MARIANO GIAQUINTA GIUSEPPE MODICA ji SOUdEK
Volume 37
Ergebnisse der Mathematik and ihrer Grenzgebiete 3. Folge
Cartesian Currents in the Calculus of Variations I
A Series
of Modern Surveys in Mathematics
Cartesian Currents
" Springer 4:s"
Mariano Giaquinta Giuseppe Modica Jir i Soucek
Cartesian Currents in the Calculus of Variations I Cartesian Currents
Springer
Mariano Giaquinta Dipartimento di Matematica University di Pisa Via F. Buonarroti, 2 1-56127 Pisa
Italy Giuseppe Modica
Dipartimento di Matematica Applicata University di Firenze Via S. Marta, 3 1-50139 Firenze
Italy Jiii Soucek Faculty of Mathematics and Physics Charles University Sokolovska, 83 i86oo Praha 8 Czech Republic Library of Congress Cara log tng-,n-Publication Data Giaquinta. Marlano. 1947Cartesian currents in the calculus of variations G,aqu,nta, Giuseppe Modica, Juui Soucek. o.
Mar, and
cm. -- iErgemnsse der Mathematlk and ,hrer Grenzgeb,ete
3. Fdlge, v. 37-38) Includes b,bl,ograph)ca I references and index. ISBN 3-540-64009-6 (v hardcover alk. paper). -- ISBN 3-540-64010-X in 2 hardcover alk. paper) I. Calculus of variations. I. Medica. Giuseppe. II. Soucek. Jlli. III Title. IV. Series Ergebnisse der Mathemat)k and hrer Grenzgeblete 3. Folge. Bd. 37-38QA316.G53 1998 515'.64--dc21 98-18195 CIP 1
,
Mathematics Subject Classification (1991): 49Q15, 49Q20, 49Q25, 26B30, 58E20, 73C50, 76A15
ISSN 0071-1136
ISBN 3-540-64009-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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To
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Preface
Non-scalar variational problems appear in different fields. In geometry, for instance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of o-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i.e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial. Keeping in mind the direct methods of Calculus of Variations, this amounts roughly to the question of identifying weak maps or fields, and weak limits of sequences of weak maps with equibounded energies. As we shall see, the choice of the notions of weak maps and weak convergence is very relevant, and different choices often lead to different answers concerning equilibrium points.
The aim of this monograph is twofold: discussing a homological theory of weak maps, and in this context treating several typical and relevant variational problems. The basic idea in defining a weak notion of vector valued maps is to think of them not componentlywise but globally, i.e. as graphs. In other words we define
weak maps between two oriented and boundaryless Riemannian manifolds X and y similarly to distributions or Sobolev functions, using a standard duality approach, but testing with functions which live in the product space X x Y. Thus one is naturally forced to move from the context of Sobolev maps, for instance, to that of currents and to allow vertical parts, and one is naturally led
viii
Preface
to the basic notion in this monograph of Cartesian currents. One should think of Cartesian currents as weak limits in the sense of currents of graphs of smooth maps, although this is not always true. In particular Cartesian currents satisfy the homological condition of having zero boundary in the cylinder X x y, and they induce a homology map which is continuous for the weak convergence of currents. As the natural context for those notions is geometric measure theory, we first develop an elementary introduction to the theory of currents of Federer and Fleming to provide all needed information. In particular we prove the deformation and closure theorems of Federer and Fleming, that, as we shall see, play a relevant role not only to study parametric but also non parametric integrals. In the first part of our monograph, after a preliminary chapter about measure theory and the phenomenology of weak convergence, we discuss integer rectifiable and normal currents, differentiability properties of the graphs of maps, continuity of Jacobian minors, and how those notions are related in terms of approximate tangent planes, area or mass, and the homological notion of boundary. We also discuss related topics as, for instance, higher integrability of determinants, functions of bounded variations and degree theory, and of course closure, compactness and structure properties of special classes of Cartesian currents. Finally in Vol. I Ch. 5 we deal with the homology theory for currents. There we present classical topics as for instance Hodge theory, Poincare-Lefschetz and de Rham dualities and intersection numbers, and we conclude by discussing the homology map associated to a Cartesian current in terms of periods and cycles. In doing this we have tried to keep our treatment elementary, illustrating with simple examples the results, their meaning and their typical use, and we always give detailed proofs. Also, at the cost of some repetition we have tried to make each chapter, and sometimes even sections, readable as far as possible indipendently of the general context, so that parts of this monograph can be easily used separately for example for graduate courses. This we hope justifies the size of our monograph. Open questions are often mentioned and in the final section of each chapter we discuss references to the literature and sometimes supplementary results. In the second part of our monograph we deal with variational problems. In Vol. II Ch. 1 we discuss general variational problems, their parametric and non parametric formulations, and in connection with them, different notions of ellipticity (parametric ellipticity, polyconvexity, and quasiconvexity). The rest of the second part of this monograph is then dedicated to specific variational problems in the setting of Cartesian currents: in Vol. II Ch. 2 we deal with weak diffeomorphisms and non linear elasticity, in Vol. II Ch. 3, Vol. II Ch. 4, and Vol. II Ch. 5 we discuss some issues of the harmonic mapping problem and of related questions; and, finally, we shortly deal in Vol. II Ch. 6 with the non parametric area problem. For further information about the content of this monograph we refer the reader to the introductions to each chapter, to the detailed table of contents, and to the index.
Preface
ix
In preparing this monograph we have taken advantage from discussions with
many friends and colleagues. Among them it is a pleasure for us to thank G. Anzellotti, J. Ball, F. Bethuel, H. Brezis, F. Helein, S. Hildebrandt, J. Jost, H. Kuwert, F. Lazzeri, D. Mucci, J. Necas, K. Steffen, M. Struwe, V. Sverak and B. White. We thank also Mirta Stampella for her invaluable help in the typing and retyping of our manuscript. We would like to aknowledge supports during the last years from the Ministero dell'Universita e della Ricerea Scientifica, Consiglio Nazionale delle Ricerche, EC European Research Project GADGET I, II, III, and Czech Academy of Sciences. Parts of this work have been written while the authors were visiting different Universities. We want to extend our thanks to Alexander von Humboldt Foundation, Sonderforschungsbereich 256 of Bonn University, to the Departments of Mathematics of the Universities of Bonn, Cachan, Keio, Paris VI, Shizuoka, and to the Forschungsinstitut ETH, Zurich, to the Center for Mathematics and its Applications, Canberra and to the Academia Sinica, Taiwan. Firenze, October 1997
Mariano Giaquinta Giuseppe Modica
Jiri Soucek
Contents Cartesian Currents in the Calculus of Variations I and II
Volume I. Cartesian Currents 1
General Measure Theory ....................................
2
Integer Multiplicity Rectifiable Currents ..................... 69
3
Cartesian Maps .............................................. 175
4
Cartesian Currents in Euclidean Spaces ...................... 323 Cartesian Currents in Riemannian Manifolds ................ 493
5
1
Bibliography ..................................................... 667 Index ............................................................ 697
Symbols ......................................................... 709
Volume II. Variational Integrals 1
Regular Variational Integrals ................................
1
5
Finite Elasticity and Weak Diffeomorphisms ................. 137 The Dirichlet Integral in Sobolev Spaces ..................... 281 The Dirichlet Energy for Maps into S2 ...................... 353 Some Regular and Non Regular Variational Problems ....... 467
6
The Non Parametric Area Functional ........................ 563
2 3
4
Bibliography .....................................................
653 Index ............................................................ 683 Symbols ......................................................... 695
Contents Volume I. Cartesian Currents
1.
General Measure Theory .................................... 1
General Measure Theory ................................... 1.1 Measures and Integrals ...............................
1 1
1
(Q-measures and outer measures, measurable sets, Caratheodory mea-
sures. Measurable functions. Lebesgue's integral. Egoroff's, Beppo Levi's, Fatou's and Lebesgue's theorems. Product measures and Fubini-Tonelli theorem)
1.2
....................
10
Hausdorff Measures ..................................
12
Borel Regular and Radon Measures
(Borel functions and measures, Radon measures. Caratheodory's criterion in metric spaces. Vitali-Caratheodory theorem. Lusin's theorem)
1.3
(Hausdorff measures and Hausdorff dimension, spherical measures. Isodiametric inequality. Cantor sets and Cantor-Vitali functions. Caratheodory construction. Hausdorff measure of a product)
1.4
.....
23
Covering Theorems, Differentiation and Densities ........
29
Lebesgue's, Radon-Nikodym's and Riesz's Theorems
(Lebesgue's decomposition theorem, Radon-Nikodym differentiation theorem. Vector valued measures and Riesz representation theorem)
1.5
(Vitali's and Besicovitch's covering theorems. Symmetric differentiation and Radon-Nikodym theorem. Densities. Approximate limits and measurability. Densities and Hausdorff measure)
2
Weak Convergence ......................................... 36 2.1
Weak Convergence of Vector Valued Measures .......... (Definitions. Banach-Steinhaus theorem and the compactness theorem. Convergence as measures and L1-weak convergence. Lebesgue's theorem
36
about weak convergence in L1)
2.2
Typical Behaviours of Weakly Converging Sequences ..... 40
(Oscillation, concentration. distribution, and concentration-distribution. Nonlinearity destroys the weak convergence) 2.3
Weak Convergence in L4, q > 1 ....................... 44
2.4
spaces. Riemann-Lebesgue's lemma. Radon-Riesz theorem and variants) Weak Convergence in L' ............................. 50 (The proof of Lebesgue theorem on weak convergence in L'. Weak con-
(Weak convergence in L' and weak and weak" convergence on Banach
vergence of the product)
2.5
2.6
2.7 3
Concentration: Weak Convergence of Measures ..........
(The universal character of the concentration-distribution phenomenon. The concentration-compactness lemma)
55
Oscillations: Young Measures ......................... 58 More on Weak Convergence in Ll ..................... 64 (Equivalent definitions of Young measures. Examples)
(Convergence in the sense of biting and convergence of the absolutely continuous parts) Notes .................................................... 67
Contents Volume I. Cartesian Currents
Integer Multiplicity Rectifiable Currents ..................... 69 Area and Coarea. Countably n-Rectifiable Sets ................ 69 Area and Coarea Formulas for Linear Maps ............. 69 1.1
1
(Polar decomposition theorem. Area and change of variable formula for linear maps. Coarea formula for linear maps. Cauchy-Binet formula)
1.2
Area Formula for Lipschitz Maps ...................... 74
(Area formula for smooth and for Lipschitz maps: curves, graphs of codimension one, parametric hypersurfaces, submanifolds, and graphs of higher codimension)
1.3
Coarea Formula for Lipschitz Maps ....................
82
(Coarea formula for smooth and for Lipschitz maps. C'-Sard type theorem)
1.4
1.5 2
Rectifiable Sets and the Structure Theorem ............. 90
(Countably n-rectifiable and n-rectifiable sets. The approximate tangent space of sets and measures. The rectifiability theorem for Radon measures. Besicovitch-Federer structure theorem)
The General Area and Coarea Formulas ................ 99
(Area and coarea formulas for sets on manifolds and for rectifiable sets. The divergence theorem)
Currents ................................................. 103 2.1
Multivectors and Covectors ........................... 104
(k-vectors, exterior product of multivectors, simple k-vectors. Duality between k-vectors and covectors. Inner product of multivectors. Sim-
ple k-vectors and oriented k-planes. Simple n-vectors in the Cartesian product 1R" x 1RN. Characterization of simple n-vectors in IR" x RN. Induced linear transformations)
2.2
Differential Forms ..
... .......
................. 118
(Exterior differentiation, pullback, forms in a Cartesian product. Integration of differential forms)
2.3
Currents: Basic Facts ................................ 122
(Currents and weak convergence of currents. Boundary and support of currents. Product of currents. Currents with finite mass, currents which are representable by integration. Lower semicontinuity of the mass. Compactness-closure theorem for currents with locally finite mass. ndimensional currents in R" and BV functions. The constancy theorem. Examples. Image of a current under a Lipschitz map. The homotopy formula)
2.4
Integer Multiplicity Rectifiable Currents ................ 136
2.5
Slicing ............................................. 151 (Slices of codimension one. Slices of codimension larger than one)
2.6
(Currents carried by smooth graphs. Rectifiable and integer multiplicity rectifiable currents. The closure theorem for integer multiplicity 0dimensional and 1-dimensional rectifiable currents. Examples. Image of a rectifiable current under a Lipschitz map. The Cartesian product of rectifiable currents)
The Deformation Theorem and Approximations ......... 157
(The deformation theorem. Isoperimetric inequality. Weak and strong polyhedral approximations. The strong approximation theorem for normal currents)
2.7 3
The Closure Theorem
................................ 161
(The classical proof: slicing lemma, the boundary rectifiability theorem, the rectifiability theorem. White's proof) Notes .................................................... 173
Cartesian Maps .............................................. 175 1
Differentiability of Non Smooth Functions
.................... 179
Contents Volume I. Cartesian Currents 1.1
xv
The Maximal Function and Lebesgue's Differentiation Theorem
........................................... 180
(The distribution function. Hardy-Littlewood maximal function and inequality. Lebesgue's differentiation theorem. Lebesgue's points. The Hausdorff dimension of the set of non Lebesgue points. Calderon-Zygmund decomposition argument. The class L log L)
1.2
Differentiability Properties of W 1,P Functions
........... 192
(Differentiability in the LP sense. Calderon-Zygmund theorem. MorreySobolev theorem)
1.3
Lusin Type Properties of W''P Functions ............... 202
(Kirszbraun and Rademacher theorems. Lusin type theorems for Sobolev functions. Whitney extension theorem. Liu's theorem)
1.4 1.5
Approximate Differential and Lusin Type Properties ..... 210
(Approximate continuity and approximate differentiability. Lusin type properties are equivalent to the approximate differentiability)
Area Formulas, Degree, and Graphs of Non Smooth Maps 218
(Area formula, graphs and degree for non smooth maps. Rado'-Reichelderfer theorem) 2
Maps with Jacobian Minors in L' ........................... 228 2.1 2.2
2.3 2.4
The Class AI (.(l, RN) , Graphs and Boundaries .......... 229 (The class A'(0, 1RN ). The current integration over the graph. Convergence of graphs and minors) Examples .......................................... 233 (The map x/Ixl. Homogeneous maps: boundary and degree)
Boundaries and Integration by Parts ................... 238
(Approximate differential and distributional derivatives. Analytic formulas for boundaries. Boundary and pull-back )
More on the Jacobian Determinant .................... 247
(Maps in Wl,"-1 The distributional determinant. Isoperimetric in-
equality for the determinant. The class A,,,,, and Higher integrability of the determinant. BMO and Hardy space 1 (]R"))
2.5
Boundaries and Traces ............................... 265 (The boundary of a current integration over a graph and the trace in
3
the sense of Sobolev. Weak and strong anchorage) Cartesian Maps ........................................... 276
3.1
Weak Continuity of Minors ........................... 277 (Weak convergence of minors in L1, as measures, and convergence of
graphs. Examples Reshetnyak's theorem)
3.2
The Class cart' (.fl, RN): Closure and Compactness ...... 285
(The class of Cartesian maps. The closure theorem. A compactness theorem)
3.3
The Classes cartP(f?, R'`'), p > 1 ....................... 293 (The class cart P (0,1RN) closure and compactness theorems.)
4
Approximability of Cartesian Maps .......................... 296 The Transfinite Inductive Process ..................... 299 4.1 (Ordinal numbers. The transfinite inductive process and the weak sequential closure of a set)
4.2
Weak and Strong Approximation of Minors ............. 303
(Sequential weak closure and strong closure of smooth maps in the class of Cartesian maps. Cart1(D,RN) = CART' (f2, RN) C cart' (S2, RN))
4.3 5
The Join of Cartesian Maps .......................... 313 (Composition and join of Cartesian maps. Weak continuity of the join)
Notes .................................................... 318
Contents Volume I. Cartesian Currents
xvi
4.
Cartesian Currents in Euclidean Spaces ...................... 323 Functions of Bounded Variation ............................. 327 1 1.1
1.2
The Space BV (Q, R) ................................ 329
(The total variation. Semicontinuity. Approximation by smooth functions A variational characterization of By. Sobolev and Poincare inequalities. Compactness. Fleming-Rishel formula)
Caccioppoli Sets .................................... 340 (Sets of finite perimeter. Isoperimetric inequality. Approximation by
smooth open sets)
1.3
1.4
1.5
De Giorgi's Rectifiability Theorem ..................... 346
(Reduced boundary. De Giorgi's rectifiability theorem. Measure theoretic boundary. Federer's characterization of Caccioppoli sets)
The Structure Theorem for BV Functions .............. 354
(Jump points and jump sets. Regular and singular points. The structure of the measure total variation. Lebesgue's points and approximate differentiability of BV-functions)
Subgraphs of BV Functions .......................... 371
(Characterization of BV functions in terms of their subgraphs; jump and Cantor part of Du in terms of the reduced boundary of the subgraph of u)
2
Cartesian Currents in Euclidean Spaces ...................... 379 Limit Currents of Smooth Graphs ..................... 380 2.1 (Toward the definition of Cartesian currents)
2.2 2.3
The Classes cart (,fl x IRN) and graph(.f2 x ISBN) .......... 384
(Cartesian currents and graph-currents)
The Structure Theorem .............................. 391
(The structure theorem. The map associated to a Cartesian current. Weak convergence of Cartesian currents)
2.4 2.5
2.6
Cartesian Currents in Codimension One ................ 403
(BV functions as Cartesian currents and representation formulas. Cantor-Vitali functions. SBV functions)
Examples of Cartesian Currents ....................... 411
(Bubbling off of circles, spheres, tori. Attaching cylinders. Examples of vector-valued BV functions. A Cartesian current with a Cantor mass on minors. A Cartesian current which cannot be approximated weakly by smooth graphs)
Radial Currents ..................................... 439
(Currents associated to radial maps u(x) = U(IxI)x/Ixi. A closure theorem)
3
Degree Theory ............................................ 450 3.1
n-Dimensional Currents and BV Functions ............. 451 (Representation formula and decomposition theorem for n-dimensional
currents in F. Constancy theorem and linear projections of normal currents)
4
3.2
Degree Mapping and Degree of Cartesian Currents ...... 460
3.3
The Degree of Continuous Maps ....................... 471
3.4
h-Connected Components and the Degree .............. 474
(Definition and properties of the degree for Cartesian currents and maps)
(The degree of continuous Cartesian maps agrees with the classical degree for continuous maps)
(Homologically connected components of a Caccioppoli set) Notes .................................................... 479
Contents Volume I. Cartesian Currents
5.
xvii
Cartesian Currents in Riemannian Manifolds ................ 493 1
More About Currents ...................................... 494
1.1
The Deformation Theorem ........................... 494
(Proof of Federer and Fleming deformation theorem and of the strong approximation theorem)
1.2 1.3
Mollifying Currents .................................. 505
(e-mollified of forms and currents. A representation formula for normal currents)
Flat Chains .........................................512
(Integral flat and flat chains and norms. Image of a flat chain. Federer's flatness theorem. Mollification and a representation formula for flat chains. Federer's support theorem. Cochains) 2
Differential Forms and Cohomology .......................... 527 2.1
Forms on Manifolds .................................. 528
(Tangent and cotangent bundle. Null forms to a submanifold)
2.2 2.3
Hodge Operator ........... ........ ................531
(Interior multiplication of vectors and covectors. Hodge operator. The L2 inner product for forms)
Sobolev Spaces of Forms ............................. 536 (The classes Lr2,(X) and WP'2(X))
2.4 2.5
Harmonic Forms .................................... 538
(The codifferential S. Laplace-Beltrami operator on forms and the Dirichlet integral. Their expressions in local coordinates)
Hodge and Hodge-Kodaira-Morrey Theorems ........... 543
(Gaffney Lemma. Hodge-Kodaira-Morrey decomposition theorem. De Rham cohomology groups. Hodge representation theorem for cohomology classes)
2.6
2.7
2.8
3
Relative Cohomology: Hodge-Morrey Decomposition ..... 549
(Collar theorem. Tangential and normal part of a form. Coboundary operator. The lemma of Gaffney at the boundary. Hodge-Morrey decomposition. Hodge representation theorem of relative cohomology classes)
Weitzenbock Formula ................................ 559
(Connections and covariant derivatives. Levi-Civita connection. Second covariant derivatives. Curvature tensor and Laplace-Beltrami operator on forms)
Poincare and Poincare-Lefschetz Dualities in Cohomology 565
(De Rham cohomology groups Poincare duality. Relative cohomology groups on manifolds with boundary. Cohomology long sequence. Poincare Lefschetz duality)
Currents and Real Homology of Compact Manifolds ........... 570 3.1 3.2
Currents on Manifolds ............................... 572
(Currents in X. Flatness and constancy theorems)
Poincare and de Rham Dualities ...................... 574
(Isomorphism of (n - k) cohomology groups and k homology groups. Integration along fibers Poincare dual form. de Rham duality between cohomology and homology. Periods. Normal currents and classical real homology)
3.3
Poincare-Lefschetz and de Rham Dualities .............. 589 (Relative homology. Homology long sequence. Poincare-Lefschetz duality
theorem. De Rham theorem for manifolds with boundary. Relative real homology classes are represented by minimal cycles)
3.4
Intersection of Currents and Kronecker Index ........... 599
(Intersection of normal currents in R' and on submanifolds of R'. Intersection of cycles is the wedge product of Poincare duals. Kronecker index. Intersection index)
Contents Volume I. Cartesian Currents
xviii
3.5
4
Relative Homology and Cohomology Groups ............ 608
(Homology and cohomology in the Lipschitz category. Closure of cosets. Generalized de Rham theorem)
Integral Homology ......................................... 615 4.1
Integral Homology Groups ............................ 615
(Integral homology groups. Integral relative homology groups. Isoperimetric inequalities and weak closure. Torsion groups. Integral and real homology)
4.2
5
Intersection in Integral Homology ..................... 624
(Intersection of cycles on boundaryless manifolds. Intersection of cycles on manifolds with boundary. Intersection index in integral homology. An algebraic view of integral homology)
Maps Between Manifolds ................................... 631 5.1 5.2
Sobolev Classes of Maps Between Riemannian Manifolds . 632
(Density results of Schoen-Uhlenbeck and Bethuel. d-homotopy White's results)
Cartesian Currents Between Manifolds ................. 640
(Approximate differentiability. Area formula. Graphs. Cartesian currents. The class carta,l(!2 x y))
5.3 5.4 6
Homology Induced Maps: Manifolds Without Boundary .. 648
(Homology and cohomology maps associated to a Cartesian current)
Homology Induced Maps: Manifolds with Boundary...... 658
(Homology and cohomology maps associated to flat chains)
Notes .................................................... 663
Bibliography ..................................................... 667 Index ............................................................ 697
Symbols ......................................................... 709
Contents Volume II. Variational Integrals
1.
Regular Variational Integrals ................................ 1 The Direct Methods .......................................
1.1
2
The Abstract Setting ................................
3
Some Classical Lower Semicontinuity Theorems .........
10
(Lower semicontinuous and coercive functionals. Weierstrass theorem. r-relaxed functional)
1.2
1
(Lower semicontinuity with respect to the weak convergence in L1. The
role of Banach-Saks's theorem and Jensen's inequality. Regular and smooth integrands in the Calculus of Variations)
1.3
A General Semicontinuity Theorem ....................
19
(Lower semicontinuity with respect to the weak' convergence in L1) 2
Polyconvex Envelops and Regular Parametric Integrals ......... 23 2.1
2.2 2.3
Polyconvexity and Polyconvex Envelops ................ 26
(A few facts from convex analysis. n-vectors associated to the tangent planes to graphs. Polyconvex functions and polyconvex envelopes)
Parametric Polyconvex Envelops of Integrands .......... 37
(The parametric polyconvex l.s.c. envelop of an integrand. Mass and comass with respect to the integrand f)
The Parametric Extension of Regular Integrals .......... 44 (The parametric extension of an integral as integral of the parametric polyconvex l.s.c. envelop)
2.4 3
The Polyconvex l.s.c. Extension of Some Lagrangians .... 45
(Area of graphs. The total variation of the gradient. The Dirichlet integral. The p-energy functional. The liquid crystal integrand)
Regular Integrals in the Class of Cartesian Currents ........... 74 Parametric Integrands and Lower Semicontinuity ........ 75 3.1 (Parametric integrands and lower semicontinuity of parametric integrals)
3.2
Existence of Minimizers in Classes of Cartesian Currents . 82
(Lower semicontinuity of the parametric extension of regular integrals. Existence of minimizers in subclasses of Cartesian currents)
3.3
Relaxed Energies in the Setting of Cartesian Currents .... 88
3.4
Relaxed Energies in the Parametric Case ...............
(The relaxed functional in classes of Cartesian currents and maps)
(The approximation problem for parametric integrals. A theorem of Reshetnyak. Flat integrands and Federer's approximation theorem. In-
90
teger multiplicity rectifiable and real minimizing currents)
4
Regular Integrals and Quasiconvexity ........................ 106 4.1 4.2
Quasiconvexity
...................................... 107
(Quasiconvexity as necessary condition for semicontinuity. Rank-one convexity and Legendre-Hadamard condition)
Quasiconvexity and Lower Semicontinuity
.............. 117
(Quasiconvexity as sufficient condition for semicontinuity in classes of Sobolev maps or Cartesian currents)
Contents Volume II. Variational Integrals
roc
4.3 5
2.
