

ˆ

1∗ ψ dσ

ˆ ∇ ψ + Hψ dσ. ≤C T

1/1∗

M

M

M

(C.17) Now, ∇ ψ + Hψ = ϕ t−1 t ∇ ϕ + Hϕ ≤ t ϕ t−1 ∇ ϕ + Hϕ . T T T By Hölder’s inequality, it follows that

ˆ ∇ ψ + Hψ dσ ≤ T M

ˆ

∇ ϕ p + Hϕ p dσ T

C(p)

1/p ˆ

M

(t−1)p0 ϕ dσ

1/p0 .

M

Observe that by the definition of p∗ , 1 1∗

−

1 p∗

=

1 1

−

1 p

=

1 p0

.

In particular, (t − 1)p = 0

p∗

1 1 ∗ −1 p = p − ∗ p0 = p∗ . ∗ ∗ 1 1 p 291

(C.18)

Stable solutions of elliptic PDEs Hence,

ˆ ∇ ψ + Hψ dσ ≤ T M

ˆ

∇ ϕ p + Hϕ p dσ T

C(p)

1/p ˆ

M

p∗ ϕ dσ

1/p0 .

M

Plugging this in (C.17), we obtain ˆ

p∗ ϕ dσ

ˆ

1/1∗ −1/p0

∇ ϕ p + Hϕ p dσ T

≤ C(m, p)

M

1/p .

M

By (C.18), 1/1∗ − 1/p0 = 1/p∗ and the desired inequality follows. It remains to prove (C.13) in the case p = 1. Step 2. We now assume that p = 1 and, without loss of generality ϕ ≥ 0. Using a covering argument and Lemma C.2.4, we prove that σ({x ∈ M : ϕ(x) ≥ 1}) ≤ 4 Let

m

ˆ ω−1 m

ϕ dσ

1/m ˆ

M

[|∇ T ϕ| + |H|ϕ] dσ. M

A = {x ∈ M : ϕ(x) ≥ 1}

and assume for the time being that A is nonempty. For x ∈ A, let ϕ x , ψ x be 1/m ´ as in Lemma C.2.4 and let J = 4m ω−1 . For i = 1, 2, . . . , let M ϕ dσ m −i ρi = 4.2 J and 1 . Ai = x ∈ A : ϕ x (4ρ) ≤ Jψ x (ρ) for some ρ ∈ ρi , ρi 2 It follows from Lemma C.2.4 that A = ∪+∞ A. i=1 i Next, define inductively a sequence F0 , F1 , . . . of subsets of A as follows: (i) F0 = ;. (ii) Let k ≥ 1 and assume that F0 , . . . , Fk−1 have been defined. Let Bk = k−1 Ak \ ∪i=0 ∪ x∈Fi S2ρi (x). If Bk = ;, then set Fk = ;. If Bk 6= ;, then choose Fk to be a finite subset of Bk such that Bk ⊂ ∪ x∈Fk S2ρk (x) and such that the sets Sρk (x), x ∈ Fk , are pairwise disjoint. Then, it is not difficult to check that the following properties hold: 292

Appendix C. Geometric tools (a) Fi ⊂ Ai , for i = 1, 2, . . . , (b) A ⊂ ∪+∞ ∪ x∈Fi S2ρi (x), and i=1 (c) the sets of the countable collections Sρi (x), x ∈ Fi , i = 1, 2, . . . , are pairwise disjoint. By property (a) we have, for each x ∈ Fi , ϕ x (4ρ) ≤ Jψ x (ρ) for some ρ ∈

1

ρ , ρ i . Thus, 2 i ϕ x (2ρi ) ≤ Jψ x (ρi ),

for each x ∈ Fi . Summing over all x ∈ Fi , i = 1, 2, . . . , and using properties (b) and (c), we then have ˆ σ(M1 ) ≤ J [|∇ T ϕ| + |H|ϕ] dσ, (C.19) M

where M1 = {x ∈ M : ϕ(x) ≥ 1}, provided that A 6= ;. When A = ;, (C.19) is trivially true. Step 3. Completion of the proof. Let α, " > 0 be arbitrary constants, let λ ∈ C 1 (R) be a nondecreasing function such that λ(t) = 0 for t ≤ −", λ(t) = 1 for t ≥ 0, and use (C.19) with λ(ϕ − α) in place of ϕ. Then, (C.19) gives σ(Mα ) ≤ 4

m

ω−1/m m

ˆ

1/m

M

λ(ϕ − α) dσ ˆ × [λ0 (ϕ − α)|∇ T ϕ| + λ(ϕ − α)|H|] dσ, (C.20) M

where Mα = {x ∈ M : ϕ(x) ≥ α}. Multiplying each side of (C.20) by α1/(m−1) and using the fact that λ(ϕ − α) = 0 for α ≥ ϕ + ", we then get α