Ellipticity and Quasiconvexity ........................ 127 (Ellipticity, quasiconvexity and lower semicontinuity)
Notes .................................................... 131
Finite Elasticity and Weak Diffeomorphisms ................. 137 1
State Space and Stored Energies in Elasticity ................. 139 1.1
Fields and Transformations ........................... 139
1.2
Kinematics ......................................... 140 (Bodies, states, and deformations of a body. Deformations as 3-surfaces
(Non local and non linear structure of transformations) in R6, or as graphs of diffeomorphisms)
1.3
Local Deformations .................................. 143
1.4
Perfectly Elastic Bodies: Stored Energy, Convexity and
(Infinitesimal deformations as simple tangent vectors to the deformation surface)
Coercivity .......................................... 147
(Stored energy: different forms and constitutive conditions. Polyconvexity and coercivity)
1.5
Variations and Stress ................................ 152 (Infinitesimal variations and the notion of stress. Piola-Kirchhoff and
Cauchy stress tensors. Energy-momentum tensor.) 2
Physical Implications on Kinematics and Stored Energies ....... 155 Kinematical Principles in Elasticity: Weak Deformations . 156 2.1 (Material body and its parts. Impenetrability of matter. Weakly invertible maps. Weak one-to-one transformations. Existence of local deformations. Weak deformations. Elastic bodies and absence of fractures. Elastic deformations)
2.2
Frame Indifference and Isotropy ....................... 169 (Frame indifference principle. Energies associated to isotropic materials)
2.3
2.4 2.5 3
Convexity-like Conditions ............................ 170
(Convexity is not compatible with elasticity. Noll's condition. Polyconvexity and Diff-quasiconvexity)
Coercivity Conditions ................................ 175
(A discussion of the coercivity conditions)
Examples of Stored Energies .......................... 179 (Ogden-type stored energies for isotropic materials)
Weak Diffeomorphisms .....:; .............................. 182 3.1
3.2
3.3 3.4 3.5
3.6
The Classes dif p'q (,f2, ,fl) ............................. 183
(The class of (p, q)-weak diffeomorphisms. Weak convergence. Closure' and compactness properties. An example of a discontinuous weak diffeomorphism)
The Classes dif
p 4 ,
(0, Rn) ............................ 191
(Weak diffeomorphisms with non-prescribed range. Closure and compactness properties. Elastic deformations as weak diffeomorphisms)
Convergence Theorems for the Inverse Maps ............ 199 (Convergence of the ranges and of the inverse maps)
General Weak Diffeomorphisms ....................... 204
(Weak diffeomorphisms with vertical and horizontal parts: structure, closure, and compactness theorems)
The Dif-classes ...................................... 213
(The approximation problem for weak diffeomorphisms)
Volume Preserving Diffeomorphisms ................... 214 (The Jacobian determinant of weak one-to-one maps and of weak deformations)
4
Connectivity Properties of the Range of Weak Diffeomorphisms . 216
Contents Volume II. Variational Integrals 4.1
Connectivity of the Range of Sobolev Maps
(Connected sets and d,_1-connected sets.
............. 217
"mapped" into connected sets by Sobolev maps)
4.2
xxi
sets are
Connectivity of the Range of Weak Diffeomorphisms
..... 219
(Weak diffeomorphisms "map" connected sets into essentially connected sets. Weak diffeomorphisms do not produce cavitation. Examples)
4.3
Regularity Properties of Locally Weak Invertible Maps ... 229
(Weak local diffeomorphisms. Local properties of weak local diffeomorphisms. Vodopianov-Goldstein's theorem. Courant-Lebesgue lemma)
4.4
Global Invertibility of Weak Maps ..................... 238
(Conditions ensuring that weak local diffeomorphisms be one-to-one and homeomorphisms)
4.5 5
(A.e. open sets and a.e. continuous maps. Weak diffeomorphisms in W1' ', p > n - 1, "are" a.e. open maps)
Composition .............................................. 247 5.1
6
An a.e. Open Map Theorem .......................... 243 Composition of weak deformations ..................... 248
(Composition of one-to-one maps and of weak diffeomorphisms)
5.2
On the Summability of Compositions .................. 250
5.3
Composition of Weak Diffeomorphisms ................. 254
(Binet's formula and the summability of the composition)
(The action of weak diffeomorphisms on Cartesian maps and the pseudogroup structure of weak diffeomorphisms. Weak convergence of compositions)
Existence of Equilibrium Configurations ...................... 260 6.1
Existence Theorems ................................. 261
(The displacement pressure problem. Deformations with fractures)
6.2 6.3 7
3.
Equilibrium and Conservation Equations ............... 264
(Energy-momentum conservation law and Cauchy's equilibrium equation)
The Cavitation Problem .............................. 268
(Elastic deformations do not cavitate)
Notes .................................................... 278
The Dirichlet Integral in Sobolev Spaces ..................... 281 1
Harmonic Maps Between Manifolds .......................... 281 1.1 First Variation and Inner Variations ................... 283 (Euler variations and Euler-Lagrange equation of the energy integral. Inner variations, energy-momentum tensor, inner and strong extremals. Conformality relations. Stationary points. Parametric minimal surfaces)
1.2
Finding Harmonic Maps by Variational Methods ........ 293
(Existence and the regularity problem. Mappings from B" into the upper hemisphere of S". Mappings from B" into S"-1)
2
Energy Minimizing Weak Harmonic Maps: Regularity Theory ... 296 2.1 Some Preliminaries. Reverse Holder Inequalities ......... 297 (Some algebraic lemmas. The Dirichlet growth theorem of Morrey. Reverse Holder inequalities with increasing supports)
2.2
Classical Regularity Results .......................... 303 An Optimal Regularity Theorem ...................... 307 (Morrey's regularity theorem for 2-dimensional weak harmonic maps)
2.3
2.4
(A partial regularity theorem and the existence and regularity of energy minimizing harmonic maps with range in a regular ball: results by Hildebrandt, Kaul, Widman, and Giaquinta, Giusti.)
The Partial Regularity Theorem ....................... 319
(The partial regularity theorem for energy minimizing weak harmonic maps: Schoen-Uhlenbeck result)
Contents Volume II. Variational Integrals
)odi
3
Harmonic Maps in Homotopy Classes ......................... 333 The Action of Wl'2-maps on Loops .................... 334 3.1 (Courant-Lebesgue lemma. In the two-dimensional case the action on loops is well defined for maps in W122)
3.2 3.3 4
4.2
4.
(Energy minimizing maps with prescribed action on loops. Schoen-Yau, Saks-Uhlenbeck, Lemaire, Eells-Sampson and Hamilton theorems)
Local Replacement by Harmonic Mappings: Bubbling .... 337 (Jost's replacement method and existence of minimal immersions of S2)
Weak and Stationary Harmonic Maps with Values into S2 ...... 339 4.1
5
Minimizing Energy with Homotopic Constraints ......... 336
The Partial Regularity Theory ........................ 339 (An alternative proof. More on the singular set) Stationary Harmonic Maps ........................... 345 (Partial regularity results for stationary harmonic maps)
Notes .................................................... 350
The Dirichlet Energy for Maps into S2 ...................... 353 1
Variational Problems for Maps from a Domain of R2 into S2 .... 354 1.1
Harmonic Maps with Prescribed Degree ................ 354
(Homotopic equivalent maps and degree. Bubbling off of spheres. The stereographic and the modified stereographic projection. e-conformal maps)
1.2
The Structure Theorem in cart2'1(Q x S2), .f2 C JR2 ...... 362 (The structure and approximation theorems in cart2.1(1? X S2))
1.3
2
Existence and Regularity of Minimizers ................ 366
(The relaxed energy and existence of minimizers. Energy minimizing maps with constant boundary value: Lemaire's theorem. The simplest chiral model and instantons. Large solution for harmonic maps: BrezisCoron and Jost result. A global regularity result)
Variational Problems from a Domain of JR3 into S2 ............ 383 2.1
The Class cart2'1(,f? X S2), ,f2 C R3 .................... 385 (The D-field and homological singularities)
2.2
Density Results in W1'2(B3, S2) ....................... 392 (Approximation by maps which are smooth except at a discrete set of points)
2.3
2.4
Dipoles and Gap Phenomenon ........................ 400 The Structure Theorem in cart2'1(.f2 x S2), Q C J 3 ...... 409
(Dipoles and the approximate dipoles. Lavrentiev or gap phenomenon)
(Structure of the vertical part of Cartesian currents in cart2'1(S2 x S2))
2.5
Approximation by Smooth Graphs: Dirichlet Data ....... 412
(Weak approximation in energy by smooth graphs. The minimal connection and its continuity properties with respect to the W1,2-weak convergence. Cart2,1(1 X S2) = cart2 1(12 x S2). Weak approximability by smooth maps in W,P'2(S2 X S2))
2.6
Approximation by Smooth Graphs: No Boundary Data... 419 (T belongs to carte"1(S2 x S2) if and only if it can be approximated weakly and in energy by smooth graphs Guk possibly with uk = UT on
an) 2.7
The Dirichlet Integral in cart2'1(S? X S2), 0 C JR3 ....... 423 (The parametric polyconvex extension of the Dirichlet integral is its relaxed or Lebesgue's extension. The relaxed of V(u, f2) in W 1,2 (D, S2))
2.8
Minimizers of Variational Problems .................... 429 (Variational problems and existence of minimizers)
2.9
Contents Volume II. Variational Integrals
xxiii
A Partial Regularity Result ...........................
433
(The absolutely continuous part uT of minimizers T is regular except on a closed set whose Hausdorff dimension is not greater than 1. Tangent cones)
2.10
The General Dipole Problem .......................... 449
(The coarea formula and the minimum energy of dipoles)
2.11
Singular Perturbations
............................... 452
(Trying to solve Dirichlet problem by approximating by singularly perturbed functionals of the type of Ginzburg-Landau) 3
5.
Notes .................................................... 458
Some Regular and Non Regular Variational Problems ....... 467 1
The Liquid Crystal Energy ................................. 467 1.1 The Sobolev Space Approach ......................... 470
(Existence and regularity of equilibrium configuration)
1.2
1.3
The Relaxed Energy ................................. 470
(Existence of equilibrium configurations for the relaxed energy. The dipole problem. Relaxed energies in Sobolev spaces and Cartesian currents. Equilibrium configurations with fractures)
The Dipole Problem ................................. 477
(Approximation in energy: irrotational and solenoidal dipoles. The general dipole problem) 2
The Dirichlet Integral in the Regular Case: Maps into S2 ....... 485 2.1
Maps with Values in S2 .............................. 485
(Maps from a n-dimensional space into S2 and the class carte' 1(S2 x S2).
The (n - 2)-D-field) 2.2
The Dipole Problem ................................. 489 (Degree with respect to a (n - 3)-curve and the dipole problem)
2.3
The Structure Theorem .............................. 494
(Structure theorem for currents in cart2'1(0 x S2S2 C LQ") 3
The Dirichlet Integral in the Regular Case: Maps into a Manifold 496 3.1
The Class cart2" (.(2 x y) ............................. 497
(The structure theorem for currents in cart2'1(D x y), 0 C E2)
3.2
Spherical Vertical Parts and a Closure Theorem ......... 501
(Reduced Cartesian currents. Closure theorem. Vertical parts of currents
in Cart2.1(!2 x y) are of the type S2)
3.3
4
The Dirichlet Integral and Minimizers .................. 506
The Dirichlet Integral in the Non Regular Case: a Homological Theory ................................................... 508
4.1
(n,p)-Currents ...................................... 509 ((n, p)-graphs and the classes A(P),'. rectifiable (r, p)-currents, (r, p)mass, (r, p)-boundary. Vertical (r, p)-currents and cohomology. Integer multiplicity rectifiable vector-valued currents)
4.2
Graphs of Sobolev Maps ............................. 516 (Singularities of Sobolev maps and the currents P(u; a) and D(u; a). The class me - W1'p (S2, y))
4.3
p-Dirichlet Graphs and Cartesian Currents ............. 525 (The classes Vp-graph(S2 x y), red-Dp-graph(S2 x y), and carte" '(Ox y): closure theorems)
4.4 4.5 5
The Dirichlet Integral ................................ 534
(Representation. Minimizers and homological minimizers of the Dirichlet integral)
Prescribing Homological Singularities .................. 543
(s-degree. Lower bounds for the dipole energy) Notes .................................................... 546
Contents Volume II. Variational Integrals
xxiv
6.
The Non Parametric Area Functional ........................ 563 Area Minimizing Hypersurfaces ............................. 564 1 1.1
1.2
2
Non Parametric Minimal Surfaces of Codimension One ... 579
(Solvability of the Dirichlet problem. Bombieri-De Giorgi-Miranda a priori estimate. The variational approach. Removable singularities. Liouville type theorems. Bernstein theorem. Bombieri-De Giorgi-Giusti theorem on minimal cones)
Problems for Maps of Bounded Variation with Values in Sl ..... 590 2.1
2.2 2.3
3
Parametric Surfaces of Least Area ..................... 564
(Hypersurfaces as Caccioppoli's boundaries: De Giorgi,s regularity theorem, monotonicity, Federer's regularity theorem. Surfaces as rectifiable currents: Almgren's regularity theorem. Minimal surfaces as stationary varifolds: a survey of Allard theory. Boundary regularity: Allard's and Hardt Simon's results)
Preliminaries ....................................... 594
(Forms and currents in .fl x S1. BV(fl,E): a survey of results)
The Class cart(,fl x Sl) .............................. 600
(The structure and the approximation theorems)
Relaxed Energies and Existence of Minimizers .......... 610 (The area integral for maps into S1: the relaxed area. Minimizers in cart(Q x Sl). Dipole-type problems)
Two Dimensional Minimal Surfaces .......................... 619 3.1
3.2
Plateau's Problem ................................... 619
(Morrey's e-conformality theorem. Douglas-Rado existence theorem. Hildebrandt's boundary regularity theorem. Branch points and embedded minimal surfaces: Fleming, Meeks-Yau, and Chang results)
Existence of Two Dimensional Non Parametric Minimal
Surfaces ............................................ 625 (Rado's theorem and existence, uniqueness, and regularity of two-dimensional graphs of any codimension)
3.3
4
The Minimal Surface System ......................... 627
(Stationary graphs are not necessarily area minimizing. Existence and non existence of stationary Lipschitz graphs. Isolated singularities are not removable in high codimension. A Bernstein type result of Hildebrandt, Jost, and Widman)
Least Area Mappings and Least Mass Currents ................ 632 4.1 4.2
Topological Results .................................. 633
(Representation and homology of Lipschitz chains) Main Results .......................................635
(Least area mapping u : B" - W' and least mass currents "agree" if n > 3. If n > 3 the homotopy least area problem reduces to the
homology problem) 5
The Non-parametric Area Integral ........................... 639 5.1
5.2
The Mass of Cartesian Currents and the Relaxed Area ... 641
(Graphs of finite mass which cannot be approximated in area by smooth graphs)
Lebesgue's Area ..................................... 649
(The mass of 2-dimensional continuous Cartesian maps is Lebesgue's area of their graphs) 6
Notes .................................................... 651
Bibliography ..................................................... 653
Index ............................................................ 683 Symbols ......................................................... 695
1. General Measure Theory
This chapter deals with general measure theory. In Sec. 1.1 we collect definitions
and results from general measure theory that will be freely used later on; in Sec. 1.2, due to its relevance for the sequel, we shall discuss in more details weak convergence of functions and of measures and we shall illustrate some of its main features. In Ch. 2 we shall then develop some of the basic theory of n-rectifiable sets and integer multiplicity rectifiable currents. Of course we do not aim to completeness, for instance Sec. 1.1 of this chapter contains no proof; and in principle, we supply proofs essentially when claims or
their proofs are especially relevant for the sequel; sometimes, proofs are postponed to later chapters. Our goal in these first two chapters is to state precisely results and notations (though we shall usually adopt standard notations) and to illustrate them mainly by examples. In some sense, the first two chapters may be regarded as a simple, and in some regard, rough introduction to the elementary part of geometric measure theory, the right context in which the content of the following chapters lives.
At first lecture the reader can start from Ch. 3 and use Ch. 1 and Ch. 2 as reference chapters.
1 General Measure Theory In this section we collect some basic definitions and results from general measure
theory
1.1 Measures and Integrals Let X be a set and let 2X denote the collection of all subsets of X.
Definition 1. A collection. of subsets of X, F C 2X, is said to be a o--algebra in X if F has the following three properties
(i) X E F,
(ii) IfE,FEF, then E\FE.F,
1. General Measure Theory
2
(iii) If {Ek} C F, k = 1, 2, ... , then U' 1Ek E Y.
Definition 2. Let.F be a a-algebra in X and let µ be a function defined on .F, whose range is [0, +oo],
µ :.F , [0, +oo]. We say that µ, or better (µ, F), is a a measure on X if µ is countably additive on F, that is, if {Ek} is a disjoint countable collection of members of F, then 00
iµ(Ek)
µ U Ek)
(1)
k=1
k=1
If (µ,.F) is a a-measure on X, we evidently have (i) (ii)
µ(O) = 0,
p is in particular finitely additive, i.e., if El, ... , E,, is a finite collection of disjoint sets in.F, then µ(E1 U ... U E,) = 4(E1) + ... + p(E.."), (iii) if E1, E2 E.F, E1 D E2, then µ(E1 \ E2) = µ(E1) - µ(E2) > 0, hence µ is monotone µ(E1) D µ(E2). Given a a-measure µ on X, we may define the measure of any subset of X by trying to measure it in the best possible way by means of the elements of .F, i.e., by setting 00
00
µ* (E) := inf { E µ(Fk)
(2)
I
Fk E .F , U Fk D E}
k=1
k=1
and, if there is no countable collection {Fk } in F covering E, by setting µ* (E) _ +oo. This way we define a new set function µ*
:
2X -> [0, +oo]
which satisfies, as it is not difficult to see, the following properties (i)
(0) = 0,
µ* is monotone, i.e., if A C B then µ*(A) < µ*(B), (iii) p* is countably sub-additive, i.e., if {Ek} is a countable collection of subsets of X, then (n)
(3)
(iv) (v)
µ( UEk <_00p(Ek), k=1 k=1
µ* is an extension of µ, i.e., p* (E) = p(E) whenever E E F, µ* can also be computed as (4)
p* (E) = inf{µ(F)
F E F F D E} ,
Notice that in general µ* is not countably additive.
.
1.1 Measures and Integrals
3
Definition 3. A set function A : 2X --3 [0, +oo] is said to be an outer measure, or simply a measure on X if A(O) = 0 and A is monotone and countably subadditive.
Therefore we can say that every o--measure (µ, F) can be extended by (2), or by (4), to a measure u* on X. In a sense to be made precise, also the converse is true. To see this, we introduce, following Caratheodory, the important notion of measurable sets. Very roughly they are those sets which divide well every subset of X, more precisely
Definition 4 (Caratheodory). Let A be a measure on X. A subset E of X is called A-measurable if for every subset F of X we have
A(F) = A(F-E)+A(FnE)
(5)
The collection of all A-measurable subsets of X is denoted by MA,
MA :_ {E C X I E is A-measurable} Observe that, for any set E and any set F, we have
A(F) < A(F - E) + A(E n F);
also X and 0 are measurable and every A-null set E, i.e., every set of zero measure, A(E) = 0, is measurable. Actually we have
Theorem 1. Let A be a measure on X. Then Ma is a o--algebra; moreover (AIMS, Ma) is a a-measure. In particular A is countably additive on MA. As an immediate consequence of this we get
Proposition 1. Let p be a measure on X and let {Ek} be a sequence of measurable sets. (i) If El C E2 C E3 C ..., then µ(Ek) T u (Uk iEk) (ii) If El D E2 D E3 D ... and µ(E1) < oo, then lc(Ek)
µ (nk 1Ek). The process (2) of generating an outer measure from a o--measure is essentially due to Caratheodory and it is known as Caratheodory's method; for this .
reason outer measures are often called Caratheodory's measures. Such a method in fact works starting from any set function.
Let g be any sub-family of 2X, which contains the empty set, and let a -p [0, +oo] be any set function with a(0) = 0. Define 00
a* (E) := infa(Gk) 0" I Gk E k=1
,
U Gk D E} k=1
if there exists {Gk} c 9 such that UGk D E, or a* (E) :_ +oo otherwise. Then it is not difficult to verify that a* is a measure on X; but, this time, in general we have 9 0 Ma., a*(G) < a(G) and (4) does not hold in general.
4
1. General Measure Theory
Definition 5. A measure A on X is said to be c-regular, where 9 is a subcollection of 2X, if for every E C X we have
A(E) = inf{A(G) I E C G, G E 9}
.
If 9 = M), we say A regular instead of Ma-regular.
As we have seen, if a* is the extension of the v-measure (p, F), then µ* is ,F-regular and even regular, but in general a measure on X need not be regular. However to any measure p we can associate a regular measure j) defined by
µ(E) := inf{µ(F)l In this case one has (t)
If A E Mv, then A E Mµ; and µ(A) = µ(A),
(ii) if A E Mr, and A (A) < +oo, then A E M.
Regular measures are very important; later we shall return on that concept. For the moment we just mention the following:
Proposition 2. If A is a regular measure, then (i) and (ii) of Proposition 1 hold for all sequences {Ek} not necessarily in Ma. Moreover the following measurability criterion holds: if A U B is measurable and A(A U B) = A(A) + A(B), then A and B are A-measurable
Finally, we observe that, if 9 is a v-algebra, then 9-regularity of A. is equivalent to the existence, for every E C X, of an element F E 9 such that A(F) = A(E). Observe that if A is regular and E ¢ Ma, then there is F E Ma with A(F) = A(E), but A(F - E) > 0, otherwise F - E would be measurable, so that also E = F - (F - E) is measurable. We can now conclude our general discussion on the definition of outer measure
by saying: If (p, .F) is a o -measure, then µ* defined by (2) is a measure on X, which is .F-regular and regular; moreover the elements of F are measurable sets and u* = p. Conversely, if A is an (outer) measure on X, which is F-regular with respect to a o--algebra F contained in Ma, then (A 1,F, .F) is a v-measure on X and (A1 )* = A.
Definition 6. Let p be a measure on X and let A C X. The restriction of p on A is defined as the measure
(pLA)(B) := µ(An B)
dB C X.
It is easily seen that p-measurable sets are also p L. A-measurable; moreover, if p is regular and A is measurable, then p L A is regular.
We recall that a property P(x) is said to hold for p-a.e. x c X, p-almost every x in X, if there is a null set A C X, p(A) = 0, such that P(x) holds for
each xEX\A.
1.1 Measures and Integrals
5
Finally let us recall that Lebesgue's outer measure in RI is the result of Caratheodory's construction by taking for instance as family G the family of intervals in R' and as set function a the function which associates to each interval its standard measure. Let us now introduce the integral with respect to a measure A. Here the key point is Lebesgue's point of view which in some sense reverses the classical approach of Cauchy and Riemann. The integral is defined not as limit of sums of terms of the type p((x1,x2))f(X1), but instead as limit of sums of terms of the type y1 µ(f-1((yi,y2))) Therefore the notion of measurable functions plays a fundamental role.
Definition 7. Let-p be a measure on X, and Y a topological space. A function f : X -* Y is said to be p -measurable if and only if f -1(U) is µ-measurable whenever U is an open set in Y.
If Y = [-oo, +00] := ft, and fk, k = 1, 2, 3, ..., are µ-measurable functions from X into [-oo,+oo], then f, +f2, fife, f1j, min(fl, f2), max(fl, f2), infk fk, SUPk fk, liminfk_,,,, fk, limsupk-OO fk are all µ-measurable functions whenever they are defined.
Definition 8. Let µ be a measure in X. A function g : X -+ [-oo, +oo] is called a it p -step function if g is µ-measurable and its image g(X) {g(x) I x E X} is a countable set. A µ-step function is said to be µ-integrable, respectively µ-summable, if the sum
E yµ(g-1(y))
I (g; µ)
yEg(X)
exists in [-oo, +oo], respectively in (-oo, +oo). Here we use standard conventions on 12, in particular 0.o0 = 0, while oo - 00 is undefined. It is not difficult to prove
Proposition 3. Let f be a non-negative p-measurable function. Then f is the pointwise limit of a non decreasing sequence of steps functions each of which takes only a finite number of values.
Let f : X -+ [-oo, +oc] be any function and let Sµ denote the set of all µ-integrable step functions. We define
ffdµ := inf{I(g; µ) I g E SN, f< g µ -a.e}
6
1. General Measure Theory
ffdµ := sup{I(g; µ) 19 E Sµ
g < f µ -a.e}
(as usual we set inf 0 :_ +oc, sup O:= -oo).
Definition 9. The function f : X --+ A is said to be µ-integrable if and only if
r
f
Jfdµ := J
fdµ=
f fdµ;
f is called µ-summable if and only if f f dx E R. Evidently, for every µ-integrable step function g : X -* Il8 we have
I(9;µ) =
fgd#.
One also verifies that every non-negative µ-measurable function is µ-integrable and that every µ-summable function is µ-measurable. It is usual to identify functions which are equal for µ-almost every x E X. The space of all (classes of equivalence of) functions which are µ-summable is denoted by L' (X ; p) . We have (i) L' (X; µ) is a linear space; moreover If I E L' (X ; µ) if f E Ll (X; µ) . (ii) The operator
f E L1(X; µ) ->
ffdµ
E
is linear and continuous in the sense that
ff dµ < f IfI dµ. (iii) Also, the integral is monotone, i. e., f f dp > 0 whenever f > 0 a. e. in X.
Next theorem collects some relevant results concerning Lebesgue's integration.
Theorem 2. Let µ be a measure in X and let { fk}, fk : X - [-oc, +oc], be a sequence of µ-measurable functions. Then we have (i)
Egoroff's theorem. Suppose that the measurable functions fk are finite, i.e., fk : X -* R, and let e > 0 and A C X with µ(A) < +oo. If fk(x) ----p f (x)
for a. e. x E A,
then there exists a µ-measurable set B C X such that µ(A \ B) < E and
fk(x) -> f (x)
uniformly in B
.
1.1 Measures and Integrals
7
(ii) Beppo Levi theorem. If 0 < fl < f2 < ... and f (x) := limk-,,. fk(x), then
ffkdP T
Jfd.
(iii) Fatou's lemma. If fk > 0, k = 1, 2,..., then
f (iv)
lim
r
f fk dµ < lim n f fk dµ
.
Lebesgue's dominated convergence theorem. If fk(x) -+ f (x) for pa. e. x E X and if there exists a p-summable function g such that
then
bk=1,2,...
9,
Ifkl
f lfk - fIdtt f fkdµ --> ffd. 0,
consequently
(v)
Absolute continuity theorem. If f : X -* [-oo, +oo] is p-summable, then for every e > 0 there exists b > 0 such that
Jfdi < C A
for all p-measurable sets A with p(A) < J.
Definition 10. We say that fk converge to f in L1(X; p) if and only if
f f-fd
0.
We say that fk converge to f in measure p if and only if for every e > 0 p{x E X I Ifk(x) - f(x) I > E} -> 0
as k -> 00
Since for every E > 0 we have
p{x E X I I fk (x) - f(X)I> E} < E-1
f
l fk - f I dp ,
we deduce that fk - f in measure p provided f I fk - f I dp -+ 0, i.e., Llconvergence implies convergence in measure. Also, because of Egoroff's theorem, convergence a.e. in a set of finite measure implies convergence in measure.
1. General Measure Theory
8
Proposition 4. Let {fk} be a sequence of p-measurable functions, which converges to f in measure p. Then there exists a subsequence { fk, } such that fk. (x) -+ f (x) for p-a. e. x E X
Notice that in general the entire sequence fk does not converge to f a.e., even if fk converge to f in L1. 1 For instance let f (x) = 0 for x E R. For each k E N, write k = 21 + m, where 0 < m < 2n, k = 0,1.... ; then n and m are uniquely determined by k. Let
_ A (x)
'
r SI
1
if 2 < x< 2±1
0
otherwise .
Then
f fk(x)-f(x)dx = 2-n ->0
k->oo,
but
lim fk(x) koo exists for no x E [0, 1]. If we instead set
_ irk (x)
2"`
{ 0
if 2 < x < 2±1 otherwise
.
we see that for every positive e
meas{xER I but
fk(x)- f(x)I >e}<2-"-+0
ask -->oo
f Ifk(x) - f(x)ldx = 1,
i.e. the convergence of fk to f in measure does not imply Ll convergence.
Every function f : X -> Y induces a map f# which associates to each measure p on X the image measure f#1L
on Y by the formula
f#p(B) := p(f -1(B)) One readily verifies that f -1(B) is p-measurable if and only if B is f# (ii L. A) -
measurable for every A C X. One also sees that if µ(X) < +oo, p is regular, and B C Y is f#p-measurable, then f -1(B) is p-measurable. To state our next results we need the concept of countably p-measurable sets and functions.
Definition 11. Let p be a measure on X.
1.1 Measures and Integrals
9
(i) A set A C X is called countably p-measurable or µ-o -finite if and only if it is expressible as the union of some countable family of p-measurable sets Ak, A = Uk=1 Ak, with µ(A k) < +00(ii) A µ-measurable function f with values in a topological vector space is called countably ,u-measurable or µ-a-finite if and only if the set {x c X I f (x)
0} is µ-o finite. Observe that every p-measurable function is µ-o--finite.
Definition 12 (Product measures). Let µ be a measure on X and let v be a measure on Y. For any M C X x Y define
(µ x v)(M) := inf { >µ(Ak) x v(Bk)} k
where the infimum is taken over all sequences of µ-measurable sets Ak C X and v-measurable sets Bk C Y such that M C UkAk x Bk. The set function µ x v is called the product measure of p and v.
Theorem 3 (Fubini). Let µ and v are respectively measures on X and Y. (i) µ x v is a regular measure in X x Y (ii) If A C X is p-measurable and B C Y is u-measurable, then A x B is
µ x v-measurable and (µ x v)(A x B) = p(A)v(B). (iii) If M C X x Y is countably it x v-measurable, then the set
M. :_ {y E Y I (x, y) E MI,
XEX
is v-measurable for p-a.e. x E X and (µ x v) (M) =
fv(Mx)d(x) x
(iv)
If f: X X Y --p [-oo, +oo] is a µ x v-integrable and countably µ x vmeasurable function, then f (x, ) : y E Y -+ f (x, y) is v-integrable for p-a.e. x E X, the function h(x) := ff(x,Y)dv(Y)
Y
%s p-integrable and
f f(x,y)dµxv = XxY
fhd µ = X
J
X
I
f (x, y) dv(y)
dµ(x)
Y
However finiteness of iterated integrals does not imply in general that of the multiple integral. But for non-negative f and more generally we have
10
1. General Measure Theory
Theorem 4 (Tonelli). Let f : X X Y -+ [0, +oo] be a (p x v) -measurable and countably p x v-measurable function. Then the existence of one of the two integrals
f f(x,y)dy x v XXY
1 X
(ff(xv)()) dp(x) Y
implies the existence of the other and equality as well.
1.2 Borel Regular and Radon Measures In this subsection we assume that X is a metric space with distance d and the induced topology.
Definition 1. Let p be a measure on the metric space X. (i) The smallest o-algebra containing all open sets in X is called the Borel o-
algebra and is denoted by B(X). A function f : X -> Y, with values in a topological space Y is called a Borel function if, for every open set U C Y,
f-1(u) E B(X). (ii) p is called a Borel measure if every Borel set is p-measurable, i.e., B(X) C
Mµ. (iii) W is called Borel-regular if p, is Borel, i.e., B(X) C MN,, and for every A C X, there exists B E B(X) such that B A and µ(B) = µ(A). Definition 2. A measure p over a locally compact and separable space X is called a Radon measure if p is Borel-regular and for every compact subset K of X we have µ(K) < +oo. An important example of Radon measure in JRY is given by Lebesgue's ndimensional outer measure Cn-.
One easily verifies that: If p is Borel-regular and A is p-measurable, then p L A is Borel-regular.
If µ is Borel-regular and A is a p-measurable subset of a locally compact and separable space X with µ(A) < +co, then p L A is a Radon measure. Borel measures, in contrast to measures on a generic set, have a quite rich family of measurable sets. The following theorem gives a criterion ensuring that a measure over a metric space is a Borel measure.
Theorem 1 (Caratheodory's criterion). Let p be a measure over a metric space X. If
µ(A U B) = µ(A) + µ(B) for all A and B with positive distance, i. e.,
dist (A, B) := inf{d(x, y) I x E A, y E B} > 0, then p is a Borel measure, i.e., B(X) C M,,.
1.2 Borel Regular and Radon Measures
11
Next theorem states the basic approximation property in terms of Borel measures
Theorem 2. Let p be a Borel measure on the metric space X and let B C X be a Borel set.
If µ(B) < +oo and e > 0, then there exists a closed set C C B such that µ(B \ C) < e. (ii) If B C U' iUk, where Uk are open sets with p(Uk) < +oc, and e > 0, then (i)
there exists an open set U D B such that lc(U \ B) < E.
Moreover, if p is Borel-regular, then (i) and (ii) hold for all p-measurable sets
BCX.
The following two propositions follow at once from the previous theorem.
Proposition 1. Let p be a Borel-regular measure on the metric space X. Suppose that X C Uk 1Uk where Uk are open sets with p(Uk) < +oo. Then
p(A) = inf{µ(U) U D A, U open} for every subset A C X, and
µ(A) = sup{ µ(C) Cc A, C closed} for every p-measurable subset A C X.
Proposition 2. Let p be a Radon measure on a locally compact and separable space X. Then
p(A) = inf{p(U) U D A, U open} for every subset A C X, and
µ(A) = sup{µ(K) I K C A, K compact} for every p-measurable A with µ(A) < +oc. Let µ be a Borel-regular measure in X. Of course continuous or semicontin-
uous functions f : X -> R are p-measurable and in fact Borel functions; it is also easily seen that the inverse image of a Lebesgue measurable set in ]I8 by a semicontinuous map is measurable. Notice instead that the direct image of a measurable set by a continuous map f in general is not measurable. For instance Cantor-Vitali function in Sec. 1.1.3 below maps the Cantor set, which is a null set, into a set of positive measure, therefore we can obtain non-measurable sets as continuous image of a null set.
Definition 3. We say that a µ-measurable map f : X -> Rm, m > 1, satisfies Lusin's property (N) if p-null sets in X are mapped into null sets in Rm.
1. General Measure Theory
12
The following is then easily proved taking into account Lusin's theorem below: Let µ be a Borel-regular measure in X and let f : X -, ]R'n be a Borel map which has Lusin property (N). Then f (E) is Lebesgue measurable provided E is p-measurable, compare Step 1 of the proof of Theorem 1 in Sec. 2.1.2. Recall that f : X R is a Borel function if the inverse image of an open set is a Borel set.
Also recall that not every Lebesgue measurable function f : Rn -, JR can be obtained as a monotone limit of continuous functions, but we have
Theorem 3 (Vitali-Caratheodory). Every Lebesgue-measurable function f
:
I[Pn -, Ifs is the a.e. pointwise limit of a non-decreasing sequence of functions cpk which are upper semicontinuous and bounded from above. If moreover f is integrable, we have
lim
f
(f - Wok) dx = 0 .
Finally next theorem shows an important relation between measurable and continuous function.
Theorem 4 (Lusin). Let p be a Borel-regular measure on a metric space X and let f : X -+ III be a p-measurable function.
If A C X is p-measurable with u(A) < +oo and e > 0, then there exists a closed set C C A such that µ(A \ C) < e and fic is continuous. (ii) If X is moreover p-o--finite, then there exists a Borel function g : X -, JIB such that f = g p-a.e. in X. (iii) Let X = Ian, let A C Rn be p-measurable with µ(A) < +oo, and let e > 0. (i)
Then there exists a continuous function g : IRn ---, R such that
p{x E A I f (x)
g(x)} < e.
Notice that (iii) follows from (i) by means of Tietze's extension theorem for continuous functions on closed sets. Remark 1. Observing that RI is the union of a countable family of closed cubes with disjoint interiors and with side one, one easily sees that Lusin's theorem holds for any measurable set A of R', not necessarily of finite measure, if p is the Lebesgue measure C. This remark will be used later.
1.3 Hausdorff Measures Hausdorff measures are among the most important measures. They allow us to define the dimension of sets in IRn and provide us with s-dimensional measures in JR'1 for any s, 0 < s < n and actually in any metric space. They will be continuously used in the sequel. Let s be a nonnegative real number. We denote by w9 the volume of the unit ball in IRS f o r s = 1, 2, 3, ..., we set wo := 1 and we let w9 any convenient fixed constant for non-integers s. Since the measure of the unit n- ball is given by
1.3 Hausdorff Measures
rn/2
13
n=1,2,...
(n/2)I'(n/2) where F(t) is Euler's gamma function 00
j
xt-1 exp(-x) dx
,
we may and do take as ws
WS :_ (s/2)F(s/2) .
Fig. 1.1. k (E) - 2, f' (E) - 4. Definition 1. Let X be a metric space. For 6 > 0 and A C X we set (1)
'H'(A) := inf { ws (diamCk)8 2
k
UkCkJA, diamCk<6, CkCX} The Hausdor$ measure ?i is then defined by 7-ls(A)
(2)
Observe that 71
bola(A),
AC X
> 7-1o2 for 61 < b2i thus 7-1 is well defined and
'H'(A) = supW (A)
.
5>0
Notice that it is essential to use the approximating measures 7-lb in order to define H'. The naive definition
ECUCk}
7ls(E) k
k
1. General Measure Theory
14
in fact would have very unpleasant features. In (1) we may also require that the sets Ck be closed or open without changing the value (2). By the same procedure we can define the so-called s-dimensional spherical measure Ss by requiring in (1) that the sets Ck be balls. Then we obviously have
Hs < S8 <
25'Hs
but there exist sets A for which 'H' (A) < 88(A). For X = Rn, one easily verifies (i) 71s(.A) = As71s(A), \ > 0, where AA :_ {Ax I x E Al (ii) 7{° is the so-called counting measure , i. e.,
7H° (A) _ #A := number of elements of A (iii) 7{s = 0 on Rn for s > n (iv) 7{s is not a Radon measure on IIBn for 0 < s < n. (v) If 7l6(A) = 0 for some b > 0, then 'H" (A) = 0.
Finally, observe that for 0 < s < r
xb (A) <
(8)r8718(A)
thus 7{s (A) = +oo if 7lr (A) > 0 and ?ir (A) = 0 if 'Hs(A) < oo. This in particular shows that 71s (A) can be positive and finite for at most one (possibly none) s > 0. This motivates the following definition
Definition 2. The Hausdorff dimension of a set A C X is defined by
dim%(A) := inf{s > 0
:
1s(A) = 0}
From the previous remark it follows that if dim HA > 0, then diin (A) = sup{s > 0 1 7-ts(A) _ +oo}
and that, if 0 <7{s(A) < oc, then dimHA = s. It is easily seen that 7{a and 7{s are measures over X. Borel sets are not in general 71b-measurable for b > 0: for instance the half-space {(x, y) E R2 1 x > 0} c R2 is not 7{b-measurable. However, observe that 7-I = xn on III and we have
7{a (A U B) = li (A) + 7{b (B) if dist (A, B) > 26; therefore Borel sets are 7{s-measurable by Caratheodory criterion. Actually one proves
Proposition 1. 7-a and 7{s are measures on X; moreover 1s is Borel-regular
for alls>0. If X = RI and A is a Hs-measurable subset of X with 7-ls(A) < oc, then Hs L A is a Radon measure on X.
1.3 Hausdorff Measures
15
Using the so-called isodiametric inequality
Ln (A) <(diarn w2 A '
VA C R,
which says that among sets A C ][fin with a given diameter p, the ball with diameter p has the largest Lebesgue measure, one also proves the following important theorem
Theorem 1. The Hausdorff measure H-ln in R' coincides with the Lebesgue measure Ln* on IEBn. Moreover if M is an s-dimensional smooth submanifold in 1R , 0 < s < n, then ?-l9 L M is the standard volume measure in M, induced by the Euclidean metric in M.
0 Cantor sets. It is often difficult to determine the Hausdorff dimension of a set and even harder to find its Hausdorff measure. Of course the most difficult part is to give a lower estimate. The most familiar sets of real numbers of non-integral Hausdorff dimensions are Cantor's sets. These are obtained by the following construction. Fix 8 E (0, 1/2), begin with Eo :_ [0, 1] and in the first step remove an open interval of length 1 - 25 in the middle. Inductively then define Ek+1 by removing in each interval of Ek a centered open interval of length dk (1 - 26). Of course Ek D Ek+i for all k and each Ek is the union of 2k intervals of length b. The Cantor set associated to b is then defined as the closed, non denumerable, and dense in itself set cc C :=
I
IEk.
k=0
Classically one chooses b = 1/3 and the resulting set C is known as Cantor's middle thirds set.
{
F-]
F-]
F-I
F- -4
F--I
H-I
7--1
Fig. 1.2. Cantor's middle thirds set.
To be more precise and for future references we can describe Cantor set C as follows. For each k = 0, 1, ... and j = 1, ... , 2k define the base points bk,j of the Cantor set C inductively by bo,1
:= 0
16
1. General Measure Theory
and
Then
if j = 1, ... , 2k
8 bk,9
bk+l,j
t (1- 8) + 8 bk,j
if j = 2k + 1, ... , 2k+1
'k-l,j := bk_1, j + 8k-1(6,1 - 8)
j = 1, ....
2k-1
are the intervals removed from Ek_1 to get Ek at the step k, and Jk,j
bk,j + 8k [0, 1]
j = 1, ... , 2k
are the intervals whose union yields Ek. Consequently 00
C
2k
= k=0 n uj=1Ji,k
From this expression it is easily seen that Cantor set C is self-similar in the
Fig. 1.3. Self-similarity of Cantor sets in two dimensions.
sense that certain homothetic expansions of C are locally identical to the original set; more precisely we have
bk,j + 8kC = C n Jk,j .
(3)
In fact it is readily seen that x E Jh, j
if
bkX E Jh+k,j
BkEh
= Eh+k n [0,
for all h, k, j
hence Sk]
1.3 Hausdorff Measures
17
bk,j + BkEh = Eh+k n Jk,j for all h, k > 0, j = 1, ... , 2k, so that (3) follows by taking the intersection in h. By introducing the map rr : P([0,1]) --, P([0,1]) defined by
-r(A) := (8A) U (1- 8 + 8A)
,
we also see that rrk([0,1]) = Ek, rk({0}) _ {bk,l, ... , bk,2k }. Therefore the selfsimilarity of C can be also expressed by cc
c = n rk([O,1]) ,
7- (C) = C .
i.e.,
k=0
Proposition 2. The Hausdorff dimension of the Cantor set C, actually C5, is s := log 2/ log(1/8). Moreover
fs(C) = 2-8ws
.
Finally for all x, y E [0,1], x < y RI (C n [x, y]) = k iin £1(Ek n [x, y] ) Proof. Since C c Ek and Ek is the union of 2k intervals of length 8k we see that
lbk(C) < 2-sws2kdks = 2-sws(28s)k = 2-sws Letting k --> oc, 7-ls(C) < 2-sws. To prove the opposite inequality we consider
an open covering F of C. As C is compact we may assume that F is finite F = {K1, ... , KK} and we have Kl U ... U KK D Ek for some k E N. For any e > 0 choose k large enough so that 48k$ < e, and let K := (a, b) denotes one of the intervals in T. For convenience denote by R(I) and L(I) respectively the right and left end points of any interval I. We have 2k
[0, 1] _
k -1
2 "`
(U Jk,j) U ( U U j=1
m=0 i=1
where the J-type intervals yield Ek and the I-type intervals are the ones which are taken out up to step k. First we slightly change K as follows
if a E I,,,,,i, we set a' = R(I00,i); a' > a
if aEJk,j, we set a'=L(Jk,j); a'>a-8k Similarly
if b E I,,,,,i, we set b' = L(I,,i); b' < b if b E Jk,j, we set b' = R(Jk,j ); b' < b + 8k
.
1. General Measure Theory
18
This way for K' _ (a', b') we find
8K' C aEk
I JK'j - IKI I < 2bk ,
and of course K' covers K fl C. From now on we can therefore assume that for each Kr in F
aKr C aEk ,
R(Kr) = R(Jk,i)
L(Kr) = L(Jk,j),
for some j < 2k and j 2k Next we note that, by the construction of Cantor set, each Jk,j is preceded and followed by one of the I-type intervals, one taken out at the k-step and one at least at step k - 1. More precisely there are (m1, i1) and (m2i i2) with
ml, M2:5 k - 1, it < 211, i2 < 212 such that
R(Iml,il) = L(Jk,j) ,
R(Jk,.i) = L(Im2,i2)
and moreover either ml = k - 1, m2 < k - 2, or m1 < k - 2, m2 = k - 1. Consider now the largest I-interval Im,i in Kr = [ar, br]. We claim that (4)
ar > L(Im i) - bm+1
In fact we have the disjoint decomposition bm,i + bm [0, 1] = Jrn+1,.7 U Im,i U J,,,,+1,j+1
for some j, where
= bm,i + [0, Sm+1] Jm+l,j+1 = bm,i + (1- b)81 + [0,6-+'] Jm+1,.7
Suppose that ar < brn,i = L(I,,,,,i) - 51+1 Then there exists I,,,,',i' such that
R(Im',i') = L(Im+1,.7) C Kr. By the previous remark we then have m' < m, i.e., Im',i' I > IIm,i 1, that is Im,i is not the largest I-interval in Kr, a contradiction. Similar we proceed to show that br > R(Im,i) + 8m+1 On account of (4) we can define hence
Jl
Kr fl [0,L(Irn,i)]
J2
,
Kr n [R(Im,i),1]
and find the disjoint decomposition Kr = J1 U Im,,i U J2 with IJ1 1,
1 J21 <
1
SZb IIm,iI
1.3 Hausdorff Measures
19
Therefore using the concavity of t5 and the fact that 285 = 1 we deduce
Kris =
(IJlI + IIm,iI + IJzI)5 > [IJII + 21 b2a(IJII + IJ2l) +
IJ2I]5
-5(21J1I + 2IJ2I)5 ? 2a-5(IJlls + IJ2I5) IJl is + 1-72I5
.
Thus, replacing Kr by the two intervals Jl and J2 which still cover Ek fl Kr and satisfy aJl, 9J2 c aEk, we do not increase the sum IKl
5+...+IKKI8
.
Proceeding this way we obtain in a finite number of steps IKrls
>_
IJk,jls Jk,jCK,
and eventually after a finite number of steps IEhI
for a suitable h. This yields ls(C) > 28 w5.
2 Cantor-Vitali functions. A procedure similar to the one which leads to Cantor set with parameter 8 allows to construct a non decreasing continuous function V, which we call Cantor- Vitali function, whose derivative vanishes almost everywhere. For that, with the notation of O1 we consider for each k the two functions Vk, gk : [0,1] -+ lR defined by 9k(x)
2-k8-kXEk(x)
,
Vk(x)
IYktdt
=
2-ks-(Ekn[0,x])
Clearly Vk(0) = 0, Vk(1) = 1, Vis nondecreasing and
f 9k (t) dt = 2-k
Jk,j
For x E [0, 1] we denote by j (x, k) the largest integer between 0 and 2k such that Jk,j(y,k) is contained in [0, x]; if such an interval does not exist we set j (x, k) = 0. We then see that for x Ek
1. General Measure Theory
20
f
k)
Vk(x)
=
9k (t) dt = 2-kj (x, k)
j=1 Jk,j j(x , k)
fgk+l(t)dt = 2-kj (x, k)
Vk+1(x)
j=1 Jk,j
so that
if x
Vk(x) = Vk+1(x) = 2-k.7 (x, k) Vk+1(x) =
Ek
if x E Ik, j
2k+1
while, if x E Ek, so that x E Jk,j for some j, Vk(x) is linear on each Jk,j, V,' (x) = (2b)-k in Ek and we have Vk+1(x) - Vk (x) I
(5)
G f I9k+1(t) - 9k (t) I dt < 2-k+l Jk,j
Equivalently the approximate Cantor-Vitali functions Vk can be inductively defined by
2 Vk(b-lx) 1
2
if x E [0, 6]
ifxE[6,1-b] +2Vk(d-l(x-(1-b))) ifxE [1-6,1]
Then again we readily see that Vk(0) = 0, Vk(1) = 1, Vk(x) = (26)-k in Ek. Moreover, being Jk,j = [bj,k, bj,k + 6k], we have
Vk(bj,k) _
2k
Vk(bj,k + bk) = '
j 2k
and
2j-1
Vk(x) = 2m+1
for x E 1,,,j
m = 0, ... , k - 1, j = 1,... ,
,
2,m
.
By induction we immediately see that for each k we have (6)
IVk(x)-Vk(y)I < CIx-2,/Is
where
c := (1 - 2b) -1
and
s=
log 2 log(1/b)
From (4) and (6) we infer that Vk converge uniformly to a Holder-continuous and non decreasing function V, called the Cantor- Vitali function.
1.3 Hausdorff Measures
21
Fig. 1.4. An approximation of Vitali's
+--
function.
Notice that Vk(x) = Vk+1(x)
for x E [0, 1] \ Ek
V (x)
and that, by Proposition 2
V(x) _ Hs (c n [o, x])
V x E [0,1]1
xs(C) self-similarity of V is expressed by
V(x) =
V(bkJ)+2-kV(b-k(x-bk,j))
forxEJk,.7:_[bk,9,bk,3+8kl
Finally it is easily seen, compare for more details Ch. 4, that (i)
V has bounded variation in (0, 1), V E BV(0,1), since it is non-decreasing,
andV(0)=0,V(1)=1. (ii)
On each interval in [0,1] \ Ek we have V (x) = Vk (x) = constant. In particular, V (x) is differentiable on each x E [0, 1] \ E, and (approximately) differentiable almost everywhere in [0, 1] since El = 0, and
apV'(x) = 0 (iii)
for a.e. x E [0,1]
.
Since V is continuous, V' has no jump part but only a Cantor part
V' = Va(c). and V' has support in E. (iv) Obviously V([0,1]) = [0, 1], V maps [0, 1] \ E into the denumerable dyadic
set
D :_ {y E lI
I
y=
2k
, j E [0, 2k], j, k E N}
.
Therefore V maps E onto [0, 1] \ D. In particular V has not Lusin property (N).
22
1. General Measure Theory
Let f : Rn --> R'm', m > 1, be a map and let A be a subset of ln. The graph off over A is given by
cf,A := {(x, f(x)) I x E Al
.
Recall also that f is said to be Lipschitz continuous if
Lip f := sup{ If (x)
- f(Y) I
Ix - yI
I x, y E Rn , x
y} < co
.
Then we have the following theorem
Theorem 2. Let f : ][8n -. R' and let A C lien (i) If 7-ln (A) > 0, then dim -h9 f,A >_ n (ii) If f is Lipschitz-continuous, then for s > 0
7{s(f(A)) 5 (Lip f)s1s(A) (iii) If f is Lipschitz-continuous and 7-Ln(A) > 0, then
dim%gf,A = n
.
Remark 1. Hausdorf measure 7i is the result of a special case of a construction known as Caratheodory's construction,conpare Sec. 1.1.1. Given any collection F of parts of X with 0 E F and any function V) :.7= --> [0, +ooJ with 0(0) = 0, we define a measure p = pV,,.F on X as follows. First, for each S > 0, we set
ps(A) := inf{Ezb(Ak) I Uk Ak D A , Ak E .7= diarn Ak < 6} ,
,
ACX
,
and then
µ(A) := sup pi(A)
,
b>0
AcX
Of course Hausdorff measure 7-[s corresponds to the choice F = 2X and This way one may construct several interesting geometric measures, but for our purposes it suffices to work with Hausdorff measure. 0 (A) = w9 (2 diam A)
Having in mind Lebesgue's measure, a natural question is whether 7-ls+t(A x
B) and 7-ls(A) x 7dt(B) are equal, if A C R' and B C R"n. It turns out that such a result is far from being true and in general we only have
Theorem 3. Let A C Wt and B C III". Then 7-Ls+t(A x B) > b H" (A) x 7-lt(B) for some constan b > 0. In particular dim %(A x B) > dim HA + dim HB.