1/(m−1)

σ(Mα ) ≤ 4

m

ω−1/m m

ˆ

ˆ

(ϕ + ")

m/(m−1)

1/m dσ

M

[λ0 (ϕ − α)|∇ T ϕ| + λ(ϕ − α)|H|] dσ. (C.21)

× M

293

Stable solutions of elliptic PDEs Finally, we obtain the desired inequality by first integrating over (0, +∞) with respect to α, using the fact that

ˆ

+∞

ˆ λ(ϕ − α) dα =

0

+∞

αλ0 (ϕ − α) dα

0

ˆ ≤ (ϕ + ")

+∞

λ0 (ϕ − α) dα ≤ ϕ + ",

0

and

ˆ

+∞

α

1/(m−1)

0

σ(Mα ) dα =

m−1 m

ˆ ϕ m/(m−1) dσ, M

and then letting " → 0.

C.3

Geometry of level sets

Consider an open set Ω ⊂ RN and a value a ∈ R. The level set of the real valued map u ∈ C 1 (Ω) is given by L a = { y ∈ Ω : u( y) = a}. For example, if Ω is the unit ball and if u is a radial function, the level set L a is the union of all the hyperspheres of radius R such that u(R) = a. The classical Morse-Sard theorem asserts that for any u ∈ C N (Ω) and almost all a ∈ R, the level set L a is a regular hypersurface. Theorem C.3.1 ([164], [192]) Let N ≥ 1 and let Ω denote an open set in RN . Assume that u ∈ C N (Ω). Let X denote the set of critical points of u, that is, X = {x ∈ Ω : ∇u(x) = 0}. Then u(X ) has Lebesgue measure 0 in R. For a proof, see, for example, [162]. Definition C.3.1 A value a ∈ R such that ∇u(x) 6= 0 for all x ∈ L a is called a regular value of u. Using the implicit function theorem, we obtain the following corollary. Corollary C.3.1 Let u ∈ C N (Ω). For almost every a ∈ u(Ω), the level set L a ⊂ RN is an (N − 1)-dimensional submanifold of RN . 294

Appendix C. Geometric tools In particular, if u ∈ C N (Ω), for almost every a ∈ u(Ω), the level set L a has zero N -dimensional Lebesgue measure. It is interesting to note that if there exists a value a, such that the level set L a has nonzero Lebesgue measure, then the gradient of u must vanish at almost every point of L a . 1,1 Lemma C.3.1 Let Ω be an open set of RN , N ≥ 1, u ∈ Wloc (Ω) and a ∈ u(Ω). Then, ∇u(x) = 0, for almost every x ∈ L a .

Proof. We may assume that a = 0 without loss of generality. For " > 0, take a cutoff function χ" ∈ Cc (R), such that´ 0 ≤ χ" ≤ 1, χ" ≡ 1 in (−", ") and χ" ≡ 0 t in R \ (−2", 2"). Also let Φ" (t) = 0 χ" ds, for all t ∈ R. Then, given any function ϕ ∈ Cc1 (Ω), ˆ ˆ (C.22) ϕχ" (u)∇u d x = − Φ" (u)∇ϕ d x. Ω

Ω

Note that |Φ" (t)| ≤ 2" for all t ∈ R. Hence, the right-hand side of (C.22) converges to zero as " → 0. Now, χ" (u) is bounded by 1 and converges pointwise to χ[u=0] . By dominated convergence, we conclude that ˆ ϕ ∇u d x = 0. [u=0]

This being true for all ϕ ∈ Cc1 (Ω), the result follows.