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
23
Theorem 3 is a special case of the following inequality
(7)
Jfl8(Aflf_1(y))dt
< Wswt(Lipf)s7-[t+s(A) Ws+t
which holds for any Borel set A, any Lipschitz function f, and non negative real numbers s and t. Its proof is presented in Ch. 2, Theorem 2 in Sec. 2.1.3, step 2, in case s, t are integers, but it works as well for general s, t. The opposite inequality in (7) does not holds. For instance in Federer [226] one finds for any s general Cantor-type sets C1 and C2, 7-l8 (Cz) = 1, i = 1, 2, for which 7.12, (Cl x C2) = +oo. However, it is easily seen that 7-128(C x C) < CH, (C) x R, (C)
if C is the Cantor set in 1 A better result holds if s = n Theorem 4. For any B C JR'n with 7{t(B) < oc there exists a number c such that Wn 1 < cwt Lo-1
< 2-'(n+ 1)(t+n)/2
and
,Hn+t (A X B) = c7.n (A)7.1t (B) for every Lebesgue measurable set A.
In general the constant c in Theorem 4 is different from 1 but it is 1 if B has good tangential properties, i.e. if B is t-rectifiable, compare Theorem 3 in Sec. 2.1.5.
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems In this subsection we shall assume that X is a locally compact separable metric space. We shall denote by B,(X) the collection of all relatively compact Borel sets of X. If X is not compact the collection B, (X) is not a a-algebra, it is only a local a-algebra in the sense that
BC(X) L A :_ {B E Bc(X) I B C Al is a a-algebra in A for each A E Bc(X). A set function v : Bc(X) -- JR is called a a-additive measure if 00
v
(UA) k= 1
00
= k=1
for all sequences {Ak} of disjoint sets in Bc(X) such that Uk 1Ak E Bc(X), i.e., if v restricted over BC(X) LA is a-additive on BC(X) LA for all A E B, (X). For any f : X --* [-oo, +oc] and A C X we set
1. General Measure Theory
24
f inA 0 inX\A
fLA
f fdp :=
and
ffL.AdP. X
A
Let lc be a Radon measure over X. We denote by L ,:(X; i) the collection of all locally µ-summable functions f : X --+ [-oo, +00], i.e.,
L o_(X;,u) := { f I f L K E L1(X; A) VK C X, K compact}. For any f E L' Lc (X; p), the expression
(µL f)(A) :=
(1)
r
JA
f da
A E 13,(X)
defines a o--additive measure A L f .
Definition 1. Let p and v be two Radon measures over X. We say that v is absolutely continuous with respect to p, and write v << u, if v(A) = 0 for every A for which µ(A) = 0.
The measures µ and v are called mutually singular, and we write v1µ, if there exists a Borel set A C X such that lU L (X \ A) = 0 and v L A = 0. One of the most important result of measure theory is contained in the next theorem.
Theorem 1. Let i and v be two Radon measures over X (i)
Lebesgue's decomposition theorem. There exist unique Radon measures v(') and v(9) on X such that v = v(a) .+ U(s)
v(s) l
(a)
,
U(s) j. tL
,
U(a) << P.
v(a) is called the absolutely continuous part of v with respect to µ, and v(8) the singular part of v with respect to p. (ii) Radon-Nikodym's theorem. If v << µ, then there exists a unique Borel
function f in Lioc (X; u) such that v = it L f , i.e.,
v(A)=Jfdy
`dAEt3a(X).
A
The function f is called the Radon-Nikodym derivative of v with respect to p and is denoted also by dµ .
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
25
Obviously, we can write the claim of Theorem 1, as
V = b IL
(2)
dy (a)
dµ
+v
or, equivalently, for any Borel function f with compact support
f f dv =
(3)
r
J
f dy(-a) dµ + ffdi/8).
Moreover, we have forA E 8, (X) (a) (A)
(4)
:= inf{v(B) I BE ]3,(X), µ(A \ B) = O}
Definition 2. We say that the set function v : 13,(X) -4R is a signed measure if there exists a Radon measure µ on X and f E L o,_(X; µ) such that
v = µLf
onB,,(X).
The set function v : Ba(X) -> R n, m > 1 is called a vector valued measure if there exists a Radon measure µ and a vector valued function f = (f 1, ... , f m) E L1 a (X, Rm; µ), i.e., with f k E L1(X ; µ) for k = 1, ... , m, such that v = u L f, i. e.,
(v1 ...,Um) _ (µL f1,...,pLfm) We define the total variation vI : B,(X) -+ [0, +oo) of the vector valued mea-
sure v:Ba(X) ->R"n, v=µL f, as IuI
where I f I = (>2k
:= µLIfI
IfkI2)1/2
We define the positive and negative parts v+ and v- of a signed measure
v= p L f,f :X-,R,as
v+ = ILL f+,
v
= fLLf
where f+=max(f,0), f- =min(f,0). The collection of Radon, signed, and vector valued measures on X will be denoted respectively by M(X), M(X), and M (X, R'n) . A few remarks are now in order. Trivially we have p({x I f (x)
0}) = 0 if (µ L f) (A) = 0 for all A. We also remark that vI, v+ and v- do not depend on the decomposition v = µ L f. For instance, if v = µ1 L f1 = 42 L f2i we have µl,µ2 << µ := Al + µ2 Hence by Radon-Nikodym's theorem we have, for k = 1, 2,
Pk = {1,L
Ak, µkLfk = (µL aµ)Lfk = pL(fk dµ
Thus f 1 t' = f 2 dµ =: f p-a.e. in X and
1. General Measure Theory
26
UkLlfkI = JLL(Ifk W k) = ULifl The same argument yields the result for v+ and v-. If v is a signed measured, we obviously have v = v+ - v-, I vl = v+ + vthus v+
Therefore if we define
, v- <<
'_
dv d-Ivl
Iv.1
dv+
dv-
dlvl
dIvl '
then we have
v = Ivl L
dv
dlvl
and by definition dv
+
( dJvJ )
dv+
du
dv-
d-lvl
(dlvl)
dI vI
'
If vk, k = 1, ... , m, are signed measures, vk = Ak L fk, defining p ;_ Ek µk
we
find
/k = /LL (fk
dd/G
).
thus we conclude
Proposition 1. If vI, ... , v' are signed measures, then v = (vl, ... , vm) is a vector valued measure in the sense of Definition 2. Therefore µ = (41,.. ., Ak) is a vector valued measure if and only if each µk is a signed measure. We also have
Proposition 2. The total variation I vl of a vector valued measure v can by equivalently defined as
I vI (A) = sup{E00I v(Ak) I
I
Ak E B ,(X) , Ak disjoint , UkAk C A} ,
k=1
for A E 15,(X).
If v is a signed measure, we have, for A E Bc(X):
v7'(A)=sup{±v(B) I BCA, BE13,(X)} I vl (A) = sup{ lv(AI) I + I v(A2) I I AI n A2 = 0, AI UA2 C A, Al i A2 E 5,(X)}-
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
27
Let v = (v' , ... , v') and µ be respectively a vector valued measure and a Radon measure on X. We say that v is absolutely continuous with respect to p, V << µ, if lvl
<< µ .
or equivalently if l vi l << µ for all i. For two vector valued measures v and A we define
vIA
iff
vI I IAL
Then Theorem 1 readily extends to vector valued measures. In particular we see that, if p is a Radon measure on X, the set of all vector valued measures v E M (X; IE8") such that v << ,v is equal to { p L_ f I f E Llc (X, RI; by definition v << l v I, we also get
Since
Proposition 3. Let v be a vector valued measure. Then v = lvI L i7, where V
dv
E Lloc(X,Pm; lvl)
and lvl = 1 IvI-a.e. in X. Given a vector valued measure v = (vl, ... , v') and a vector valued function g = (gl,... , gm) in Ll (X, lRm; vl ), we define the integral of g with respect to v by
f
:= E J
gk dvk
k
where
J gk dvk :=
J
gk dvk+
-
fokdvk_
.
By Proposition 3 we then get
f
(5)
=
fdivi ,
Denoting by Cc (X, II8m) the space of all compactly supported and continuous functions on X with values in Rm, evidently
A. : f E Cc (X, III') - fdivI defines a linear functional on Cc (X, R"'') for every v E M (X, RI). Next theorem roughly inverts this observation.
Theorem 2 (Riesz). We have (i) Let A : CC(X, Ii') -+ R be a linear functional which is continuous in the sense that it has locally finite norm, i.e., (6)
Sup{) (f) l f E Cc(X, IRm) , If 1 < 1
,
spt f C K} < oo
1. General Measure Theory
28
for each compact K C X. Then there exists a unique vector valued measure v= IV I L W E M (X, ]I8m) such that
A(f) =
f = fdH
df E CC(X,Rm)
(ii) Let A : C, (X, R) -, ][8 be a linear functional which is positive, i.e., A(f) > 0
whenever f > 0. Then there exists a Radon measure µ on X, z E Nl+(X), such that
A(.f)=
ffdy
VfECc(X)
Notice that in (ii) there is no condition on the norm of A, the positivity of A suffices. This expresses the well-known fact that every positive distribution is a measure. A consequence of Theorem 2 is that M (X, Rm) is the dual space of C°(X, ]IPm). We moreover remark that a distribution A : C°° (X, ]1800) -> II8 is a measure (or better extends to a measure) if and only if we have sup{.1(f)
I
f EC°°(X,R''), IfI<1,sptf CK} < 00
for all compact K C X. In this case it is common to identify A with p and write A(A) for fA dµ.
We also readily deduce that, for all open sets U in 13,(X), we have (7)
IvI(U) = sup{
f < f, dv> I f E C°(X, l[8m) ,
If 1
<_ 1, spt f C U}
.
This can in fact be used as an equivalent definition of I vI . Given v E M (X, R-) and an open set U in X, one defines the mass of v in U by (8)
Mu(v) := sup{
f
< f, dv> I f E CC(X, Rm) , If I < 1, spt f c U}
and one simply writes M(v) for Mx(v). Of course, if we extend the set function
M(v) : U --+ Mu(v)
U open
to the Radon measure on X given by
MA (V) := inf{Mu(v) I A C U, U open} we find
MA (11) = I vI (A) .
Actually, we should call Mu(v) in (8) the Euclidean mass, because we used the Euclidean norm in If I < 1. If we use another norm I in R00, the resulting mass M,(v) can be expressed as II
Mu(v) =
II v II'(U)
,
II v II'(A)
f
II
dvl
II' dI vI
1.5 Covering Theorems, Differentiation and Densities
where I
II' is the dual norm of II
29
II, i.e.,
IIy1I' := sup{ 111 X 11 <_ 1, x Exam}
Finally we remark that any linear functional A : CC (X, Rm) -> R satisfying
(6), being a vector valued measure A = IvI L v, v = 1 J vI-a.e., extends to a linear functional on the space LI(X,RI;IvI). In particular if X = UkFk, Fk compact and I vI (X) = M(v) < oc, then constant functions are v-summable and A extends to all Borel bounded functions X ---+ 112m. Such an extension can be computed by Lebesgue's theorem.
1.5 Covering Theorems, Differentiation and Densities A classical theorem of Vitali, actually in dimension n = 1, states the following
Theorem 1 (Vitali). Let E be a subset of R which is covered by a collection of closed balls 13 with the property that for each x E E and E > 0 there exists B E 13 such that x E B and diam B < E. Then we can choose a disjoint sequence {Bk} of balls in 13 such that
meas (E - UkBk) = 0 . It turns out that the property of covering a set in the sense of the previous theorem is the key point for differentiating a measure v with respect to another measure p. To state precisely such a theorem we need a few preliminaries. In this subsection X will denote , if not otherwise states, a locally compact
separable metric space. We denote, as usual, the ball of radius r and center x E X by B(x,r). Given a collection 13 of balls in X, the set of the centers of balls in 13 is denoted by C(B)
C(B) := {x E X 12r > 0 so that
B(x, r) E 13}
.
The collection B covers a set A C X if A C UBEBB.
Definition 1. We say that 13 covers A C X finely if for every x E A and e > 0 there exists B(x, r) E 13 with r < E, i.e., if for all x E A
inf{r > 0
1
B(x,r) E 13} = 0
.
Definition 2. Let µ be a Radon measure over X. We say that X has the symmetric Vitali property relative to µ if for every collection B of balls in X satisfying
(i) 13 covers finely the set of its centers C(13) (ii) µ(C(13)) < +oo there exists a sub-collection 13' of B satisfying
30
1. General Measure Theory
(i) the elements of B' are disjoint (ii) 13' covers p-almost all of C(B), i.e.,
h (C(B) \ UBES'B) = 0 . Next theorem provide us with two important cases in which X has the symmetric Vitali property relative to µ.
Theorem 2. We have (i) Let p be a Radon measure over X. If there exists a constant c such that
E.c(B(x, 5r)) < cy(B(x, r)) for all balls B(x, r) in X, then X has the symmetric Vitali property relative to p. (ii) The Euclidean space X = Rn has the symmetric Vitali property relative to p for any Radon measure p on 1E$n. The proof of Theorem 2 uses the following
Lemma 1 (Vitali's covering theorem). Let B be a collection of closed balls in X with equi-bounded radii, i.e., sup{r a sub-collection B' of B such that
I
B(x, r) e 13} < oo. Then there exists
(i) the elements of B' are disjoint, therefore B' is numerable (ii) for all B E B there exists a ball B' E B' such that B n B' 0 0 and f3' D B,
where for B = B(x, r) we have set B = B(x, 5r). In particular
UB C UB BES
BES'
From Lemma 1 one deduces that, if A is covered finely by B, then one has
A \ U{B I BEBill C U{B I BEB'\Bill for each finite sub-collection B" of B', and this yields Theorem 2 (i). The proof of Theorem 2 (ii) uses instead the following
Lemma 2 (Besicovitch's covering theorem). Let E be a subset of 1R
and
let r : E --a R be a positive bounded function defined on E (for instance, let E = C(B) where B is a collection of closed balls with center in each point of E and radius r(x)). Then one can choose an at most countable family of points A := {xi}iEN in E such that
(i) E C UiB(xi, r(xi)) (ii) The balls B(xi, Ir(xi)), xi E A, are mutually disjoint (iii) Moreover, the balls B(xi, r(xi)), xi E A, can be distributed in e(n) families Bk of disjoint closed balls, where l; (n) is a constant depending only on n (iv) No point of E belongs to more than C(n) balls of the family {B(xi, r(xi)) xi E Al I
1.5 Covering Theorems, Differentiation and Densities
31
We notice that Lemma 1 with X = R' is a consequence of Lemma 2. Later we shall refer to Lemma 1 and Lemma 2 as to Besicovitch's covering theorem. Let now µ and v be Radon measures over X. For x E X we set 17µv(x) := limsupµ(B(x,T)) r-O inf " a x,r Dµv(x) :_ lim r-O M(B(x,r))
whenever p(B(x, r)) > 0 for all r > 0, otherwise, i. e., if A(B(x, r)) = 0 for some r > 0, we set
Dµv(x) = Dµv(x) = +00. If Dµv(x) = Dµv(x) E [0, +oo], we simply write Dv(x) for Dµv(x) =Dµv(x). Definition 3. We say that v is differentiable with respect to A at x if and only if
Dµv(x) = Dµv(x) E [0, +00] In this case, Dµv is called the (symmetric) derivative of v with respect to µ or the density of v with respect to A.
The basic facts concerning the differentiation of a Radon measure v with respect to a Radon measure u are collected in the next theorem.
Theorem 3 (Radon-Nikodym). Let y and v be two Radon measures over X. Suppose that X has the symmetric Vitali property with respect to A. Then (i) Dµv(x) exists for p-a.e. x E X and the function Dµv(x) is u-measurable. (ii) There exists a Borel set Z with µ(Z) = 0 such that
v(A) = fDvdp + (VLZ)(A) A
holds for all Borel sets A C X. (iii) Taking into account Radon-Nikodym's theorem and Lebesgue's decomposition theorem in Sec. 1.1.4 we have v(a) = /.L L Dµv ,
v(s) = V L Z
thus
Dµv =
dv(a)
µ-a.e. in X
dµ
Moreover, if v is absolutely continuous with respect to /1, we may take Z = 0 and we have
Dµv = dv
.z
-a.e. in X
1. General Measure Theory
32
(iv) If X has also the symmetric Vitali property with respect to v, then Dµv(x)
exists also for v-a.e. x E X and we may take
Z = {x E X I Dµv(x) = +oo}
.
... (iv) of Theorem 3 hold if X = Rn. The proof of Theorem 3 is based on the following lemma whose proof uses in turn Vitali's property. In view of Theorem 2 (ii), the claims (i)
Lemma 3. Let A be any subset of X. (i) If Dµv(x) < a in A, then v(A) < ay(A) (ii) If D,,v(x) > a in A, then v(A) > ap(A)
A simple consequence of Theorem 3 is the following theorem to which we shall return in Sec. 3.1.1.
Theorem 4 (Lebesgue-Besicovitch's differentiation theorem). Let a be a Radon measure over R' and let f be a function in L oc (R 2; µ) with p > 1. Then
lim
r-D (B(x, r))
for µ-a.e. x
f f dµ = f (x)
B(x,r)
and even lim
1 r)) r-.O p,(B(x,
J B(x,r)
If
-f(x)
rdµ = 0
for p-a.e. x
If we apply Theorem 4 to the characteristic function XE of a p-measurable subset E of Rn, we obtain that lim
r-o
p(B(x, r) n E) µ(B(x, r))
(B
lim
µ(B(x, r))
r-O
=
d(ti L E) dµ (x)
equals 1 for p-a.e. x c E and equals 0 for µ-a.e. x E Rn \ E. The sets IntA(E)
dµL
{x
(x) = 1}
I
Extµ(E)
{x
1
d( E (x) = 0}
might be referred as to the measure-theoretic interior and exterior of E with respect top and we may refer to
am(E) := Rn \ (Int (E) U Ext (E)) as to the measure-theoretic boundary of E. Of particular relevance is the case in which µ is the Lebesgue measure in R', or equivalently the n-dimensional Hausdorff measure in Rn. We then set
1.5 Covering Theorems, Differentiation and Densities
33
Definition 4. Let E be a Lebesgue measurable set in R'. The upper and lower densities of E at x E E are defined by limsup7-ln(Ef1B(x,r))
9*(E,x) B* (E, x)
_ Dy{n(,HnLE)(x)
x (B (x, r))
r- O
:= lim inf r-.O
7-ln (E fl B(x, r)) 1 (B (x, r))
x
_ D n (q-ln L E) (x)
and the density of E at x by 8(E, x) = 0* (E, x) = 0* (E, x) if the upper and lower densities are equal at x. Of course 7{n-a.e. x E E is a point of density 1 and 7-ln-a.e. x E IISn \ E is a point of density 0 for E. In terms of densities we can recover interesting relations between measurable and continuous functions
Definition 5. Let f : R --> R be a Lebesgue measurable function. The approximate upper, respectively lower, limit of f at x is defined as
aplimsup f (y) := inf{t E R
y-x
I
0*({z E IRn I f (z) > t}, x) = 0}
respectively
apliminf f (y) y-ix
sup{t E R 10* ({z E IRn I f (z) < t}, x) = 0}
.
We speak of approximate limit of f at x in case aplim f (y) := aplimsup f (y) = apliminf f (y) y-+x
'Y-+x
and we say f to be approximately continuous at x if
aplim f (y) = f (x) y-x We then have, compare Sec. 3.1.4 (i)
f is approximately continuous at x if for every open set U C we have
R
with f (x) E U
0(f-1(u), x) = 1 (ii) f is approximately continuous at x if there exists a measurable set E with
x E E such that 0(E,x) = 1 and fj E is continuous at x, (iii) if f E Lioc(R ; µ) and aplimy-x f (y) = f (x), then lim.
1
o Gn(B(x, r))
f If (y)-f(x)Idµ = U. B(x,r)
1. General Measure Theory
34
and finally
Proposition 1. Let f : I[8n - IR be a function which is a.e. defined in RI. Then f is Lebesgue measurable if and only if it is approximately continuous at a.e.
xEIR'. The previous results do not apply, at least directly, to lower dimensional Hausdorff measures, as fl9 is not a Radon measure in IRn for 0 < s < n. For this reason one introduces lower-dimensional densities which in some sense extend the notion of Radon-Nikodym derivative to Hausdorff measures.
Definition 6. Let p be a Borel measure over a metric space X. For any subset A C X, and any point x E X, we define the n-dimensional upper and lower densities by On* (p, A, x)
lim sup
9*(p,A,x)
liminf
r-.0 r--.0
it (A n B(x, r)) Wnrn
p(AnB(x,r)) Wnrn
In case A = X we simply write Bn* (p, x) :=
9n*
(p, X, x) ,
9 (p, X, x)
9 (A, x)
In case On* (p, A, x) = 0, (p, A, x) we write on (p, A, x) for the common value. Clearly we have
Bn*(p,A,x) = Bn*(pL A, x)
On (p, A, x)
=
on
(pLA,x)
and, since p(A n B(x, r)) is an upper semicontinuous function in x for r fixed, On* (p, A, x) and 9* (p, A, x) are both Borel functions. Notice moreover that the same definition of densities can be used for any measure p. Next theorem provides useful connections between p and fn based on information about the upper density of p.
Theorem 5. Let p be a Borel regular measure on a metric space X and let
t>0.
(i) If A C B C X and 9n* (p, B, x) > t for all x E A, then p(B) > t7in (A). The
-
case A = B is important, we have: if 0n* (p, A, x) > t for all x E A, then p(A) > t7-ln(A).
(ii) If A C X and On* (p, A, x) < t for all x c A, then p(A) < 2ntfn(A).
A consequence of Theorem 5 is
Theorem 6. Let p be a Borel regular measure on a metric space X, for instance let p be the n-dimensional Hausdorff measure fn. If A is a p-measurable subset of X with p(A) < oo, then 9n* (p, A, x) = 0 for HI-a. e. x E X \ A.
1.5 Covering Theorems, Differentiation and Densities
35
Finally we have the following important bounds for densities with respect to Hausdorff measure
Theorem 7. Let A be any subset of X. (i) If In (A) < oc, then on* (7{n, A, x) _< 1 for Hn-a. e. x E A. (ii) I f 7lb (A) < oo for each 8 > 0, then Bn* (7 h, A, x) > 2-n for 7-ln -a. e. X E A.
Since 7-1 > 7-i > 7-(n, we deduce in particular
2-n < On* (7ln, A, x) < 1
for 7-Cn -a. e. x E A
if Nn (A) < oc.
Even in the case that X is an Euclidean space we cannot infer in general that the n-dimensional density of A is 1 at In - a.e. point of A. This in fact depends upon the rectifiability of A, compare Theorem 5 in Sec. 2.1.4
Remark 1. Consider the measures 1-tk and 7{1, k _< n, in R n. Although they can be very different, it is easily seen that 7dk (A) = 0 if and only if Hk 00 (A) = 0. Moreover, as we have seen,
lim sup r-k Hk 00 (E n Br) _> 2-kWk r-O for 7-lk-a.e. x E E, and one can show that, if Q, Qv v = 1, 2.... are compact se and if every open set A D Q contains Q for v sufficiently large, then
7{k (Q) > lim sup 7{1 (Qv) . v-oo
Notice that the last inequality is in general false for the Hausdorff measure 7-Ck. The properties of 7l"0 illustrated above sometimes make the use of 7-lOO more convenient than the use of 7{k. A consequence of the differentiation theory of measures is the following theorem
Theorem 8 (Disintegration of measures). Let p be a Radon measure on X x Y. Then there exists a measure A on X and, for A-a.e. x E X, a measure vx on Y such that for any M C X x Y we have
p(M) =
fL'(M)dA(x) X
{y E Y I (x, y) E M}. For any µ-summable function f on X x Y, the function f (x, ) is vx-summable for A-a.e. x E X and where Mx
f f dp = f f f (x, y) dvx XxY
X
(y)
da(x)
Y
Moreover, if µ(X xY) < oo, we can take. = 7r#Fp where 7r is the linear projection
rr(x, y) := x. In this case vx turns out to be a probability measure for A-a.e. x, i.e., vx (X) = 1.
1. General Measure Theory
36
2 Weak Convergence Due to its relevance for the sequel, in this section we would like to discuss weak convergence of functions and measures, with the aim of illustrating some of its main features.
2.1 Weak Convergence of Vector Valued Measures Let X be a locally compact separable metric space.
Definition 1. Let v, vk E M(X, R') be vector valued measures over X, k =
1,2,... (i)
We say that vconverge
to v, vk weaklyf
v, if
f --> (ii)
cp E C,(X,R n)
We say that vk converge to v in the mass norm , vk vk converge locally in the mass norm, vk + v, if
MU(vk-v)--p0
if M(vk - v) -+ 0;
VUCCX, U open.
(iii) We say that vk converge to v weakly with the mass iff vk M(v) < oo.
v and M(vk)
We have
Theorem 1. Let {vk} be a sequence of vector valued measures over X satisfying
supMu(vk) < oo
(1)
d U CC X
open.
k
where Mu(v) = IvI(ll) is the mass of v in U. Then there exist a subsequence {vk, } of {vk} and a vector valued measure v such that vk, - v. The converse of Theorem 1 is also valid, i.e., if vk - v then (1) holds. This is consequence of the following
Theorem 2 (Banach-Steinhaus). Let {vk} be a sequence of vector valued measures over X. If
SUP I J j < oo
V cp c CC(X,Rn)
k
then the masses of vk are locally equi-bounded, i.e.,
sup Mu(vk) < oc k
We also have
V U CC X , U open
2.1 Weak Convergence of Vector Valued Measures
37
Theorem 3. Let p and µk, k = 1, 2,..., be Radon measures over X. Then the following three statements are equivalent (i) Ak-'/1 (ii) pk(B) -* p(B) for all relatively compact Borel sets B E B0(X) satisfying
pI(aB) = 0 (izi) for all open subsets U C X u(U) < liminfk-.oopk(U), and for all compact
subsets K C X limsupk_,,,. pk(K) < µ(K). Notice that pk(B) does not converge in general to µ(B) if p(tB) # 0. For example if {xk} is a sequence in B(0,1) C RT converging to x". E 3B(0,1), we have 81k
8,_, but 8xk(B(0,1)) = 1 74 6 ,(B(0,1)) = 0
.