C.3.1

Coarea formula

When evaluating integrals on a radially symmetric domain, for example, the unit ball Ω = B, one is often led to using polar coordinates: ˆ ˆ 1 ˆ Ω

ϕ dx =

ϕ dσ 0

d r,

|x|=r

for ϕ ∈ L 1 (Ω). Extending ϕ by zero outside Ω = B and thinking of spheres as the level sets of a given radial, decreasing function u, the above formula can be restated as follows: ˆ ˆ +∞ ˆ 1 ϕ dσ d t. (C.23) ϕ dx = Ω −∞ [u=t] |∇u| The above formula remains valid more generally. 295

Stable solutions of elliptic PDEs Theorem C.3.2 Let Ω be an open set of RN , N ≥ 2, u ∈ C 2 (Ω) such that |∇u| 6= 0 in Ω, and ϕ ∈ L 1 (Ω). Then, (C.23) holds. Proof. Fix a point x 0 ∈ Ω and let α = u(x 0 ). Without loss of generality, we may assume that ϕ is compactly supported in a neighborhood of x 0 . We may also assume that ∂∂xu 6= 0 in that neighborhood. Now consider variables t = N (t 1 , . . . , t N −1 ) simply defined by t 1 = x 1 , . . . , t N −1 = x N −1 . By the implicit function theorem, the mapping x = x(t) = (t, x N (t)) given implicitly by u(t, x N (t)) = α

(C.24)

defines a representation of the level set M := [u = α] at x 0 . The metric tensor of M is given by gi j =

∂x

·

∂x

∂ ti ∂ t j

=

We claim that

N X ∂ xk ∂ xk k=1

∂ ti ∂ t j

p

|∇u|

g=

∂u ∂ xN

= δi j +

∂ xN ∂ xN ∂ ti ∂ t j

.

(C.25)

.

(C.26)

For now, take (C.26) for granted. Then, letting D denote the domain of the representation x, ˆ ˆ 1 1 ϕ d t 1 . . . d t N −1 . ϕ dσ = ∂ u [u=α] |∇u| D ∂ xN

Integrating over α, it follows that

ˆ

+∞

−∞

ˆ [u=α]

1 |∇u|

ˆ

ϕ dσ

dα =

+∞

−∞

ˆ =

Ω

  

ˆ

 1 ∂u D ∂ xN

 ϕ d t 1 . . . d t N −1  dα,

ϕ d x 1 . . . d x N −1 d x N ,

where we used the change of variable t 1 = x 1 , . . . , t N −1 = x N −1 , α = u(t 1 , . . . , t N −1 , x N ) in the last equality. ∂x ∂x It remains to prove (C.26). To this end, consider v = ( ∂ t N , . . . , ∂ t N ). Using 1 N −1 (C.25), we obtain 2 ! N −1 N −1 N −1 X X X ∂ xN ∂ xN ∂ xN ∂ xN g i j v j = vi + vj = 1 + , ∂ t ∂ t ∂ t ∂ t i j j i j=1 j=1 j=1 296

Appendix C. Geometric tools and so λ =

1+

PN −1 ∂ x N 2 j=1

∂ tj

with respect to t j :

∂u

is an eigenvalue of (g i j ). Differentiate (C.24)

+

∂ tj

∂ u ∂ xN ∂ xN ∂ t j

= 0.

So, the eigenvalue λ can be rewritten as 2  2  ∂u N −1 X  |∇u|   ∂ tj  λ=1+  ∂u  =  ∂u  . j=1

∂ xN

∂ xN

∂x

∂ xN

Now take any vector w which is orthogonal to v = ( ∂ t N , . . . , ∂ t 1

N −1

N −1

), with re-

spect to the standard inner product on R . Clearly, w can be written as ∂x ∂x ˜ = ∂ tN ek − ∂ t N el , where (ei ) a linear combination of vectors of the form w l

k

denotes the canonical basis of RN −1 . Then, N −1 N −1 X X ∂ xN ∂ xN ∂ xN ∂ xN ˜j = gi j w δk j − δl j δi j + ∂ ti ∂ t j ∂ tl ∂ tk j=1 j=1 = δik

∂ xN ∂ tl

− δil

∂ xN ∂ tk

+

∂ xN ∂ xN ∂ xN ∂ ti ∂ tk ∂ tl

−

∂ xN ∂ xN ∂ xN ∂ ti ∂ tl ∂ tk

˜ i. =w So, 1 is the only other eigenvalue of (g i j ), hence 2



 |∇u|  g = λ =  ∂u  , ∂ xN

as claimed.