Notice also that in Theorem 3 (ii) we can in fact state pk(B) --> µ(B) for all open set B with p(aB) = 0. In fact every Borel subset can be decomposed as 0
0
0
B =B U(B n 8B) with p(B) = µ(B), and a BC aB. Theorem 3 does not extend immediately to vector valued measures. For example consider the sequence of signed measures
An := 81/n - 6_1/n E M(IR) . p, p = pI = 0, thus, for B = [0, 1), we have p(aB) = JAI(aB) _ 0, but A, (B) = 81/n(B) = 1, so p (B) 74 µ(B). In order to extend Theorem 3 we need a stronger condition. Obviously µn
Definition 2. We say that a sequence of vector valued measures µk E M(X, lRn)
quasi-converges to (µ, a), µk - (µ, a), where p E M(X, Rn) and a e M+(X) if and only if µk
In general a
p and µkl - a. pI and in fact by lower semicontinuity IAI << a. We have
Theorem 4. Let {µk} be a sequence of vector valued measures over X. The following two claims are equivalent (i) Pk q' (A, a) (ii) pk(B) - µ(B) and Iµkl (B) - a(B) for all relatively compact Borel sets B E 130(X) satisfying a(aB) = 0.
Notice that, just by definition, the mass is lower semi continuous with respect to the weak convergence of measures, that is,
Mu(v) < liminf Mu(vk) k-oo
if vk - v
.
Let {µk} be a sequence of vector valued measures such that Pk -s p and pk I (X) --> Jp (X) where Jp (X) < oo. Using the lower semicontinuity of the total variation on open sets we have I p I (A) < 1iim of µk (A) I k-oo
38
1. General Measure Theory
On the other hand for any closed set F we have lim SUP IIik I (F) = lim SUP (I Irk i (X) - Iuk I (X \ F) ) k-+oo
k--+oo
= I a K (X) -
likm
-oo f l /1k I (X \ F) >- l µ l (X) -III (X \ F) = I µ l (F)
Thus by Theorem 3 we infer
Proposition 1. Let {µk} be a sequence of vector valued measures which converges with the masses to y, i.e., ILk
IA,/LkI(X) --p III (X),
Iul(X) < oo
Then Ilk quasi-converges to (µ, WD-
The next remark will be used later. O
Proposition 2. Suppose that X = U' 1Fh, Fh compact, Fh+1» Fh If µk
y with the mass then Lk (cp) ---> u(cp), 1 tck I (f) -> l pl (f) for all continuous and bounded functions cp, f on X.
Proof. In fact, let Xh E CC(X, IR), 0 < Xh < 1, Xh = 0 on X\ Fh, Xh T 1. Since
limsuplµlk(X\ Fh) < IILI(X\ Fh) k-+oo
,
we have for any h lim sup (Ilk - I%v)l <_ HM SUP I(Ilk -A)((1 - Xh)'P)I < 211 W IIILI(X\ Fh) k- oo
k-oo
limsup I(lPkl - IILI)(f)I <- 211 f IIIILI(X\ Fh). k-oo Letting h --> oo the result follows.
Often it is convenient to check the condition (2)
>
f<,du>
not for all cp E C, (X, RI), but for cp in suitable subsets A of CC (X, IRm). In this respect we can state (i)
Suppose that A has the following property. For every cp E CC(X, R m) there
exists a compact set K such that for every E > 0 there exists b E A with spt 4' C K and lcp-0I < E on X. Then vk - v if (2) holds for all cc A and (ii)
supk MU(vk) < oo for all open sets U CC X, Suppose that A is dense in C,(X, IR') in the C-norm topology. If (2) holds for all cp E A and supk M(vk) < oo, then vk - v.
2.1 Weak Convergence of Vector Valued Measures
39
The weak convergence in M(X, R') is not induced in general by a metric. But we have the following. Consider for any positive constant c the bounded set in M(X, Rm)
Kc :_ {v E M(X,
)
I
M(ii)<e}.
Since Cc (X, R') is separable in the C-norm, then there exists a metric p on Kc
such that vk -s v (vk and v in Kc) if and only if p(vk, v) , 0, i.e., the weak convergence on Kc is metrizable. A metric p which does it is for instance r_2=12-e
P(µ1,µ2)
f < - f <
1+f <We,dµ1>-f
where Wj, $ = 1, 2.... is a dense set in C0(X, ][gym).
Let µ be a Radon measure on X and let p be a real number with 1 < p < oc. The space of measurable functions which are p-summable with respect to µ, i.e., for which
f
If Ipdµ < oo
is denoted by LP(X; µ). As it is well-known LP(X; µ) is a Banach space with norm
1/p
(J If I P dµ)
II f II LP (X;N,) = II f Il p
(3)
moreover Cc (X) is dense in LP (X; µ), and the dual space of LP (X; µ) is LP' (X; µ)
where p' is the dual exponent of p, P + -, = 1 if p > 1 or p' = oc if p = 1.
Definition 3. Let µ be a Radon measure on X and let f, fl, f2.... belong to LP(X; µ). (i)
We say that fk converge weakly in LP(X; µ) to f, fk -s f in LP, iff (4)
f fk9 dµ - f f
g dµ
V g E Lp (X ; µ)
We say that fk converge weakly in the sense of measures, or as measures,
to f,fk - f,iff (5)
(ii)
ffk9dµ
'
ffd
V 9 E Cc(X )
µ
We say that the function fk, k = 1, 2,..., are equi-summable or equiintegrable if for every e > 0 there exists a 8 > 0 such that
fIfkd/L<
e
holds for all k and all Borel sets B C X with µ(B) < S. We have
1. General Measure Theory
40
Theorem 5. Let p be a Radon measure over X and let fi, f2,... be functions in LP (X; p).
(i) Suppose p > 1. If supk II fk II LP(x;u) < oo, then there exist f E LP(X; p.) and a subsequence {fk;} such that fk, - f weakly in LP(X; p). (ii) Suppose p = 1. Then
f weakly in L'(X;lc) iff fk -1 f as measures and the a) We have fk fk's are equi-integrable in L'(X;{i) b) If SUN II fk IIL'(x;,t) < oo and the fk's are equi-integrable, then there exist f c Li (X, p) and a subsequence { fk; } such that f k, f weakly in Li(X; µ).
While bounded sequences { fk } in Li (X; A) converge as measures to a mea-
sure modulo passing to subsequences, they in general do not converge, even passing to subsequences, weakly in Li (X; µ), compare Sec. 1.2.2 and Sec. 1.2.4 below. Finally, concerning the equi-integrability condition we mention the following
Proposition 3. A sequence { fk} C L' (X; µ) is equi-integrable if and only if there exists a continuous function 0 : [0, oc) -> [0, oo) such that sup
f
0(I fk1) dy < oo and 0(t) It
oo as t --> oo.
In the remaining part of this section we would like to discuss in more details weak convergence of measures and of functions.
2.2 Typical Behaviours of Weakly Converging Sequences We begin by illustrating some typical features of the behaviour of sequences which are weakly, but not strongly converging.
Oscillations. Sequences of rapidly oscillating functions provide examples of weakly, but not strongly, converging sequences.
0
Consider the sequence
uk (x) := sin kx ,
x E (0, 2ir) C R , k E N
.
then, by Riemann-Lebesgue's lemma, see e.g. Proposition 1 in Sec. 1.2.3 uk u := 0 weakly in Lq(0,27r) V q > 1 but Iluk IIL4(0,2,) = c(q) > 0 so that {uk} does not converge to u in any LQ(0, 2a), q > 1, and II U IIL4(o,2ir) < likm of II Uk IIL9(0,2,)
If Uk - u weakly in LQ, q > 1, by the weak lower semicontinuity of the L9-norm we have
2.2 Typical Behaviours of Weakly Converging Sequences
41
IIuIILI <
(1)
If Uk -> u strongly in L9, we instead trivially have equality in (1), but we may as well have equality under mere weak convergence. Let uk(x) := 1 + sin kx, x E (0, 2ir), ic E N. Then Uk - u 1 weakly in L9 for all q > 1. Clearly 27r
277r
fIUkdx =
J0
0
dk
I u l dx = 27r
fIuk_uI dx = J Isinkxldx = 4 0
0
so that uk - u weakly in L', Uk 74 u strongly in L' but II uk IILI(0,27r)
--;
II U IIL1(0,27r)
Notice that instead 27r
lim inf k-.c
27r
I uk I4 dx >
J 0
Vq>1
fiu l4
,
0
because otherwise, passing to a subsequence, we would have Uk
u weakly in
LI, II Uk II Ly - 11 U II L9 , q > 1 which implies Uk --> u strongly in L9, see Sec. 1.2.3.
1 + t2 is strictly convex, for k = 1, 2, .. .
Notice also that, since 27r
2
27r
1+ukdx = fVFI+U2dx >
1+ I J ul I
=f
0
so that
21r
277r
liminfJ k--oo
1+ukdx > J
1+u2dx
0
0
Concentrations. This is for instance the way Dirac mass appears. But concentration phenomena appear also just in the context of functions.
2 Let Uk : (-1,1) C R , R be defined by uk (x) ['hen
1 k ifO<x<1/k 0
otherwise.
1. General Measure Theory
42
uk
(2)
8o
weakly as measures in (-1, 1)
.
Of course, we may smooth Uk and still have (2). Notice that uk measures in (0,1) while Uk 7 0 weakly in L1(0, 1). By considering instead vk : (-1, 1) , JR given by
0 weakly as
1-k if - 1/k < x < 0 k if 0 < x < 1/k
Vk :=
otherwise
0
we clearly have Vk -s v := 0 weakly as measures in (-1, 1) while no subsequence of {vk} does converge weakly in L'. In fact for cp := X(o,1) E L°°(-1, 1) we have 1
1
J vkcpdx=1 74 fvdxU. This shows that {vk} is not pre-compact in the weak convergence of L', though f 11 IvkI dx = 2
`d k. Notice also that we have a loss in energy: IIVIIL1
while I
<
lira inf 11 vk 11L1
k-.oc
vk41-+ooifq>1.
3 A similar in certain respect, and different for others, concentration phenomena shows also in Lq-spaces for q > 1.
Letq>1,andletuk : (-1,1)-+R be defined by u k (x) :_
We have II uk IIL9 = 1 V k, uk strongly in Lq, since the norms LP(0,1) for any p < q. In fact 11
J
Iu)
k1/9 0
if 0 < x < 1/k ot herw i se
0 weakly in Lq but {uk} does not converge concentrate. But uk --f 0 strongly in
uk
asP-1<0.
dx = k9-1 --+ 0
q
In particular the norms II uk I ILr I p < q, do not concentrate.
Distributions. Concentrated measures may distribute producing a distributed measure in the weak limit. But also functions may concentrate and produce in the limit a function. 4
Consider the Radon measures in (0, 1) C ]R given by 1
Ak
k
ai/k
4.
i=1
2.2 Typical Behaviours of Weakly Converging Sequences
43
ft = An opposite phenomenon to concentration then appears, as clearly uk L1 L (0, 1) weakly as measures, in fact for any smooth ¢ with support in (0, 1) k
1
ki=1E0() -i
f
Odx
0
Replacing each Dirac mass Ji/k with an "almost Dirac mass" a similar phenomenon may appear in the context of functions. Consider for instance the sequence of functions in L1(0, 1) 1
Uk := k
k
k2X(i/k_i/k2 ilk) i=1
Then we again have Uk - u := X(o,1) weakly as measures. This is a typical concentration-distribution phenomenon. In fact L1(spt uk)
k
so that, outside a small set the uk's are zero. One could expect then that L1(spt u) = 0, but this is not true. The uk's concentrate in sets of small measure which are almost densely distributed in such a way that the weak limit is distributed. Note also that Uk, u > 0, and 11 Uk IILI(o,1) = 11 u IILl(0,1) = 1. Notice also that the sequence {uk} is not equi-absolutely integrable, therefore by Theorem 5 in Sec. 1.2.1, {uk} (as well as any of its subsequence) does not converge weakly in
.
L1.
Nonlinearity destroys the weak convergence. If {uk} converges weakly but not strongly, then for a generic non linear function f the sequence M(uk)} U01 does not converge weakly to f (u) 5 Let uk (x) = sin kx, x E (0, 2ir) and let f (t) = t2. Then we have Uk - u := 0 in L4, q > 1 while
f (uk) =sin2 kx = 2 (1 - cos 2kx) - 2
Au) = 0 .
Notice that {uk} is equibounded in L°°. Similarly, if we let f (t) := max(0, t), we have f (uk)
or, if we let f (t) = Jtl, jukI
f (0)
1
weakly in L1(0, 27r)
Jul weakly in L1(0, 2ir).
1. General Measure Theory
44
We notice instead that, if for instance f (t) is a continuous function satisfying
0 < f (t) < t1r,
p>1
and uk -* u strongly in LP(Sl), then
f (uk) - f (U) strongly inLp(.f2). This is readily seen taking into account Egoroff theorem and the absolute continuity of Lebesgue integral.
2.3 Weak Convergence in L4, q > 1 Let Sl be an open set in R' and let 1 < q < oc. For Lebesgue summable functions weak convergence just amounts to: (i) Let uk, u E Lq(Sl). We say that uk --* u weakly in Lq iff
J uk cp dx
(1)
for all cp E
Lq'(Sl), q'
fucodx
= q/(q - 1).
(ii) Let uk,u E L°°(Q). We say that Uk -+ u weakly* in LO°, Uk
-
U, if (1)
holds for all cp E L' (Sl). (iii)
Let uk, u E L1(0). We say that a) Uk ---* u weakly in Ll if (1) holds for all cp E L°O (Sl),
b) Uk -- u as measures in ( if (1) holds for all cp E C°(Q), c) Uk - u as distributions in Q iff (1) holds for all cp E C°O(Q). Clearly
(ii)
=>
(i)
(iii)a ' (111)6
(iii),
.
As it is known the previous definitions can be subsumed to general definitions in the context of Banach spaces. Let B be a Banach space and let B* denotes its dual with duality <x*, x > _ x*(x) for x E B, x* E B*. (a) Let xk, x E B. One says that xk _ x weakly iff x* (xk) --> x* (x) V x* E B*. (b) Let xk, x* E B*. One says that x* - x* weakly* if x* (xk) --4 x* (x) V x E B. Thus the weak convergence is defined in every Banach space, while the weak* convergence is defined only on dual Banach spaces. Of course B** B, hence the weak convergence in B* implies the weak* convergence. In particular, if B is reflexive, B** = B, then weak and weak* convergence in B* (and in B) agree. Banach-Steinhaus theorem ensures that weakly (weakly*) converging sequences are equibounded in B (in B*). The norm in B (B*) is lower semicontinuous with respect to the weak (weak*) convergence. Moreover we recall that bounded
2.3 Weak Convergence in Lq, q > 1
45
sequences in a reflexive Banach space B (in the dual B* of a separable Banach space B) admit weakly (weakly*) converging subsequences. Finally let us mention Banach-Saks or Mazur lemma which states that for any e > 0, if xk - x in some Banach space B, then there exist an integer n and real al, ... , a,,, with
a1 +... + a,,, = 1 such that n
Il X - E ajxi 11 < E . i=1
A similar result is instead false in general for the weak* convergence. The definitions in the beginning, as well as many results on weak convergence of functions may then be seen as applications of the previous abstract facts to LP-spaces, taking into account that
I. If 1 < q < oo, then Lq(Q) is reflexive and (Lq(Q))* = Lq' (Q), q' = q/(q-1). II. (L1(.(2))* = L°°(Q) and (LOO (Q))* # L1(,f2).
This explains why one defines only the weak* convergence in L°°. In particular we see that we can apply Banach-Steinhaus theorem to all L1'spaces, we can infer a simple weak compactness theorem in Lq, 1 < q < cc, and
also in LO° = LI*; while a compactness theorem in L1 turns out to be more delicate.
Finally we mention that in order to have xk - x, (xk - x*) it suffices to check that supk 11 Xk II < CC (supk II xk II < oo) and z*(xk) --+ z*(x) (xk(z) x*(z)) for a dense set z* E B* (of z E B).
-i
From the previous remark or directly we can now easily deduce
Theorem 1. Let 1 < q < oc, and let uk, u E L" (D). Suppose that
supJ IukIgdx < oc k S2
then the following statements are equivalent (i) Uk u weakly in Lq(Q). (ii) Uk -s u as measures in 0. (iii) Uk -s u as distributions. (iv) For any Borel set B C ,(2 with 1 B I > 0 f-B Uk dx -1 f -B u dx. (v)
For any cube Q C .f2 fQ uk dx
fQ u dx.
Moreover one sees that a similar equivalence holds also if q = oe, replacing in (i) the weak convergence with the weak* convergence in L. We can then say that the weak convergence in L' , q > 1, is characterized by the equi-boundedness of the Lq-norms and the convergence in the sense of distributions. Condition (v) expresses precisely the intuitive idea of weak convergence (in Lq, q > 1) as convergence of the mean values.
46
1. General Measure Theory
The meaning of the weak convergence as the process of smoothing oscillations is expressed by the next proposition which extends Riemann-Lebesgue's lemma that we have already used in Sec. 1.2.2. fT (ai, bi) C Rn and let u E Lq(,fl), where 1 < q < oc.
Proposition 1. Let 1?
i=1
Extend u by periodicity from .fl to Rn and let
x E Rn, k E N.
uk (x) := u(kx) Then Uk
U:= l u dx weakly in LQ (.fl), if 1 < p < oo
uk
U:= fu dx weakly* in Loo (Q)
92
n
and uJluxdx.
Proof. Trivially we have 11 uk(x)
f
=
Next we show that (2)
f uk(x) dx
fu(x)dx=ii.
Q
Q
This of course concludes the proof in the case 1 < q < oo. Of course it suffices to show (2) in the case Q has faces parallel to those of .f2. In this case we can find a,,3 E 1n such that n
Q=a+Q0=fl(ai+Oiai,ai+Oibi) i=1
and we have ([x] = integral part of x), using the periodicity,
f(nk_I)d= J
(u(kx) - v,) dx = kn
ka+kpn
a+ 3Q
Q
kn
ka+(k3-[k3])12
([kll)' f (u(y) - u) dy + k,n n 1
kn
(u(y) - u) dy
(u(y) - u) dy +n
J ka+[ka]Q
=
1 (u(y) - u) dy
f ka+(k0- [k0] )
f k«+(k3-[k3])S7
(u(y) - u)dy
(u(y) - i) dy
2.3 Weak Convergence in Lq, q > 1
47
therefore
f(uk - i) dy
kn
f I u(y)
- uI dy
which implies (2).
In order to conclude the proof in the case q = 1, we need to show that the sequence Uk is equi-absolutely integrable, compare Sec. 1.2.1 and the next subsection. For 8 > 0 and x E (1 define vb(x) := max(min(u, 8), -8),
W'5 (X) := u(x) - vb(x).
Of course for any E > 0 we can find 8 = 8(E) such that fn I wb I dx < 2 . By periodicity we extend v6 and wa to ]R'.
J
w8 (kx) I dx
=
Twe find for k = 1, 2,
J I wb (x) I dx <
2
Therefore
f Iuk(x)Idx < f Ivs(kx)I dx + J lwb(kx)l dx < 8IEI + 2 E
E
E
and we find f Iuk(x)Idx < E E
if we choose meas E < E/28.
Notice finally that Theorem 1 does not hold for q = 1. In fact as we have seen in Sec. 1.2.2 there exist {uk} with sup II uk IILI < oo ,
uk - 0 as measures , uk f' 0 weakly in L'
k
The simple remark contained in the next proposition is often quite useful.
Proposition 2. Let 1 < q < oc, and let uk, u E L9 (Q), vk, v E Lq' (Q), q' _ q/(q - 1). Suppose that
Uk -k u weakly in Lq(Q) and vk -k v strongly in L9 (.f2). Then ukvk
uv weakly in L1(.fl).
The next proposition shows that if a weakly converging sequence in Lq, q > 1, does not converge strongly in Lq, then there must be a loss in the q-norm energy.
Proposition 3 (Radon-Riesz). Let uk,u E Lq(Q), 1 < q < oo, be such that
Uk - u weakly in Lq(Q), Then Uk -> U strongly in Lq(.Q).
II Uk IILQ((2) -> II U L9(12).
1. General Measure Theory
48
We notice that Proposition 3, is false for q = 1, compare 01 in Sec. 1.2.2. It is usually stated as a consequence of the so-called Clarkson's inequalities, which express the uniform convexity of the La-spaces for q > 1. We refer for instance to Riesz and Sz.-Nagy [556, n. 37] for a proof. u in L2(12) we have The case q = 2 is /parfticularly simple. In fact if Uk limes I
(3)
k
J
I uk 12 dx -
J
auk - ul2 dx)
=
f
I ul2 dx
u12 Iu12 weakly in L1(Q). And clearly (3) = -Iu12 + 2uku since IukI2 - I uk implies II uk - U IIL2 (O) --+ 0 under the further assumption II uk II --> 11 u II.
As corollary of Proposition 3 or by a direct proof we can also state the following weaker form of Proposition 3
Proposition 4. Let uk,u E Lq(.Q), 1 < q < oo, be such that Uk --> u
a.e. and II uk IIL(Q) - II U II L9 (n)
Then Uk --* u strongly in L4(.f2).
Proof. In fact passing to subsequences, Uk, converge weakly to some v, and v must agree a.e. with u. More directly, by Egoroff theorem we infer that uk _4 u uniformly except on an open set of small measure b. Choosing S so that by the absolute continuity of the integral
Jn
Iul9 dx < e
for sets A with meas A < 5, and observing that also fA auk I9 dx < e V k, we easily conclude.
Notice that we have in fact proved that Proposition 4 holds also for q = 1. Since the convergence in measure implies a.e. convergence passing to a subsequence, it is also easily seen that Proposition 4 remains true replacing the a.e. convergence by the convergence in measure. However, under this assumption, it does not hold anymore for q = 1. There is an analogous of (3) also for q 54 2.
Proposition 5. Let 1 < q < oo. Suppose that (4)
Uk - u a.e. in 12 and Uk
u weakly in L4(,2).
Then
kliM
(.(2).
oo(IIukIILr(57)-
2.3 Weak Convergence in L', q > 1
49
Proof. Fix e > 0 and let vk := sup(0, I Iuklq - Iuk - uNq - Iulq I - eluk - uIq). Clearly vk -4 0 a.e.. From the elementary inequality < elalq + c(e, q)Iblq
I Ia + blq - I aI q I
we also infer vk < Sup(0, I Iuklq - Iuk - ul q I + I uI q - eI uk - uIq) < (c(e, q) + 1) Iulq
Thus the dominated convergence theorem yields limk-,,,, f v' dx = 0, and we n conclude
lim sup k-boo
f
- Iuk - ulq - Iul'Idx
IIuklq
< e lim sup k--oo
f
Iuk - uI°) dx = O(e)
D 71
It is easily seen, compare the proof of Proposition 4, that, if 1 < q < oo, we can replace the assumptions (4) in Proposition 5 by (5)
u a.e.
Uk
sup II Uk IIL9(Q) < o0
,
and the conclusion still holds. However (5) does not suffices, if q = 1. For example consider for A > 0 and for k > A vk (x)
k
-k
if x E (0,1/k) if x E (-1/k, 0)
0
otherwise
and set uk(x) :_ A+vk(x). We have uk(x) --> A a.e. and II Uk IILl = II Uk - A IIL1 = 2. Finally we remark that a key ingredient in the previous statement is convexity, as we shall see better in Vol. II Ch. 2.
We conclude this subsection by restating the compactness property of Lqspaces, 1 < q < oo.
Theorem 2. Let 1 < q < oo. Then (i) If uk - u weakly in Lq(,f2) (uk - u if q = oo), then SUP II tL IIL4(Q) < oo .
(ii)
If SUP I I k
u I I La (.a) < 00
then there exist a subsequence {uk;} of {uk} and u E Lq(.f2) such that
uk; - u uki
u
weakly in Lq(s2), if q < oo weakly* in L°°(.Q), if q = oo
.
50
1. General Measure Theory
2.4 Weak Convergence in L' Most interesting for us will be the notions of weak convergence and convergence as measures in L'(fl) where Si is a bounded open set in RI. As we have seen, compare in particular Sec. 1.2.2, bounded sets in Ll (.fl) are not compact with respect to the weak convergence of L'. This is the reason for regarding L'(0) as embedded in the Banach space of Radon measures with bounded total variation in .f2
L'(Q) - M(Q)
.
Since A4 (0) is the dual space of Co ((7), we can then consider in L' (Q) the weak* convergence of M = (CO ((l))*, which is nothing else than the convergence as measures we have already considered. The important outcome is then
Theorem 1. Let {uk} be a bounded sequence in LI(Q). Then there exist a subsequence {uk; } and a Radon measure a E M (.f2) such that Uk - µ as measures in 0 or, equivalently, weakly* in M((). Next theorem, due essentially to Lebesgue and which we have already stated in Sec. 1.2.1, yields a necessary and sufficient condition for the more delicate compactness with respect to the weak-L'-convergence.
Theorem 2. Let {uk} be a sequence in LI(Q) such that (i) the Uk's are equibounded in L1(S?), supk 11 uk IIL1((l) < 00 (ii) the uk's are equi-integrable, or in other words the set functions E E C (2 are equi-absolutely continuous, i.e., for any e > 0 fE ukI dx ,
there is b > 0 such that fE Iuk I dx < e for all k provided meas E < J. Then there exists a subsequence of {uk} which converges weakly in L'(f2). Conversely, if {uk} converges weakly in L'(0), then (i) and (ii) hold true. Proof. Since step functions are dense in L°°(Q), and taking also into account
the Radon-Nikodym theorem, it is easy to see that it suffices to show that for a subsequence uk, the limit
lim f uk; dx
i
00
B
exists for all measurable set B C Q. Suppose (i) holds. From Theorem 1 we infer that there is a subsequence {uk; } and a measure p such that
fukdx --4
fd:)
E C° (Q)
Using (ii) we shall now show that for such a subsequence and all measurable B C .fl { f e uk; dx} is a Cauchy sequence.