C.4

Spectral theory of the Laplace operator on the sphere

Definition C.4.1 A function F defined on RN \ {0} is homogeneous of degree k ∈ Z if F (t x) = t k F (x), for any t > 0, x ∈ RN \ {0}. 297

Stable solutions of elliptic PDEs Definition C.4.2 Given N ≥ 2 and k ∈ N, let Pk denote the space of polynomials (in N variables), which are homogeneous of degree k. Let Ak denote the subspace of Pk consisting of all harmonic polynomials. Then, the space H k = Ak S N −1 , obtained by restriction from Ak is called the space of spherical harmonics of degree k. Theorem C.4.1 Let N ≥ 2 and let S N −1 denote the unit sphere of RN . The eigenvalues of the Laplace-Beltrami operator −∆S N −1 are the numbers µk = k(k + N − 2),

k ∈ N.

Furthermore, • Let H k denote the set of spherical harmonics of degree k. Then, H k is the eigenspace associated to µk . • L 2 (S

N −1

)=

L+∞ k=0

Hk .

Theorem C.4.2 dim(H k ) =

(2k + N − 2)(N + k − 3)! k!(N − 2)!

= O (k N −2 ).

To prove the above results, we follow [200], [11] and begin with a series of elementary lemmata. Lemma C.4.1 Let u, v ∈ C 2 (S

N −1

). Then,

ˆ S

ˆ N −1

u∆S N −1 v dσ =

S

N −1

v∆S N −1 u dσ.

Proof. To see this, let U(x) = u(x/|x|) and V (x) = v(x/|x|), for all x ∈ RN \ {0}. That is, U, V are extensions of u, v, which are constant along the normal lines of S N −1 . In particular, ∆S N −1 u = ∆U|S N −1 and a similar formula holds for v. In addition, since U, V are homogeneous of degree 0, ∆U, ∆V are homogeneous of degree −2 and so we have

ˆ

ˆ (V ∆U − U∆V ) d x = B2 \B1

ˆ

2

r N −3 d r S

1

298

N −1

v∆S N −1 u − u∆S N −1 v dσ.

Appendix C. Geometric tools Integrating by parts the left-hand side of the above expression, we obtain

ˆ B2 \B1

(V ∆U − U∆V )d x = ˆ ˆ ∂V ∂V ∂U ∂U −U dσ − −U dσ. V V ∂r ∂r ∂r ∂r ∂ B1 ∂ B2

Since U, V are homogeneous of degree 0, lows.

∂U ∂V , ∂r ∂r

vanish, and the lemma fol

Lemma C.4.2 Given U ∈ C 2 (RN \ {0}), ∆U =

∂ 2U ∂r

2

+

N −1∂U r

∂r

+

1 r2

∆S N −1 U.

In particular, if U ∈ Pk , then ∆U = k(k + N − 2)|x|k−2 u(x/|x|) + |x|k−2 ∆S N −1 u(x/|x|), where U(x) = |x|k u(x/|x|). Proof. By the Stone-Weierstrass theorem, it suffices to prove the identity when U is a homogeneous polynomial of degree k, so that U(x) = |x|k u(x/|x|) for some u ∈ C 2 (S N −1 ). Now, ∆(H G) = H∆G + 2∇H · ∇G + G∆H; we apply this with H(x) = |x|k and G(x) = u(x/|x|). With this H and G, ∇H is perpendicular to the sphere |x| = constant, and ∇G is tangential to that sphere, so ∇H · ∇G = 0. Since u is homogeneous of degree 0, ∆u is homogeneous of degree −2, and so |x|k−2 ∆S N −1 (u(x/|x|)) = |x|k ∆u(x). Finally, a direct calculation shows that ∆|x|k = k(k + N − 2)|x|k−2 , so that, letting r = |x| and ω = x/|x|, ∆U(x) = k(k + N − 2)|x|k−2 u(x/|x|) + |x|k−2 ∆S N −1 u(x/|x|) 2 N −1 ∂ 1 k ∂ k + r u(ω) + r ∆S N −1 u(ω) = ∂ r2 r ∂r r2 =

∂ 2U ∂ r2

+

N −1∂U r

∂r

+

1 r2

as desired.