2.4 Weak Convergence in L1
51
By Lusin theorem we can find a sequence {cph } C C° (,f2) with cOh
---+ XB
a.e. in (2 ,
IIWh IILOO(Q) < 1
In correspondence of S = b(e) in (ii) we find an open set Bb C (2 such that IB8I < 6 and {cph} converges uniformly in (2 \ Bb to XB. Now we have
f(uk: - uk,) dx
f(uki - uk,)XB dx
B
fu ki - ukj)(XB - Wh) dx B6
J (uk;-uk;)(XB-Wh)dx + <2
f B6
+
S-AB6
(uki - uk; )Wh dx
12
(Iuk I + Iuk; I) dx + Sup IXB - OhI \B6
f
f f
(uk1 - ukj) cOh dx
(Iuk I + Iuk, I) dx
12
.
S2
In dependence of e we now find some ho such that
Sup IXB- ohI < E
.n\B6
dh>ho.
Since cphc, E C°(0) for all i, j larger than some ko depending on ho and e we have
f(zLk-uk3)cohadx < 6Therefore we infer
f(uk -uk)dxi < B
This concludes the proof of the first part of the theorem. Suppose now that Uk -k u weakly in L' (Q) and assume for simplicity u = 0 (otherwise we would consider the sequence {uk - u}). Then (i) follows from Banach-Steinhaus's theorem. In order to verify (ii), we first prove the following claim:
If (ii) does not hold, then there exist a positive number z, a sequence of disjoint measurable sets Ei C .f2 and an increasing sequence of integers vi such that
Jlttvidx > E;
z
foralliEN.
52
1. General Measure Theory
By assumption, there exists an & > 0 such that, for all b > 0, we can find some set F C ,(2 with meas F < 8, and some v E N which may be taken arbitrarily large such that
fkidx > e. F
Since u, E L' (,fl), for any or > 0 there is some 77 > 0 such that
J
I
B
holds true for all measurable sets B with meas meas B < 77. Choose now o = of = e/2, b = b1 = mess ,f2; then we find 77i > 0, F1 C .f2 and v1 E N such that I dx < o1 for all B with meal B < 771. I dx > e and fB meal F1 < 51, fF, Next we choose 0'2 = z , b2 = Sl /2; then we find 772, F2 C f2 1/2 E N such that dx > e and fB Iu, 2 I dx < o2 for all B with meas B < 772. meas F2 < b2, fF2 , bi = min{ , ... , '7121 }; then Similarly for all i > 2, we choose of = e2
2 meas Fi < bi, fF; u,,, J dx > e, we find 77i > 0, Fi c .f2, vi > vi_1 such that
fB u,,, I dx < of for all B with meas B < 77i. Set now
Ei := Fi - UFq. q>i
We have
meas I U Fq I < q>i
/
meas F. < q>i
24-ti = 77i q>i
and also
J
E;
lu,,,Idx=
lu,,Idx>e-2i>E/2
J
for alli>1.
Fi\Uq>iFq
Since the Ei are disjoint, the claim is proved with z = 6/2. We now observe that replacing Ei by Ei fl {x : u,,, (x) > 0} or by Ei n {x u,,, (x) < 0} and z by z/2 we can modify the claim as follows: There exist Ei C ,f2 and vi such that
f
u,,,dxI
>z
for alliENN.
Ei
We shall finally construct a function cp E L°° (,(2) for which the sequence
f
ukcp dx
does not converge to zero. This will conclude the proof. The function cp will be defined as 1 on the union of suitable Ei and zero otherwise. We set E(1) := E1 and v(1) := vl, and we choose 6(1) so that
2.4 Weak Convergence in L'
JB
for all B with meas B < E(l)
iu (1) I dx < z/3
53
.
If fE(1) uk dx does not converge to zero, the proof is complete as we can take (p = XE(1). Otherwise we choose E(2) as the first E2 for which the remaining ones, i.e., the Ej with j > i, have total measure less than E(1) (this is possible since the Ei are disjoint) and the corresponding index vi is such that
< z/3
for all k > vi
E(1)
(this is possible since fE(1) Uk dx -> 0). Denoting by A(2) the index corresponding to E(2), we choose E(2) > 0 such that
J lu, (2) 1 dx < z/3
for all B with meas B < E(2)
B
If fE(1)uE(2) Uk dx does not converge to zero, the proof is complete. Otherwise we proceed as before and in general we find E(k), v(k), 6(kl) such that
meas U Ei < E(k-1)
J Ui>kE(')
i>k
uh dx
> z/3
for all h > v(k)
and fB I uv(k) I dx < z/3 for all B with meas B < E(k). Finally, set (p := Xuk>1E(k). Then we find U,(k)
dx
=
J
U,(k) dx +
>
,I
U,(k)
U,(k) dx +
E(k)
Ui
S2
J
dX
- 3 - 3> 3 Z
Z
J
U,(k) dx
Ui>kE(')
for all k E N
E(k)
and this gives a contradiction to the weak convergence of Uk to zero.
Often Theorem 2 is referred to as the Dunford-Pettis theorem, although it was first proved by Lebesgue [425].
It is not difficult to show that paralleling the case q > 1, in L'(0) we have
Theorem 3. Let Uk, u E L1(Q). Suppose that SUPk 11 uk IIL1(s2) < oc and {uk} is equi-integrable. Then the following statements are equivalent (i) Uk - u weakly in L'(,R) (ii) Uk -k u as measures (iii) Uk - u as distributions (iv) for any Borel set B C ,R, f-B uk dx -> f-B u dx
54
1. General Measure Theory
(v) for any cube Q C .R, fQ uk dx ---> fQ u dx.
Next proposition, though simple, is very useful
Proposition 1. Let ak, a, bk, b be measurable functions on a measurable set
0CRn. (i) If ak(x) -* a(x) a.e., ak bk
(ii)
A V k, and bk
II ak
ab weakly in L' (.(2) .
If ak -* a in L', II ak I.,n 5 A `d k, and bk ak bk
b weakly in L1(,(2), then
b weakly in L'(s7), then
ab weakly in L'(.Q).
Proof. From the weak convergence of bk to b, we deduce that {bk} is equiabsolutely continuous, i.e., for any e > 0 there exists SE > 0 such that if mess E <
bE then fE Ibk I dx < e d k. In correspondence of SE, Egoroff's theorem yields a closed set F such that meas (.(2 \ F) < bf and ak --, a uniformly in F. Thus for W E L°°(,(2) we get
f cp(ak-a)bkdx <
f Ibkldx < 2AIIcpIIe S2\F
,(2\F
fco(ak-a)bkdx ---> 0 ask---*oo. F
0 weakly in L'(Q). On the other hand, From this we conclude (ak - a)bk 0 weakly in Ll ((2). Thus (i) is proved. since a E LOO, a(bk - b) Part (ii) now follows from (i). In fact from any subsequence of {ak} we can extract a subsequence which converges a.e. to a. Part (i) then yields that any subsequence of akbk has a subsequence which converges weakly in L' to ab. Since the limit is independent from the chosen subsequence, (ii) follows.
Of course (iv) of Theorem 3 does not hold if we merely have Uk -s u as measures in 0. However one can prove, compare Sec. 1.2.1,
Theorem 4. Let uk, u E L' (Q) be such that supk 11 uk 41(fl) < oo and uk - k u as measures in ,(2. Assume moreover that there exists a function v E Ll (12) such that uk > v d k E N. Then (i) for any compact K C 12, fK u dx > lim supk-. fK uk dx, (ii) for any open set U c 12, fu u dx < lim infk-,,. fu Uk dx, (iii) for any Borel set B C .(2 with I aB I = 0, fB Uk dx ---+ fB u dx.
The second example in 2 in Sec. 1.2.2 shows that the assumption Uk > v is essential. The example in 4 in Sec. 1.2.2 shows that strict inequality may occur in (i) and (ii). As we have seen in the previous subsection Uk -i u a.e.
and
11 uk IILl(S2) - 11 U IILl(12)
2.5 Concentration: Weak Convergence of Measures
55
imply strong convergence in Ll (.rl) of Uk to u. Examples in Sec. 1.2.2 show that one cannot replace the a.e. convergence with the weak convergence in L1. However we shall prove in Proposition 1 in Vol. II Sec. 1.3.4, as consequence of a theorem due to Reshetnyak the following interesting result Proposition 2. Let Uk, u E Ll (.(l). Suppose that Uk - u as measures in IZ and
f/1+u2dx.
f/1+udx -+ n
.a
Then uk converge strongly in Ll (Si) to u.
2.5 Concentration: Weak Convergence of Measures Here we would like to show that concentration-distribution phenomena associated to weak convergence of measures are in some sense universal, and shortly discuss weak convergence of measures in all of R''2.
0 Every (concentrated) Radon measure p E M(Rn) can be obtained as weak limit (in the sense of measures) of a sequence of smooth functions with equibounded L1-norm. In fact regularizing A.c, i.e., setting
uE(x) := A*,-)
(x)
where 0,,,,(y) := En o , 0 being a standard mollifying kernel, we have uE E C°° (Rn) and Ln L uE - p weakly in M (W'). 2 Let u E C° ([0, 1] ). Then the distributed measure Ln L u is the weak limit of the concentrated measures k
/ ku
Pk
kJilk
E .M((0,1))
Similarly the sequence of functions 1
Uk
k
2 k RL 2=1
W X(k-ky k
converge in the sense of measure to u. Notice that the measures Ll L Uk are 'almost-concentrated' in fact L1(spt µk) < 1/k. If .11 is not bounded, C°(.(l) is not a Banach space and the space of Radon
measures with bounded total variation in P turns out to be the dual of the separable space
56
1. General Measure Theory
Co (f2) :_ {u E C°(Q) I V e> 0 there exists a compact K C Q
such that I u l < e in 0 \ K I
.
Independently of the boundedness or the unboundedness of .f2 one says that µ locally in M(n) 1ff µk
fWdµk ->
r J
dµ
V
EC°(Q).
n
.a
The Banach-Steinhaus theorem then says that
sup1µkI(V) < 00
(1)
VVCCf2,
k
while, as consequence of the compactness theorem, every sequence {µk} satisfying (1) has a subsequence which converges locally to some µ with locally finite total variation. Consider now a sequence of probability measures µk in IE81, i.e.,
µk?0,
fdk=1.
Then there is a subsequence µki such that µk; - µ locally in M(Rn) with µ(I[8') < 1 and, in general, µ(R'2) < 1. The following result, due to P.L. Lions, describes modulo subsequences what may happen
Theorem 1 (Concentration-compactness lemma). Let {µk} be a sequence of probability measures in R1. Then there exists a subsequence {µk;} such that one of the following three conditions holds (i) (Compactness) There exists a sequence {x,} C R such that for any e > 0
there is a radius R with the property that
µki (B(xi, R)) > 1 - e
`d i
(ii) (Vanishing) For all R > 0 we have s up
µki (B ( x, R))
0
as i -* oo
xEII2n
(iii) (Dichotomy) There exists A E (0, 1) such that for any e > 0 there is a radius
R > 0 and a sequence {xi} satisfying the following property: for any R' > R there exist non-negative measures µk; and µk; such that
0 < /4, + µk; , spt µk; C B(xi, R) , spt µk. C R' \ B(xi, R') limsup(IA-µk;(IRn)l+ (1-A)-µk.(Rn)l <e i- 00
2.5 Concentration: Weak Convergence of Measures
57
Proof. The proof is based on the notion of concentration function
Q(r)
sup p(B(x, r)) XER-
of a non-negative measure, introduced by Levy [431]. Let Qk denote the concentration function of µk. Clearly {Qk} is a sequence of non-decreasing, non-negative, bounded functions on [0, oc) with
lim Qk(R) = 1
R-oc
.
In particular {Qk} is a sequence which is locally bounded in BV(0, oc). Thus there exist a subsequence uki and a bounded, non-negative, non-decreasing function Q with
Qki(R) ---> Q(R)
as i -+ oo and for a.e. R R.
Let
A := lim Q(R).
R
Clearly 0 < A < 1. If A = 0, we have vanishing, case (ii). Suppose A = 1. Then for some Ro > 0 we have Q(Ro) > 1/2. Choose xi such that Qki (Ro) <- µki (B(xi, Ro)) +
a
Then for 0 < e < 1/2 fix R such that Q(R) > 1 - e > 1/2 and let yi satisfy Qki (R)
Akj (B(yi, R)) + 1
For large i we have
l1k<(B(yi,R))+µk<(B(yi,Ro)) > 1=/.Lki(R") It follows that for such i B(yi, R) fl B(xi, Ro)
0
.
That is B(yi, R) C B(xi, 2R + Ro), and hence 1 - E <- NJki (B(xi, 2R + Ro) for large i. This proves (i). If 0 < A < 1, given e > 0 we choose R and a sequence {xi}, depending on e and R, such that
µki (B (xi, R)) > A - e
for all i > io (e)
.
Enlarging io (e), if necessary, we can also find Ri --> oe such that
58
1. General Measure Theory
Qki (R) < Qk. (Ri) < A + e
for all i > io (e)
.
Now let µki
µk; L B(xi, R) ,
{ski := /lki L (lR' \ B(xi, Ri,))
Obviously 0 < ck. + µki < µki and, given R' > R, for large i we also have spt FLki C R"' \ B(xi, Ri) C ]lgn
spt Ak. C B(xi, R) ,
B(xi,
Finally, for i > io(e) we get
I), -A',(Wn)I+I1-A-µki (Rn) = IA - JLki(B(xi,R))I + l-Lk; (B(xi, Rj) - Al < 2e which concludes the proof.
2.6 Oscillations: Young Measures A very important and fruitful idea, due to L.C. Young in the context of Calculus
of Variations and revived more recently among others by L. Tartar, is that one can understand better oscillation phenomena by looking at graphs, or more precisely at the associated Young measures. For instance those measures will give information on the weak limit of the square of a weakly converging sequence. A similar idea is the starting point in the sequel of this monograph for studying currents associated to graphs of maps u. In fact Young measures may be seen as the first component of currents associated to graphs, compare Vol. II Ch. 5.
The main difference is that in the context of currents one regards graphs as 1-graphs, i.e., as sets which carry at (almost) all of their points tangent planes, while in our present context of measures one regards graphs just as sets.
Definition 1. Let Q be a bounded open set in Rn and let u E L°° (Q, RN). The measure Yu E M+(Q X RN) defined for each continuous and bounded function, cp : ,(lXRN --->R, cpEC°(nXRN), by
f
o dYu
nxRN
f(x,u(x))dx Q
is called the Young measure associated to u. Clearly
7r#Yu = G' L 0
if 7 : (x, y) E .fl x Rn --p x E 12 denotes the standard projection and given by 7r#A(E)
µ(E x RN) .
7r#µ is
2.6 Oscillations: Young Measures
O(x) we have
In fact for O(x) E Cb (Q) and cp(x, y)
J O(x) d7r#yu = J S2
59
7r#0 dy,
=J
W(x, u(x)) dx = f co(x) dx
S2XRN
S2XRN
.
S2
This motivates the following
Definition 2. A Young measure in Q x RN is a non-negative Radon measure µ E .M+(fl x RN) satisfying 7r#p = Ln L .fl i.e.
µ(E x RN) = L (E)
for all Borel set E C .fl
The disintegration theorem for measures then yields at once
Proposition 1. Let p E JV1+(.f2 x RN) be a Young measure. Then for Lna.e. x E 12 there exists a (Radon) probability measure vx on RN, i.e., vx > 0, vx(RN) = 1, such that for all bounded and continuous cp E C°(Q x RN) we have
(i) the map
f
W (x, y) dvx (y)
RN
is Lebesgue-measurable (ii) the following equality holds w (x, y) dµ (x, y) =
fdxJco(x)dvx() RN
f2
This way to every Young measure we can associate a collection of probability measures on RN (1)
{1-'x}a.e.xG0
.
Conversely, to every collection (1) we can clearly associate the Young measure defined by (ii), provided (i) holds. Thus one can also define Young measures as a collection (1) for which (i) holds.
Let {uk} be an equibounded sequence in L°°(.l x RN), and let yuk be the Young measures associated to uk. We have
/ = .Cn(,f2) n / 0 < yuk (D X RN) = 7r#yuk (n) Therefore for a subsequence we have yuk. - p in M+ (!2 x RN). Moreover it is easily seen that p is a Young measure too. It is often called the Young measure associated to the sequence {uk;} In fact for any open set V we have 7r#A(V) = p,(V x RN) < limt yuk. (V X ]RN) = 7r#Y,
.
(V) = Gn(V)
and for any compact K C (2 and R such that spt Yuk, C (1 x B (0, R) we have
1. General Measure Theory
60
7r#µ(K) =,u(K x RN) = µ(K x B(0, R)) > lim sup Y,,,,. (K x B (O, R)) = lim sup YU,ki (K X RN ) z-00 i-00
= lr#Yuk (K) = Ln(K). Let {vx} be the collection of probability measures associated to p, i.e. let {v,} be the disintegration of p. For any co(x, y) of the type cp(x, y) = p(x)V(y), cp E E C° (/RN ), we then get from Proposition 1(ii) C° (fl ),
il J
(x)) dx
S?
=lim
-' O J
co(x, y) dYuki (x, y)
J
fl xRf'N
= lim
00
/
P(x, y) dp(x, y) =
f(x)( fb(y) dv(y))1dx 1
1RN
We c an therefore state
Theorem 1. Let {uk} C L°°(.(2, IRN) be such that SUPk 11 uk 11,,. < oo. Suppose
that the Young measures Yuk associated to uk converge weakly Yuk -i p in M+(0 x RN). Then p is a Young measure. Moreover for any b E C' (RN) we have
o uk -
weakly* in L°° (0)
v
where for almost every x E .(2
fb(Y)dvx(Y),
v(x)
RN
{vim} being the disintegration of p.
The probability measures {vv} can be thought of as giving the limiting probability distribution as k -+ oc of the values of Uk near x. This shows that all information on the weak* limit of 0(uk), 0 E C°, are just contained in the Young measure p ' {vx} associated to the sequence {uk}: any information on {vv} yields some information about the weak* limit of 1'(uk).
0
Suppose for instance that for the limiting Young measure {vx} we know that
vu is a Dirac mass, say v. = Su(x) for a.e. x E (2. Then we can infer Uk -4 u strongly in L2(.(2). In fact for b(y) := y2 we get
fuk2dx
--->
n and for V) (y) := y, p E C°(!2)
f f dx
n
RN
Iy12dvx(y) =
fu(x)2dx
2.6 Oscillations: Young Measures
fp(x)(f y dvx (y)) dx = fp(x)u(x)dx.
fPuk dx -> f2
61
RN
S?
D
Let us discuss now oscillations and associated Young measures in the simple
casen=N=1, Q=(0,1).
Proposition 2. Let u E L°°(0,1). Extend u by periodicity, u(x + m) := u(x) for x E (0, 1), m E Z, and set
uk(x) := u(kx)
k E N, x E (0,1) .
,
Then we have YUk - L1 L (0, 1) x A E M+((0,1) X IR) where A is the measure defined by 1
fdA
(2)
:= J 0(u(x)) dx
R
E C°(R)
b'
o
Proof. For cp E C°(0,1) we have
f
f k
cO(x)V) (uk(x)) dx =
0
k
dl;
0
f
k_1
k_1 Tn+1
=k
1:
cP( /k) (u(C))
m=0 m
1
= k E f W(x km)
(u(x)) dx.
k=0 0
The result then easily follows, since k
for x E (0, 1).
Lim o co ("`k
x)
d uniformly
- fo
The next proposition allows us to compute the measure in (2) for special classes of functions u
Proposition 3. We have (i) Suppose u E C1(0,1) and u'# 0 a.e. in (0,1). Then the measure A in (1) is given by
A = £f,
f (y)
E XEu-l (y)
1
u (y)
(ii) Suppose that u E L°°(0,1) takes only a finite number of values
u(x) := E yiXA; (x)
y1,
, ys E
IIR
i=1
where Ui=1Ai is a decomposition of (0,1). Then A in (1) is given by
62
1. General Measure Theory
A= E'C' (U-' i=1
yE]R
Proof. (i) Assume for instance u increasing, u(0) = 0, u(1) = I. Changing variable, x = u(y), u o u = id, we get 1
f
0da = f(u(x))dx
J and u'(y) =
u
1
I
=
0
(y)u (y) dy
0
The result then easily follows.
(ma(y))
(ii) We have
f J
1
dA = /(u(x))dx V)
J
')(u(x)) dx
i=1 A,
0
1R
f
=
s
i=1
s
(yi)L' (Ai) = f d( G1(Ai)by ) i=1
0 Let u(x) :=x, x E (0,1) uk (x) = kx - [kx] = non integral part of kx
.
By Proposition 3 (i) we get Yu, a := G1 L (0,1)2. Notice that the disintegration of Yu, is given by {bu, (x) } while the disintegration of the limit Young measure µ is given by vv := G1 L (0, 1). As consequence of Theorem 1 we obtain uk
1/3 weakly* in L'; in fact
1
v(x)
fY2dJ:1L(0,1) _ f y2 dy 1
Notice that Uk
3
0
1/2 as f ydC1 L (0, 1) = 2.
0
3 Let u(x) := sin2-7rx, x E (0,1), so that uk(x) = sin2k-7rx in (0,1). We have G1 L (0,1) x A
Yu,
in M+((0,1) X I[8)
and by Proposition 1 (i) we have A
C1L f
1
f (y)
1
y2X(-"')(y)
In fact lu'(x) I = 27r V/1 - y2, y = sin27rx and the multiplicity is 2 if Iy
0 if M > 1. Of course
< 1 and
2.6 Oscillations: Young Measures
63
1
1
1
1-
7r_1
y2
dy = 1.
As f is even we have f y- dvv. = f y'n f (y) dy = 0, then we find again
(sin 2irkx)' -k 0
for m odd.
For zb(y) := y2 we instead get as we know sin 2 2kirx
-
J0
sin2 27rx dx
in fact 1
y
1-y 2
-1 4
dy =
sine x dx (-ir/2 7r/2)
Let 0
fa if0<x
ifa<x<1
a, b E
IR
and, as previously, uk(x) := u(kx) x E (0,1). In this case it is easily seen that yv, - Ll L (0, 1) x (o-b,, + (1 - a)bb) the disintegration of the Young measure associated to Uk is constant in x and given by vx := aSa + (1 - U)bb
As consequence, for instance, of Theorem 1 we instead have Uk
ca + (1 - U)b
weakly* in L°° (0, 1)
.
The weak* convergence of Uk takes into account the mean value distribution of the values, while the convergence of Young measures informs us about values, a and b, and weights, o- and (o- - 1), which indicate how much time the graph of Uk spends at a and b. . We conclude this subsection by stating without proof a quite general theorem on Young measures.
Theorem 2 (Ball). Let .(l be a bounded measurable set in IIR' and let K C RI be closed. Let {Uk}, Uk
:
(2 -+ RN, be a sequence of measurable maps satisfying
for any open set U D K
Ln ({xE(2I Uk(x)V U}) -> 0
ask --oo.
Then there exists a subsequence {uk;} of {Uk} and a family of positive measures {vx} on RN such that
64
1. General Measure Theory
(i) vx(RN) < 1 for a.e. x E 0 (ii) spt V, (RN) C K for a.e. x E 11 (iii) for any V E Co(RN) with zb(y) --* 0 for IyI -> oo we have
? o uk: - v
weakly* in L°°
where
v(x)
f(Y)dv(Y). RN
If moreover we assume that supk fn h(Iukl) dx < oo for some continuous non decreasing function h : (0, oc) --+ 1R with h(t) - + oc for t --+ oo (for instance, for h(t) = t, supk fn Iukl dx < oo), then vx is a probability measure, i.e., vx(RN) = 1.
Finally, if furthermore we assume that for some subset E C Q and some Ill E CO (RN) the sequence {g o uki LE} is weakly relatively compact in Ll(E), then g o ukt L E vLE weakly in L' (E) .
2.7 More on Weak Convergence in L1 A bounded sequence {uk} in L1(Q) is not weakly relatively compact in Ll(Q), if {uk} is not equi-integrable. To overcome such a difficulty we have discussed two possible ways (i)
regard L1((2) as embedded in the space of Radon signed measures M(1) and work with the weak convergence of measures
(ii)
associate to Uk its Young measure Yuk and work with the weak convergence of the measures Yuk .
Of course both ways force us to allow limits which do not belong to Ll (2) anymore.
Here we would like to mention two other possibilities which do not require to go outside L'(Q). (i) (ii)
the so called biting convergence the weak convergence in the sense of absolutely continuous parts
Definition 1. Let .(2 C R be a bounded domain and let {uk} be a bounded sequence in L'(Q) Sup jI Uk IIL1(S2) < 00k
We say that Uk converge in the biting sense to u E L'(.fl), Uk - u in 12 if for every e > 0 there exists a measurable set E C 12 such that C' (E) < e and Uk - u weakly in L1(dl \ E). The following compactness result turns out to hold
2.7 More on Weak Convergence in L1
65
Theorem 1 (Biting lemma). Let {uk} C Ll be such that supk II uk IILI(c) < oc. Then there exist a subsequence {uki} of {uk } and u E L1(Q) such that uk -6' u in Q. This means that removing (bitten) sets of possible concentrations (of small measure) we have weak convergence.
Theorem 1 can be easily inferred from the following lemma due to Acerbi and Fusco [2]
Lemma 1. Let {Oh} C L'(Q) be a bounded sequence. Then, for any E > 0 we can find a measurable set CE, a positive number 8 and an infinite set S C N such that meas (CE) < E and moreover for any measurable C in (1 \ CE with meas (C) < 8 we have
fhIdx
<E
`dh E S
C
Proof.
Suppose that the claim is not true. Then there exists e: > o such that
for every (C5, S, S) as in the statement we may choose a measurable set B with B n CE = 0, meas B < 8, and k E S such that
fkdx
>E
B
This implies that for every set C with meas C < E, an every infinite set S C N, there exists a set A with A n C = 0, meas (A U C) < e, and an infinite subset T of S such that
VkET.