∆S N −1 U,

299

Stable solutions of elliptic PDEs Lemma C.4.3 For k ≥ 2, any polynomial p of degree at most k can be written as p = q1 + (1 − |x|2 )q2 , (C.27) where q1 is a harmonic polynomial and q2 is a polynomial of degree at most k−2. Proof. It suffices to find a polynomial q2 of degree at most k − 2 such that ∆((1 − |x|2 )q2 ) = ∆p. To do this, let W denote the vector space of polynomials of degree at most k − 2 and define the linear map T : W → W by T (q) = ∆((1 − |x|2 )q). Observe that T is one-to-one: if T (q) = 0, then (1 − |x|2 )q is harmonic. In addition, (1 − |x|2 )q = 0 on S N −1 . By the maximum principle, (1 − |x|2 )q ≡ 0 in B and so q ≡ 0. Hence, T is a one-to-one endomorphism of W , so it is also surjective and the lemma follows. Proof of Theorem C.4.1. We begin by proving that any nonzero spherical harmonic ϕ ∈ H k is an eigenvector associated to µk . Indeed, take p = |x|k ϕ(x/|x|) ∈ Ak . Applying Lemma C.4.2 to p, we have on S N −1 , 0 = ∆p = k(N + k − 2)ϕ + ∆S N −1 ϕ and so ϕ is an eigenvector associated to µk . By Lemma C.4.1, if ϕk , ϕ j are eigenvectors associated to µk 6= µ j , then ˆ ˆ 0= ϕk ∆S N −1 ϕ j − ϕ j ∆S N −1 ϕk dσ = (µk − µ j ) ϕ j ϕk dσ S

N −1

S

N −1

and so H k is orthogonal to H j in L 2 (S N −1 ). It remains to prove that the space of spherical harmonics is dense in L 2 (S N −1 ). To see this, by the density of continuous functions in L 2 (S N −1 ) and the Stone-Weierstrass theorem, it suffices to prove that given any polynomial p, its restriction p S N −1 can be written as a linear combination of spherical harmonics. We work inductively on the degree k of p. If k = 0, then p is clearly harmonic and the claim follows. Now, assume by induction that any polynomial of degree at most k−1 is a sum of spherical harmonics, when restricted to the sphere. Take p a polynomial of degree k and let pk denote its homogeneous part of degree k. Apply Lemma C.4.3 to pk . Take the homogeneous part of degree k on both sides of (C.27). Then, pk = qk − |x|2 qk−2 , where qk is a homogeneous harmonic polynomial of degree k and qk−2 is a homogeneous polynomial of degree k − 2. By the induction hypothesis, qk−2 is 300

Appendix C. Geometric tools a sum of spherical harmonics when restricted to the sphere. This implies that pk and p can also be decomposed in spherical harmonics on S N −1 . Proof of Theorem C.4.2. Let dk,N denote the dimension of the space of homogeneous polynomials of degree k. Separating monomials of degree k into those divisible by x N and those not divisible by x N , we get dk,N = dk−1,N + dk,N −1 , and clearly dk,1 = d0,N = 1. By induction on k + N , we deduce that dk,N =

(N + k − 1)! k!(N − 1)!

.

By Theorem C.4.1, any polynomial p of degree at most k can be written as a linear combination of spherical harmonics qi , i = 0..k, when restricted to the sphere. Noting that |x|2 q = q on S N −1 , we deduce that p S N −1 = q1 S N −1 + q2 S N −1 , where q1 is a homogeneous polynomial of degree k and q2 a homogeneous polynomial of degree k − 1. Note that the decomposition is unique, since any homogeneous polynomial q of degree j must be either odd or even with j. We deduce that the space of restrictions of polynomials of degree at most k to the sphere, has dimension dk,N + dk−1,N . Finally, applying Theorem C.4.1, any of degree at most k can be uniquely decomposed in the form polynomial p S N −1 = q1 S N −1 + q2 S N −1 , where this time q1 S N −1 is a spherical harmonic of degree k and q2 is a polynomial of degree at most k − 1. We deduce that dim H k = dk,N + dk−1,N − (dk−1,N + dk−2,N ), which proves Theorem C.4.2.

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