P 7kIdx > E
A
We postpone the proof of the claim, and we prove that this leads to a contradiction. Set C = 0, S = N and let Al and T1 be as above. Starting from C = Al and S = T1 we pass to A2 and T2 where Al n A2 = 0 and
f lOkldx=IAldx+IIOkldx>2E A1UA2
Al
`dkET2
A2
Since meas (A1 U A2) < e, we may set C = Al U A2, S = T2 and continue with the same argument. If
N>
E_1
SUP IIcbkIILI k
after N iterations we obtain a contradiction. Finally, let us prove the claim. Let C and S be as stated, set Sl = S and take 81 <
e - measC 2
66
1. General Measure Theory
Then there exist B1 disjoint from C, with meas B1 < 61 i and k1 E S1 such that J , I'k I dx > e. By induction, setting 1
Si
Si :_ {k E Si-1
16i-1,
I
k > ki-1}
T={kiIiEN}
A=UhBh,
it is easily seen that A and T fulfill all requirements. The first example in 02 in Sec. 1.2.2 shows that Uk -k So as measures and 0. But uk L 0; in the second example we have Uk - 0 as measures and Uk the example in ® in Sec. 1.2.2 shows that
-
b
Uk
v:= 0
Uk - u := 1
and
as measures,
i.e., in general, there is no connection between the weak limit in the sense of measures and the biting limit. More information can be instead obtained if those limits agree. We state without proof one such a result due to Muller [501]
Proposition 1. Let .(2 be a bounded open set and let {uk} be a non-negative sequence such that b Uk - u as measures in .(Z and Uk -
1.
U.
Then for any compact set K C S? we have Uk - u weakly in L1 (K). The second example in [4] in Sec. 1.2.2 shows that the assumption uk > 0 is essentially necessary.
Definition 2. Let .(2 be a bounded open set, {uk} C L1(.(2), supk II uk IL1(n) < oo, and u c L' (0) . We say that ACP Uk
-
'
U
if for every subsequence {uk,} Of {uk} satisfying uki -k it as measures, where p E
M(Q), the absolutely continuous part of p (with respect to Lebesgue measure) agrees with u, i.e., pa = U. Remark 1. The rather complicated form of Definition 2 is necessary. In fact we cannot define
ACP- lim uk := (weak*- lim uk) a In this case in fact, by considering a sequence uk ACP Uk
But if Vk = 0, the sequence
0
ACP Uk
-'
.
Jo as measure we have 0.
3 Notes
67
{wk} = {u1, v1, v2, u3, v3, ...}
would not converge in the ACP sense. Instead, trivially we have wk
0.
Of course, if uk -k u E M(f2) as measures then Uk 2P µ(°') , and, if Uk u E L1(12) as measures then uk ASP
U.
Also a compactness theorem is trivial
Theorem 2. Let {uk} C L1(Q) with supk 11 uk IIL'(n) < oc. Then there exist a subsequence {uk=} of {uk} and u E L1(Q) such that uk; we can also assume that
uki -1 p E M(17)
A2.P
u in P. Moreover
as measures
with pa = U.
Notice that in the first example of 2 in Sec. 1.2.2 we have
Uk - 8o as measures in (-1, 1) uk
0
ACP Uk
0
in (-1,1) in (-1,1)
in the second example in 2 in Sec. 1.2.2 uk
Uk Uk
as measures in (-1, 1)
0
0
ASP
0
in (-1,1) in (-1,1)
and finally in the example 4 in Sec. 1.2.2 as measures in (0, 1)
Uk -1 1 Uk Uk
in (0,1)
- 0 ASP
1
in (0,1)
3 Notes 1 There are many books which deal with general measure theory, among them we mention e.g. Caratheodory [135], Federer [226], Halmos [338], Hewitt and Stromberg [375], Munroe [505], Rudin [566], Saks [570], Wheeden and Zygmund [662], see also Caratheodory [134]. Concerning Hausdorff measures the reader is referred to the original paper by Hausdorff [362], and to Rogers [564], Falconer [218], and to Federer [226]. Cantor-Vitali functions appears in Vitali [648], compare also Vitali [649]. Concerning the classical differentiability properties of Cantor-Vitali functions the reader is also referred to Darst [174] and its references.
68
1. General Measure Theory
2 The content of Sec. 1.2.1,..., Sec. 1.2.5 is quite classical. We only mention that Proposition 5 in Sec. 1.2.3 is taken from Brezis and Lieb [110], while Theorem 1 in Sec. 1.2.3 is due to P.L. Lions [435]. In connection with weak convergence we would like to mention the lecture notes by Evans [215]. 3 Concerning Young measures, a part from the original works by Young, see e.g. Young [678] [682], the reader may consult Tartar [626] where Young measures are developed as a tool for analyzing non linear partial differential equations, Ball [69] (and its references)
where Theorem 2 in Sec. 1.2.6 is proved, and Pedregal [530]. The proof of the biting lemma, in Sec. 1.2.7, can be found in Brooks and Chacon [114], compare also Acerbi and Fusco [2] Ball and Murat [72]. Finally, with reference to Proposition 2 in Sec. 1.2.4 the reader may consult e.g. Visintin [647], Amrani, Castaing, and Valadier [40], Balder [59], Kinderleher and Pedregal [412], Brezis [105].
4 Quite interesting results connecting weak convergence in the Hardy space 7{i (Jr) with a.e.-convergence, biting convergence and weak convergence in Ll can be found in Coifman, Lions, Meyer, and Semmes [156].
2. Integer Multiplicity Rectifiable Currents
It is the aim of this chapter to develop and illustrate some of the basic theory of integer multiplicity rectifiable currents. In Sec. 2.1 we shall discuss the important area and coarea formulas and some of the theory of n-rectifiable sets. After two preliminary subsections in which we collect relevant definitions and facts on multivectors, covectors, and differential
forms, we shall present, in Sec. 2.2, some of the basic theory of currents and especially of integer multiplicity rectifiable currents. Because of the relevance of those topics for the sequel we shall give in principle proofs of our claims, though sometimes we refer to later parts or elsewhere; in any case we try to illustrate our claims by simple examples.
1 Area and Coarea. Countably n-Rectifiable Sets In this section we shall present the important area and coarea formulas and some of the basic aspects of the theory of countably n-rectifiable sets.
1.1 Area and Coarea Formulas for Linear Maps Let f be a function from lE8n into RN and let A C R'. We distinguish two cases n < N. In this case we look for an explicit formula, the area formula, which expresses the n-dimensional measure of f (A), Nn (f (A)), (ii) n > N. In this case we consider the (n - N)-dimensional slices f -1(y) f1 A of A for y E f (A) and we want to relate their xn-N-measure with the total measure of A by a formula, called the coarea formula. (i)
Such formulas describe of course transformation properties of volume elements. Therefore we begin by considering first the case in which F is a linear map.
Let us recall a few facts from linear algebra. A linear map U : I[8n -p RN is said to be orthogonal if (Ux) (Uy) = x y for
allx,yER'.
A linear map S : lid' -> R" is said to be symmetric if x (Sy) = Sx y for all x,yElR .
70
2. Integer Multiplicity Rectifiable Currents
A linear map D : R' -> R' is said to be diagonal if there exist d1, ... , do E R such that Dx = (dlxl,... , dx''') for all x E IRn. The adjoint T* of a linear map T : Rn defined
-->
RN is the linear map T* : RN yE
--..
Rn
RN.
For linear maps R and T one easily verifies that T** = T, (R o T)* _
T* o R*, T* = T-1 if and only if T is orthogonal, i.e., T is orthogonal if and only if T *T = id, T* = T if and only if T is symmetric. Identifying the linear map T : Rn -> R" with an n x n- matrix, we also have
detU = ±1 if U is orthogonal. Finally, we recall that for every symmetric linear map S : Rn -+ Rn there exist an orthogonal map U : Rn -+ Rn and a diagonal map D : IRn -> Rn such that
S = UoDoU-1.
The following theorem will be useful
Theorem 1 (Polar decomposition). Let T : R' ---> RN be linear, (i) If n < N, there exist a symmetric map S : ]Rn _ Rn and an orthogonal map
U:Rn ->RN such that T = U o S, (ii) If n > N, there exist a symmetric map S : RN --+ RN and an orthogonal
map U : RN --+ Rn such that T = S o U*
Proof. The claim (ii) follows applying (i) to T*. Thus assume n < N. It is easily seen that C := T* o T : Rn -+ Rn is symmetric, and nonnegative definite, i.e., (Cx) . x > 0. Hence there exist µl, ... , µn > 0 and an orthogonal set {x1, ... , xk} in Rn such that
Cxk = /Lkxk
k = 1, ... , n .
,
Set µk = a', A k > 0. We claim that there exists an orthonormal set {z1, ... , z,.} in RN such that
k=1,...,n.
Txk = Akzk, For that it suffices to define 1
- k Txk
zk
if Ak # 0. As for Ak, At # 0 Zk - ze =
AkAe --
Txk ' Txe =
ae (CXk) - xe =
Ak AkAQ
Xk ' x2 =
Ak SPk Ai
we see that {zk Ak # 0} is orthonormal. Then, for Ak = 0, we define zk to be Rn any vector such that {z1, ... , zn} is orthonormal. We now define S : R' I
by
SXk = Akxk
k=1,...,n
1.1 Area and Coarea Formulas for Linear Maps
71
and U : R' -+ RN by
U2k = Zk Clearly S is symmetric and U orthogonal, and Uo SXk = )tkUxk = Akzk = Txk,
soT=UoS. We can now state
Theorem 2 (Area formula for linear maps). Let T : Rn -* RN be linear, and n < N. Then for any measurable set A C ItSn we have
7-ln(T(A)) = JTLn(A)
(1)
where
JT =
det (T*T)
Proof. By the polar decomposition theorem we can write
T=US and, being S symmetric,
S=VSdV* where Sd is diagonal, and U and V are orthogonal, that is isometries. Thus we deduce, taking into account Binet's formula det AB = det A det B, ,Hn(T(A))
= 7-nr(US(A)) = H(S(A)) =7cn(Sd(A))
J
Sd(A)
dx =
f(det SdI dCn =
Idet SdILn(A) = Idet S(Ln(A)
.
A
It now remain to observe that T*T = S* U* US = S*S, so that
(detS)2 = detT*T. It is usual to refer to JT as to the Jacobian of the map T and denote it also
by J(T). Notice that JT 0 0 if and only if T is injective, i.e., dim T(Rn) = n, therefore we can also write the area formula (1) as
fJTdx = f 7-l°(AnT-1(y))dfn(y)
(2) A
T(A)
A simple consequence of Theorem 2 is the following
.
72
2. Integer Multiplicity Rectifiable Currents
Theorem 3 (Change of variable formula for linear maps). Suppose that T is a linear map from R' into Rn , that for the sake of simplicity we assume invertible and let u : R' -+ R be a Lebesgue summable map. Then we have
fu(x)IdetTldx = f u(T-'y) dy
(3)
.
Rn
Rn
Proof. By writing u as the difference of its positive and negative part, we reduce to prove (3) in the case u > 0. Since in this case u is the limit of a non- decreasing
sequence of step functions, the result easily follows applying the theorem of monotone convergence and taking into account the linearity of the integral and the equality
2 XA(x) = 7-l°(A n T-'(y)) -ET-1(y)
for the characteristic function XA of A.
Suppose now n > N, and let T : Rn -+ RN be linear. In this case the Jacobian of T, denoted again JT or J(T), is defined by
JT =
(4)
det (TT*)
and we have
Theorem 4 (Coarea formula for linear maps). Let T : Rn -+ RN be lin-
ear, and n > N. Then for any Lebesgue measurable set A C Rn the mapping y E RN --: 7-ln-N(A n T-1(y)) is Lebesgue measurable and we have
fNn_N(AnT_1(y))dy = JTLn(A)
(5)
RN or equivalently
fn_N(AflT_1(y))dN(y) = f JT dx
(6) R
.
A
Proof.
If dimT(Rn) < N the claim is trivial. In fact for 1.tN-a.e. y E RN we have A n T-1(y) = 0 and consequently 7-ln-N(A n T-1(y)) = 0. Also, as T = SU* by the polar decomposition theorem, we have T(RI) = S(RN); thus dim S(RN) < N and hence JT = Idet S1 = 0. Suppose therefore dim T(R) = N. By the polar decomposition theorem we have
T = SU* where U : RN -4 Rn is orthogonal and S : RN --+ RN is symmetric; moreover det S = JT 0. By considering any orthogonal map V : Rn -+ Rn such that
1.1 Area and Coarea Formulas for Linear Maps
73
V*(X"...' x N,0,. . .,0) = U(x1,.. . ,xN) and denoting by P* : RN -> R' the linear inclusion
N) = (1
p*( 1
N
)
we also see that U* = PV, i.e., T = SPV. We now observe that for each y E RN P-1(y) as well as V-1P-1(y) and T-1(y) are the translate of the (n-N)-dimensional subspaces of 1Rn P-1(0), V-1P-1(0) and T-1(0), respectively. By Fubini's theorem we then infer that the functions of the variable y
,H--N(V
,Hn N(AnV-1P-1(y))
(A) n P-1(y))
fln N(AnT-1(y))
are Lebesgue measurable and
xn-N (V (A) n P-1(y)) dy
,C" '(A) = Ln (V (A)) RN
= fnn_N(Aflv_1P_1())dy RN
By changing variable z = Sy, the claims follow.
Notice that for n = N the area and coarea formula are just the same, and the Jacobian of T : ]Rn -* R' reduces simply to the determinant of the matrix associated to T (with respect to the standard basis of Rn). We conclude this subsection by stating a formula which expresses the Jacobian of a linear map T : Rn -> RN First assume n < N, and set I(n, N) :_ {A :_ (A1, ... , An)
I
Ai integers, 1 < Al < ... < An < N}
For each A E I (n, N) we denote by Sa the span of {eal , ... , e,\,,} and by Pa the projection of RN into Sa defined by 1
N) = (
al
Theorem 5 (Cauchy-Binet formula). Let T : Rn --> RN be a linear map (i) Ifn < N, then JT2 = (det Pa o T) 2 AEI(n,N)
(ii) Ifn > N, then
JT = JT. thus we can use (i) to compute JT.
;
74
2. Integer Multiplicity Rectifiable Currents
We postpone the proof of Theorem 5 till Sec. 2.2.1, here we only remark that,
analytically, it says that JT is the square root of the sum of the squares of the n x n-subdeterminants of the (N x n)-matrix associated to T, geometrically in view of Theorem 2, it says that the square of the 7f -measure of A equals the sum
of the squares of the ln-measures of the projections of A onto the coordinate planes, thus it may be regarded as a generalization of Pitagora's theorem to which it reduces if n = 1.
1.2 Area Formula for Lipschitz Maps In this subsection we shall generalize Theorem 2 in Sec. 2.1.1 to Lipschitz maps. Let f : Rn -* RN be a Lipschitz map. In Ch. 3 we shall see that
(A) f is almost everywhere differentiable in the classical sense with differential represented by Jacobi's matrix
af'
aft
Ofnfn
axl
Df(x) = afN
afN ..
ax'
Also the partial derivatives derivatives.
afn l
which exist a.e., agree with the distributional
(B) For every E > 0 there exist a closed set FE and a C'-map gE : 1Rn such that
f=gE,
Df=DgE
IAN
in FE
Gn (Rn \ FE) < E
II DgE IIo.,R, < cLip f for some constant c independent of E. Using these results we shall now prove
Theorem 1 (Area formula). Let f : Rn --+ R v be a Lipschitz map, n < N, and let A C Rn be a Lebesgue measurable set. Then (1)
f Jf(x)dx =
f 7H°(An f-I(y))d7-C(y) f(A)
A
where Jf (x) is the Jacobian of f (2)
In particular we have
Jf(x)
det(Df(x)*Df(x))
1.2 Area Formula for Lipschitz Maps
(f (A)) =
(3)
75
fJf(X)dx A
if f is injective on A. The function
y -i f°(An f-1(y)) = #{x I f(x) = y, x E A} =: N(f, A, y) is called the multiplicity function or the Banach indicatrix or the counting function of f over A. Notice that
f(A) = {y I N(f,A,y)
0}
Since a are Borel functions, being limits of continuous functions, the Jacobian J f (x) is a Borel function. Below we shall see that the multiplicity function and f (A) are Lebesgue-measurable. Therefore both integrals in (1) exist: Theorem 1 states equality of those integrals; in particular if one of them is finite, the other one is finite, too. A simple consequence of Theorem 1 will be
Theorem 2 (Change of variable formula). Let f : ][fin --> RN be Lipschitz, n < N, and let u : R" -* [0, +oc] be a Lebesgue measurable map. Then the map
y -' E u(x) xEf -1 (y)
is measurable and (4)
fu(s)Jf(x)dx Rn
f ( E u(x)) dx"(y) RN
xEf-1(y)
In particular, denoting by N(f, A, y) the counting function of f over A,
N(f, A, y) := #{x I f (x) = y, x E A }, for any H'-measurable function v : R' -+ [0, +oo] we have (5)
fv(f(x))Jf(x)dx = fv()N(f,A,y)dfln(y). A
Proof of Theorem 1. We split it into several steps. Step 1. f (A) is measurable. We may assume A bounded otherwise we write A as A = k Ak, Ak bounded, Ak C Ak+1, and we notice that f (A) = f (U Ak) _
f(Ak) k
76
2. Integer Multiplicity Rectifiable Currents
Let {Kh} be a non decreasing sequence of compact sets such that Ln(A) _ suphCn(Kh). Then
Gn(h Kh) = Ln(A)
h Kh C A.
,
Being f continuous, it follows that f (U Kh) = U f (Kh) is a Borel set. Now
f(A) = f(UKh) Uf(A\hKh), and, since f is Lipschitz, it has Lusin property (N)
7T(f(A\hKh)) < (Lipf)nCC(A\UKh) = 0; this shows that f (A) is measurable.
Step 2. H°(An f-1(y)) is H' -measurable and (Lip
f 1t°(An
f)n£72(A) .
RN
We construct a nondecreasing sequence {9k} of 7'ln-measurable functions converging a.e. to 7-l° (An f-1(y)); this shows that N°(An f -1(y)) is 7{n-measurable. Dividing the unit cube of Rn into 2n congruent cubes, iterating the process, and extending by periodicity such a subdivision, we consider a so-called dyadic decomposition of ]1V {Qi } with side Q2 = 2-k. Set now 00
E i=1
9k(Y)
(y)
As f (A n Qk) is 7-ln-measurable for all i, 9k is measurable for each k. Moreover we clearly have 9k (Y) < gk+1(y) Observe now 1. If f -1(y) n A = 0, then 9k(y) = 0 = 7-l°(f -1(y) n A) for all k. 2. If N° (An f -1(y)) < oo, then for k sufficiently large every point in f -1(y) n A belongs to exactly only one cube Qk, hence
9k (Y) =
7-l°(An f(y))
3. If 7-l° (A n f -1(y)) = +oo, we trivially have limk-.oo gk (y) = +00. In conclusion we find 9k (Y)
? 7-l°(An f-1(y))
for each y E RN.
Finally by the Monotone Convergence Theorem f 7-1°(A n f -1(y)) d7-ln (y)
= k i . f 9k (y)
d7-l"'
(y)
RN
RN
= k--.oo lim Tfn(f(AnQk)) < limsupj(Lip f)nfn(AnQk) k-oo
= (Lip f )n
Gn
(A)
1.2 Area Formula for Lipschitz Maps
77
Step 3. It suffices to prove (1) for Lipschitz maps which are also of class
C'. In
(B) above we infer
fact assuming (1) valid for Cl-maps and a
f J(x) dx =
J9(x) dx +
J
=
f
Jf (x) dx
A\Fe
FEnA
A
H°(FEnAn f(y))dxn(y)+ f
1
f (FEJnA)
Jf(x)dx.
A\FE
By the Monotone Convergence Theorem letting e -> 0 we then get (1), as .C (A\ FE) - 0.
Step 4. In view of Step 3 from now on we may assume f of class C'. Denote by B the Borel set, actually the open set
B := {x E JRn
I
Jf(x) > 0} .
We now claims: Let t > 1. There exists a sequence {Ej} of disjoint Borel sets such that
B = U E, ,
f Ej is invertible.
Moreover, there exist linear, symmetric, invertible maps TT ]E8n -+ Wn and Lipschitz invertible maps gj : Tj (Ej) --f f (Ej) with Lip gj < t, Lip gj- 1 < t such that :
fI E; = gj o TI
(6)
In particular (7)
t IdetTjI < Jf(x) < t' IdetTjl
b' x E Ef .
Of course we can split B as union of Borel sets Ej in such a way that f, Ej is invertible. Taking into account the continuity of D f we can also assume that the Ed's be small enough so that, setting
Tj := (Df(xj))* (Df(x5)) where xj is a point in Ej, the map gi := fl Ej 0 T7-1 is as close as we like to the identity. This proves the claim. We are now ready to conclude the proof. For that we distinguish two cases Such a claim could be proved directly for Lipschitz maps. But its proof is quite more delicate, compare Federer [226, 3.2.2], as in this case one cannot use the continuity
of Df.
78
2. Integer Multiplicity Rectifiable Currents
Jf(x) > 0}. Consider the dyadic decomposition {Qk} as in Step 2, and let t > 1. In view of Step 4 we write B = UjEj and we
Step 5. A C B := {x E R' define
I
F j :=AnE; nQk,
Gk(y) => Xf(Fk)(y)
ij
For all k the family {Fj} defines a partition of A, and the map Gk is H-measurable. Moreover X f(F j) = 1 iffy E f (Fkj), that is, if f (y) E F j, and f is invertible on F. It follows, compare Step 2,
Gk(y) T 7-l°(An f-1(y)) as k -> oo, and the Monotone Convergence Theorem yields
fGk(Y)d7-c(Y)
fflo(Aflf1(y))dflfl(y)
T
RN
RN
On the other hand
fGk(Y)dN(Y) =
j:xn(f(Fj)) 2,7
RN
and
xn(f(Fj))=xn(97(Tf(Fj)))
(Lip
9;)"xn(Tj(F
f
< tnG'(Tj(Fj))=tnIdetTj I < t2n
f
))
det7 dx
Jf (x) dx ,
F
if we take into account Theorem 2 in Sec. 2.1.1 and (7) above. Also
xn(Tj(Fj))
xn(9j 1(f(Fj)))
(I' 2p g3- 1)'i1
(f(Fkj))
< tnxn(f(Fj hence, similarly to the above,
xn(f(F )) ?
t-n71n(T
> t-2n
f
F k.
Therefore we conclude
=t-n IdetT;IGn(FZj)
f(.F ))
Jf(x) dx
.
1.2 Area Formula for Lipschitz Maps
t-2n
79
f J f (x) dx < fGk(Y)dn(Y) < t2n f Jf (x) dx A
RN
A
and the result follows first letting k --> oo, and then t -- 1. Step 6. Jf - 0 on A. In this case it suffices to check that
f fl°(An f-1(y)) dxn(y) = 0 RN
or equivalently that fn (f (A)) = 0. For E > 0 we define gE : A C Rn -> RN by (f(x),Ex)
gE(x)
and we factor f as f(x) = ir(gE(x))
where 7r(y, z) := y for y E ii /" and z E ][fin. Now we apply Step 5 to gE to infer
x'y(f (A)) =1-l'(-r(gE(A))) S (Lip 7r)n 7-i'(gE(A)) =
f
Jg. (x) dx .
A
Letting e - 0 we then get
xn(f(A)) <
fg0xx = 0 A
by the continuity of J9E (x) in x, as Jgo (x) = 0. To conclude the proof of the theorem it suffices now to split A as
A = {x E A
I
Jf(x) > 0} U {x E A I Jf(x) = 0}
and apply Step 5 and Step 6. Proof of Theorem 2. If u is the characteristic function of a measurable set (4) simply reduces to (1) as
Y XA (X) = 7-l°(An f-1(y)) x E,f -' (y)
Consequently (4) holds for all step functions, and therefore for all measurable functions u as they are monotone limits of step functions. In order to prove (5) it suffices to prove that u(x) := v(f (x)) is 7-la'-measurable. This follows from the fact that every Lipschitz map has Lusin property (N), therefore maps measurable sets into measurable sets. We shall now state a few important special cases of the area formula, using Cauchy-Binet formula
2. Integer Multiplicity Rectifiable Currents
80
1 Curves in ][8N. Let cp : JR -,. JRN be a smooth curve in RN. Then
Jw (t) = I0(t)I and bb
r[1(,p(a, b)) =
- oo < a < b < +oo ,
fdt a
if cp is a one-to-one parametrization. If f is not one-to-one the multiplicity of the covering of o((a, b)) comes into play. For example, if co (t) _ (cost, sin t) E 1[82, a = 0, and b = 2k-7r + b1i 0 < b1 < 21r, we have
k+1 if 0 < argy < bl
7-l°((a, b) n cp-1(y))
if b1 < arg y < 27r
k
2 Graphs of codimension one. Let u : I[Rn --+ II8 be a Lipschitz function and let P be a bounded open set in Rn. The graph of u over 12 is
Ju,sl = {(x, y) E Rn x R I y = u(x)I and can be regarded as the image of P under the injective map U : Rn --+ Rn x JR given by U(x) (x, u(x)) .
Then one easily computes
Ju(x) =
1 + DDu(x)
hence
xn(gu,.?) = rcn(u(P)) =
f+D2dx. S?
0 Parametric hypersurfaces. Let U : Rn --r Rn+1 be a Lipschitz, and one-toone into its image map. According to Cauchy-Binet formula Ju (x) = sum of the squares of all n x n-subdeterminants of Du(x)
.
Thus the area of the Lipschitz n-dimensional manifold u(Q), fl c l[8n, is given by
7L
=f R
r
n+1 i=1
a(ul
ui-1 ui+ ,...a,
,ufl+l) )21l/2 dx ,
11
1.2 Area Formula for Lipschitz Maps
81
Submanifolds in RN. Let M C RN be an n-dimensional embedded submanifold. Let A C M and let f : Q C Rn -+ M be a parametrization of A over 12, i.e., a smooth one-to-one map with f (.f2) = A. Denote by {g2j } the metric tensor 4
on M
Of gz;
of
axi - axj
and set
g := det (gij) Then we have
(Df)*Df = (gij) so that
volume of A in M = xn (A) =
J n
lg- dx .
5 Graphs of higher codimension. Let u : Rn --* II$N be a Lipschitz map and let f2 a bounded open set in Rn. The graph of u over .f2 is G,,, s-2
:_ {(x, y) E ,f2 X RN I y = u(x)}
and can be regarded as the image of S? under the injective map U : }R given by U(x) := (x, u(x)). Set JM(o) (Du) I
:= 1
and for k = I,-, n, n:= min(n, N) I M(k) (Du) 12
:= sum of the squares of the determinants of all k x k-submatrix of Du .
Applying Cauchy-Binet formula, compare e.g. (17) in Sec. 2.2.1, we then find n
E IM(k)(Du)j2 k=0
Consequently, n
I M(k) (Du) I2 dx S2
V
k=0
.
82
2. Integer Multiplicity Rectifiable Currents
1.3 Coarea Formula for Lipschitz Maps In this subsection we shall generalize Theorem 2 in Sec. 2.1.1 to Lipschitz maps, proving
Theorem 1 (Coarea formula). Let f : RI -+ RN be a Lipschitz map, n >_ N, and let A C Rn be a Lebesgue measurable set. Then
f Jf(x)dx = f Hn-N(An f-1(y))dfN(y)
(1)
A
IAN
where Jf(x) is the Jacobian of f
Jf(x)
(2)
detDf(x) (Df(x))*
A simple consequence of Theorem 1 will be
Theorem 2. Let f : Rn -> RN be a Lipschitz map, n > N, and let u : Rn -4 [0, +oo] be a Lebesgue measurable map. Then
f(x)Jf(x)dx
(3)
J IRN
,/
I
nd?-ln-N
dHN(y) J
f-1(y)
Before proving those results let us make a few remarks. As we saw in Sec. 2.1.2
Jf (x) is a Borel function, below we shall prove that y --> 7.In-N (An f -1(y)) is 7ln-measurable; therefore both integrals in (1) exist. Theorem 1 states equality of those integrals. Also, for every y E ]LAN f -1(y) is clearly a Borel set, thus f -1(y) is 7.ln-Nmeasurable. It is part of the claim in Theorem 2 that
fu
y --->
d7-ln-N
f -'(Y)
is 7-lN-measurable, too.
Finally notice that Theorem 1 and Theorem 2 may be regarded as a curvilinear generalization of Fubini's theorem. Proof of Theorem 1. We split it into several steps. Step 1. f (A) is 7-lN-measurable. This can be proved exactly as in Step 1 of the proof of Theorem 1 in Sec. 2.1.2. Step 2. We have (4)
fn_N(Anf_1(y))dy RN
<
Wn- Nn WN
(Lip f)Nxn(A)
1.3 Coarea Formula for Lipschitz Maps
83
For each j = 1, 2, ... we choose closed balls {Bi } such that A C UO°1B2 , diamBi < 1/j and >°_1 JB?1 < JAI + 1/j. Then we define Wn -N
(y)
C diam Bi
)fl_N Xf(B,)(y)
2
measurable, also for all y E RN we have
By Step 1 the functions 9- 1'
00
,Hn-N (A n f -1(y)) <
gi (y) i=1
Using Fatou's lemma we therefore get
f
I*
if-N(An f-1(y))dy= 1 limb ln-N(Anf-1(y))dy RN
N RN
<
inf f lim J~
00
00
gj3 dy < lim inf
RN
r
gi dy
J
RN
°° =liminfw_N 00
(diamBi )n_N ,HN 2
7= 1
(f (Bf ))
and, using the isodiametric inequality, compare Sec. 1.1.3, we infer
Hn-N(An f-1(y)) dy < RN 00
<1lm inf
Wn_N
diam Bi )
< In-NWN (Lip f)N
In
n-N
2
WN
diam f (Bi )
N
2
00
x n(Bi) i=1
Wn-NWN
Wn
(Iiip
f)N'
n(A)
Step 3. The mapping y -> xn-N(An f-1(y)) is .Cn-measurable, consequently
f Rn-N(An f-1(y))dy < In-NWN (Lip f)Nxn(A) Wn
(5)
RN
Of course (5) follows from (4) once we establish the measurability of the mapping y --r
xn-N(An f-1(y)).
(i)
If A is a compact or an open set, then y -->
map.
In-N (A
n f -1(y)) is a Borel
84
2. Integer Multiplicity Rectifiable Currents
First assume A compact. Fix t > 0, and for each positive integer i, consider the set Ui of all y E RN such that there exist finitely many open sets S1,. .. , SP such that
6Si
Anf -1(y) C
1
diam Sj
,
i
j=1 Wn-N
C
diam Sj 2
) n-N
1
j=1
We now claim that the sets U2 are open sets and 00
Hn-N (An f-1(y)) < t} =
(6)
flu. i=1
This clearly proves (i) in case A is compact. e
To prove that Ui is open, assume y E Ui, A n f -1(y) c U Sj. Then since j=1
f is continuous and A is compact e
Anf-1(z) C U Sj j=1
for all z sufficiently close to y. Let us prove (6). If 7-ln-N (A n f
(y)) < t, then for each 8 > 0
xb-N (An f-1(y))
< t.
Given i, choose 8 E (0, 1/i). Then there exist open sets Sj such that cc
Anf-1(y) C U Sj
diam S. < 8 < 1
j=1
DC
E Wn-N j=1
n-N
( diam Sj 2
<
t+1.
J
Since An f -1(y) is compact, a finite collection {S,. .. , SP} covers An f -1(y), and hence y E U. Thus 00
{y 17-Ln-N(An f-1(y)) 5 t} C n Ui . i=1
i=1
On the other hand, if y E n Ui, then for each i
fl?2 N(Anf-1(y)) < t+a
1.3 Coarea Formula for Lipschitz Maps
85
This proves (6) and therefore (i) when A is
hence 7-ln-1 (A n f-1(y)) compact.
Finally, suppose A is open. Then there exist compact sets K, C K2 C ... C A such that A = U°_1Ki. Thus, for each y E RN, 7..Cn-N(An f-1(y))
= lim 7-ln-N(Kin f-1(y)) i- 00
hence the mapping y --> 7-ln-N (A n f -1(y)) is again Borel measurable. (ii) If A is a Lebesgue measurable set, then y -+ 7-ln-N(A n f-1(y)) is Lnmeasurable. By writing A as union of an increasing sequence of bounded Lnmeasurable sets, we can reduce to the case in which Ln(A) < oc. Then there exist open sets V, D V2 D ... D A such that
lim £(V j\ A) = 0 00
2
Ln (V1) < oo
.
Now
xn-N(Vi n f-1(y))
<7ln N(An f-1(y)) +x"-N((V \ A) n.f-1(y))
and by Step 2
limsup 2-+00
J RN
j7-l"-N(V n f-1(y))
f-1(y)) I dxN(y)
< limsup wn-NWN (Lip f)n7-ln(V \A) = 0 i-00 Wn
.
Consequently
xn-N(V n
f-1(y)) -'
Hn-N(An f-1(y))
LN-a.e. Therefore (ii) follows from (i).2 Step 4. It suffices to prove (1) for Lipschitz maps which are also of class C'. In fact assuming (1) for C' maps and applying (B) in the previous subsection, we infer
f Jf(x) dx = FE nA
and passing to the limit for e
fn_N(FflAflf_1(y))dn(y), RN
--0
f J(x) dx = fn_N(uFflAflf_1(y))dn(y) 2 Notice that, since every measurable set can be exhausted apart from a zero-set by an increasing sequence of compact sets, one could slightly shorten the proof by using only Step 2 and the fact that y -. ?fin-N(An f-1(y)) is a Borel map, if A is compact.
86
2. Integer Multiplicity Rectifiable Currents
The result then follows since by Step 2 1
Ixn-N(EFenAnf-1(y))-xn-N(An f(y)) dxn(y) = 0
RN
as fn-N(A\(UF nA))=0. Step 5. In view of Step 4 from now on we may assume f of class C1. Denote by B the Borel set, actually the open set,
B := {x E R'
Jf (X) > 0}
and assume that A C B. The idea is to apply the coarea formula for linear maps. In order to do that it is convenient to introduce for any A E 1(n-N, n) the map ha : Rn --. RN x Rn-N given by
ha(x) := (f(x),PA(x)) where PA(x) is the orthogonal projection P'\ (x 1,. . ., xn) := (x Al ,...,x an).,
in particular f (x) = 7r o ha(x), if 7r : RN x R71-N -.. RN is the orthogonal projection ir(y, z) = y. Set also
Aa :_ {x E A and we observe that
A=
I
det Dha # 0}
U
A,
AEI(n-N,n)
so we may as well assume that A = Aa for some A E I(n - N, n). Accordingly, we write h instead of ha. By the local invertibility theorem, for any t > 1, we can find disjoint Borel sets {Dk} and linear isomorphisms {Sk}, Sk : ]Rn -> ]R', such that (i) ,Cn (A \ Uk Dk) = 0, 1 (ii)
hl Dk is one-to-one,
iii)
the maps 9k (x) := S 1 o h : Dk --i Rn are one-to-one with Lip 9k <_ t
Lip gk' < t ,
compare Step 4 of the proof of the area formula in Sec. 2.1.2. Since
DflDk = D(7roh) = lroSkoD9k we have
Jf = det((Df)*.Df)=det((Dgk)*.(lroSk)*.(7roSk).D9) 2
J7roSk (detDgk)
2
2 2 = J7roSk J9k
1.3 Coarea Formula for Lipschitz Maps
87
h% (f -'(y))
h;,
y
Fig. 2.1. Coarea: the map ha.
therefore by the area formula, setting
Gk := An Dk
fi
we get (7)
dx = JnoSk t (9k(Gk))
Gk
On the other hand, using the rough estimate for the area we have for any y E
tN-nxn-N(9k(Gk n f-1(y))) < ln'-N(Gk n f-1(y))
(8)
<
tn-Nxn-N
(9k(Gk n f
being Lip gk, Lip g- 1 < t. Integrating (8) in y, taking into account the coarea formula for the linear maps Tr o Sk, compare Theorem 4 in Sec. 2.1.1, using (7) and noting that by construction
9k(Gk n .f-1(y)) = 9k(Gk) n (r o'Sk)-1(y) we then get
tN-n fJf dx = tN-n JlrOSk Hn(9k(Gk)) Gk
tN-n f xn-N(9k(Gk) n (7r o Sk)-1(y)) dy RN
f xn-N
RN
(Gk n .f -1(y)) dy
2. Integer Multiplicity Rectifiable Currents
88
< to-N J hn-N(9k(Gk) n Or o SO -'(Y)) dy RN
to-N f Jf dx. Gk
Now since 7-ln (A
\ Uk 1 Gk) = 0 we can sum over k, use Step 3, and let t -- 1
to conclude that (1) holds in this case.
Step 6. It remains to consider the case in which A C {Jf = 0}, as in the general case we can write A = Al U A2 with Al C {Jf > 0} and A2 C {Jf = 0}. Fix e > 0 and define g : IR" x 1 [ 8 N
f (x) + Ey
g(x, y)
I[8N
?:W XRN _.RN , ?(x,y):=y.
,
x E IR", Y E IR v
Then Dg = (D f, EI), therefore eN < J9 < Ce. Also we observe that for all w
f 'H"-N(An f-1(y))dy= f fn-N(An f-1(y-Ew))dy RN
RN
hence
f ,Hn-N(An f-1(y))dy= RN
N B(0,1) f
f-1(y-ew))dy. RN
A x B(0,1) we have
We now claim that for y E RN, W E RN and B
Bng-1(y)n?-1(w)=S (Aof-1(y-ew))x{w} ifwEB(0,1). In fact (x, z) E B n g-1(y) n F-1(w) if and only if
xEA,
zEB(0,1),
f(x)+ez=y,
z=w
i.e., if and only if
xEA,
z=wEB(0,1),
f(x)=y-ez
or, if and only if
w E B(0,1) ,
(x, z) E (A n f -1 (y - Ew)) x {w}
.
Thus we can continue the computation above using Fubini's theorem and Step 5
1.3 Coarea Formula for Lipschitz Maps
89
f xn-N (A n f-1(y)) dy = IRN
=N
f
dw
f xn-N (B n g-1(y) n Fr-1(w)) dy RN
]RN
Nf
7-ln(Bng-1(y))dy=
1
RN
f J9dxdz B
< 7-ln (A) sup J9 < 7-LT(A) e B
and, for e -> 0, we obtain
I 7-ln-N (An f-1(y)) dy
=0=
f
Jf dx .
A
RN
Proof of Theorem 2. It is exactly the same as the proof of Theorem 2 in Sec. 2.1.2.
E 1 C1-Sard type theorem. A consequence of Theorem 1 is that
7{n-N({Jf = 0} o f-1(y)) = 0
for 7-ln-a.e. Y E
On the other hand if f is of class C', x E A, A open, is such that Jf (x) # 0, then as consequence of the implicit function theorem we can find a neighbourhood V of x such that V n f -1(y) is an (n - N)-dimensional C1-submanifold. Therefore we can state
Corollary 1 (C'-Sard type theorem). Suppose f : A C Rn -> RN is of class C1, where A is open. Then for 7-lN-a.e. y E f (A) we can decompose f -1(y) as
f -I(y) = (f -1(y) \ C) U (f -1(y) n C) where
C := {x E A
I
Jf (x) = 0} = {x
I
rank D f (x) < N}
f-1(y) \ C is an (n - N) -dimensional submanifold and
xn-N (C n f-1(y)) = 0 . Recall that Sard theorem asserts that, if f is of class Cn-N+1, then f -1(y)nC = 0
for HN-a.e. y E RN, so that f -1(y) is itself an (n - N)-dimensional
Cn-N+1
submanifold for 7-HN-a.e. y E RN.
0 Choosing f (x) := lxI in Theorem 2, we infer at once the following well-known formula for polar coordinates
90
2. Integer Multiplicity Rectifiable Currents
J0fdr 0
+00
u dx = J dr J f-1(r)
-oo
II2n
f
u d7{"i-1 = 0
u dHn-1
8B((0,r)
In particular we see that for 7-ll-a.e.. r
d(
f
B(0,r)
udx) = /
In the codimension one case, f :
1R'
r ud7-in-1
J 8B(O,r)
--> 1i8, as Jf = P f f 1, Theorem 1 yields at
once
f IDfIdx =
Jfl1({f =t})dt. -00
Rn
More generally, we have
f IDfldx = ffl-1({ft})dt
{f>t}
t
so that for 7-ll-a.e. t
df
dt
IDfI dx = -7ln 1/{f
t})
{f>t}
1.4 Rectifiable Sets and the Structure Theorem The notion of rectifiable sets we shall discuss in this section turns out to be the appropriate generalization of n-dimensional Cl-submanifolds. Roughly, nrectifiable sets are characterized by the equivalent properties of being countable union of measurable pieces of n-dimensional C1-submanifolds or of possessing approximate tangent space ?-l'-almost everywhere.
Definition 1. A set M C Rn+N is said to be countably n-rectifiable if there exist n-dimensional embedded submanifolds N1iN2,... and No C Rn+N with 7-ln(N0) = 0 such that cc
(1)
M cArouUNk. k=1
Let M be a countably n-rectifiable set which is also 'H-measurable. Then clearly we can write M as disjoint union CO
(2)
M = .MO U U Mk k=1
1.4 Rectifiable Sets and the Structure Theorem
91
where H '(MO) = 0, and each Mk is a Borel subset of an n-dimensional submanifold. If moreover W (M) < oo, we then have 00
,Hn(M) = E7-ln(1Vtk)
(3)
.
k=1
We may also and do assume that at each point x E Mk we have (compare Theorem 6 in Sec. 1.1.5)
O'(M \ Mk, x) = 0
B"(Mk, x) = 1,
.
Definition 2. A countably n-rectifiable set M is said to be n-rectifiable if and only if M is xn-measurable and fn (M) < 00, or equivalently, if it can be written apart for a null 70-set as disjoint union of Borel subsets of n-dimensional C'-submanifolds with finite xn measure. If M is countably ?in-measurable, then M is countably n-rectifiable if and only if it is a countable union of n-rectifiable sets. Notice also that an n-rectifiable
set apart for sets of small 7-ln-measure is a finite union of Borel pieces of C'submanifolds. Despite this nice structure however n-rectifiable sets can be everywhere dense.
Consider for instance a dense set {xk} in Rn+N and denote by Bn(xk, rk) the n-dimensional ball in Rn+N of radius rk and center xk. Choose rk so that
Lark <
00 '
k=1
then clearly 00
M := U Bn(xk,rk) k=1
is a dense n-rectifiable set in Rn+N Taking into account Lusin type property (B) of Sec. 2.1.2 for Lipschitz maps, and also the extension properties of Lipschitz and CI-maps, compare Ch. 3, it is not difficult to prove also that countably rectifiable sets are characterized by the property of being countable union of Lipschitz images of n-dimensional sets
Proposition 1. A set M C R'+N is countably n-rectifiable if and only if M C Ao U (k=1 fk(Ak)) where ?in(Ao) = 0 and fk : Ak -} IIBn+N are Lipschitz maps, AkCR W.
We shall now prove that rectifiable sets are characterized by their tangential properties. First let us define the approximate tangent space in terms of a blow-up procedure
2. Integer Multiplicity Rectifiable Currents
92
Definition 3. Let M C R'+N be an 7-I' -measurable set such that 7-ln (M nK) < 0o for every compact K C IIBn+N. We say that an n-dimensional subspace P of II$n+N is the approximate tangent space for M at x E Rn+N if slim
(4)
f 77T,A(M)
where ix,),(y) :=
Vf
f(y)dxn(y) = f f(y)d7-ln(y) 2
ECc(Rn+N)
P
for x, y E Rn+N, A > 0.
Of course P is unique, if it exists, and will be denoted by TIM. By definition it is characterized by the property
Hn L rlx,a(M) _ 7-ln L P
(5)
.
The condition that M has locally finite H'-measure can be removed; and it is convenient to do it. It suffices to suppose the existence of a positive locally W -integrable function 0 on M. This is equivalent to the requirement that M can be expressed as the countable union of lu-measurable sets with locally finite fn-measure. We then set
Definition 4. Let M C Rn+N be an 7dn-measurable set and let 0 be a positive locally'H' -integrable function on M. We say that an n-dimensional subspace P of Rn+N is the approximate tangent space for M at x E Rn+N with respect to 0 if
f
(6)l
rl.,a(M
f (y)0(x + Ay)d7-Cn(y) = 0(x) f
f(y) dxn(y) df E C.(i18n+N
P
Actually P does not depend on 0 in the sense that the approximate tangent spaces of M with respect to two different positive fl''-integrable functions 0 and
0 coincide fn-a.e. in M. In fact if f1 := H. L 8 and ME := {x E M I 9(x) > e} we have 7-ln (.ME n K) < oo for every compact K C Rn+N and, by Theorem 6 in
Sec. 1.1.5, 0* (p, M \ Me) = 0 for Hn-a.e. X E M. Hence for 7dn-a.e. x E ME the approximate space for M with respect to 8 coincide with TxME as defined in Definition 3 if the latter exists. Let M C Rn+N be an 7-ln-measurable and countably n-rectifiable set. Then we can write M as the disjoint union
M = Mo U U Mk k=1
where 7-C'(Mo) = 0, Mk is l'-measurable and Mk C Mk, where Nk is
an
embedded n-dimensional C'-submanifold of Let A := 07-Cn L M where B is any positive locally R'-integrable function on M, and assume without loss of generality that 7f (Nk) < oo V k. Since Mk is C1 by the differentiation theorem IIBn+N.
we have
1.4 Rectifiable Sets and the Structure Theorem Nl(Akrkn B(B(xx, p
Bn(P,
k,x) = P0 Hn
fl
P)))
= 9(x)
93
7ln a.e. X E Mk
while, by Theorem 6 in Sec. 1.1.5,
9"n(µ,Nk \.Mk,x) = 0 7-l''-a.e. x E Mk 9*n(lc, M \ Mk, x) = 0 7-ln'-a.e. x E Mk From this we easily infer that for a.e. x E Mk and for every f E CC(1Rn+N)
f
f (y) 0(x + Ay) dx"(y)
0
f(y)9(x+Ay)d'h'(y)
0
Thr,A(M\Mk)
f
Th,,A(Nk\Mk)
as A - 0, that is, by Lebesgue theorem,
o
f = 9(x) f
f f (y) 9(x + Ay) dy = 77r,A(M)
o
f (y) 9(x + Ay) dy
77 ,,A(Nk)
TxNk
f (y) dy
as Nk is of class C1. This shows that every 71n-measurable and countably nrectifiable set M admits 7-1' -a. e. on M approximate tangent plane with respect to any positive locally 7-ln-integrable function 0 on M, moreover
TxM = TxNk
(7)
7-ln-a.e. x E Mk
.
The converse is also true and we can state
Theorem 1. Suppose M is 1-ln-measurable. Then M is countably n-rectifiable if and only if there is a positive locally 7-ln-integrable function 0 on M with respect to which the approximate tangent space exists for 7-ln-a.e. x E M. Theorem 1 characterizes countably n-rectifiable sets as sets with good tangential properties: an approximate tangent plane exists at 7-fn a.e. point. We remark that countably n-rectifiable sets can be "fractured". We need to prove the "if' part. For future purposes we shall derive it from a slightly more general result on the rectifiability of Radon measures, applied to the measure µ := 97 C L M. Definition 4 uses the measure 7-ln L 0, but we can easily extend it to any Radon measure
Definition 5. Let y be a Radon measure on ]Rn+N, and for x c Rn+N and A > 0 let flx,a be the measure given by
94
2. Integer Multiplicity Rectifiable Currents
µ.,a(A) = )-nµ(x + AA)
(8)
.
We say that the n-dimensional subspace P C Rn+N is the approximate tangent space for µ at x with multiplicity 9(x) E (0, +oo) if
a o f f (y) d,4.,,\ (y) = 0(x) J f (y) dxn (y)
(9)
P
Clearly 0 E P, and P and O(x) are unique, if they exist. For any Radon measure µ we denote by M. the set of points in Rn+N for which there exists the approximate tangent space for jL at x with multiplicity O(x) E (0, oc).
we have
Then
Theorem 2 (Rectifiability theorem for
Radon measures). Assume that µ has an approximate tangent plane for y-a. e. x E IRn+N with multiplicity 0(x) E (0, oc), i.e.
,U(rn\Mµ)=0. Then M. is countably n-rectifiable, 9 is H -measurable, and p = O7-ln L Mµ,
Proof. Choose any ball B(0, R). For any x E B(0, R) and any bounded Borel
set A we have
(N- L (IRn+N \ B(0, R)))x,a (A) = 0
provided A is sufficiently small. From this we easily deduce that
Mµ n B(0, R) _ .M/,
L B(O,R)
Therefore replacing u by p L B(0, R) we may and do assume that p(Rn+N) < co. Given any N-dimensional subspace 17 C R,,+N and any a E (0,1) let ply denote the orthogonal projection of IRn+N onto 17 and Ca (I7, x) denote the cone
0(17,x) = {yE1R.'n+N I
Ipir(y -
x)I > aly-xI}.
1.4 Rectifiable Sets and the Structure Theorem
95
having "axis" H. Also, recalling that there is a bijection between N-dimensional subspaces and projectors, define the distance between two N-dimensional subspaces 17, IF as the norm of the difference of their projectors dist (17,17') = sup I P17 (x) - P11' (x) I 111=1
This way we obtain a compact metric space. For s e N choose now 0, > 0 and a Borel subset F C 1[Bn+N such that µ(Rn+N \ F)
(10)
µ(Rn+N)
s
and such that for each x E F, µ has an approximate tangent space PP at x with multiplicity 9(x) > 9 . In particular, we therefore have lim P-0
(11)
µ(B(x, p))
> B3
lim
µ(C1/2 (Pj , x) n B(x, p))
P-.0
WnP n
Wn pn
= 0.
Fori=1,2,... andx EFdefine µ(B(x, p))
inf
O
4i(x)
Wnpn
µ(C1 /
sup
,
x) n B(x , P))
WnPn
O
Then
lim fi(x) > 9,
(12)
i- Oc
,
lim qi(x) = 0
VxEF
2-+00
and by Egoroff's Theorem we can choose a µ-measurable set E C F with µ(F \ E) < s /,r,(l[Bn+N)
(13)
and with (12) holding uniformly in E. Therefore for each e > 0 we find S > 0 such that (14)µ(B(x, p)) > 0, - e, µ(C1/2(Py , x) n B(x, p)) WnPn
WnPn
<eifxEE, 0
According to the compactness property of the metric space of N-planes, we now choose a finite number of N-dimensional planes 171 i ... , Hk in Rn+N such that for each N-dimensional plane 17 there is a j E {1, ... , k} such that dist (17, 11j) < 1/16, and let El, ... , Ek be the subsets of E defined by
Ej := {x E E I d(IIj, Pte) < 1/16}
.
Clearly E = Uj=1Ej. We now claim that for a = 0,/16n and 6 so that (14) holds we have
96
(15)
2. Integer Multiplicity Rectifiable Currents
C3/4(Ilj,x) nE; n BS12(x) = {x}
V x E Ej, j = 1,...,k
.
In fact otherwise we find x E Ej and Y E C3/4(J1 , x) n E; n aB(x, p) for some 0 < p < 6/2. From (14) we then have u(C112(PP ,x) nB(x,2p)) < ewn,(2p)' and, since
we have B(y, p/8) C C112 (Py , x) n B(x, 2p) and, again by (14) A(C1/2 (P2 , x) n B(x, 2p)) >_ A(B(y, pl8)) ? (BS - E)wn ( 8)
.
This gives 63 < (1 + 8')e, a contradiction, since E = O /16n. From (15) we also see, taking into account the extension theorem for Lipschitz maps (compare Theorem 1 in Sec. 3.1.3) that for any fixed x0 E Ej Ej n B(xo, S/2) C q(graph f )
(16)
where q is an orthogonal transformation of Rn+N with q(17;) RN and where f = (f 1 .. f N) is Lipschitz. In fact we may assume xo = (0, 0) E Rn X RN, 113 = {0} x RN. Let A denote the projection of Ei nB(xo, 6/4) on Rn x {0}, then for any (x, 0) E A we claim there is a unique (x, y) E Ej n B((0, 0), 6/2). In fact suppose that (xi, yi) E Ej n B((0, 0), S/2), i = 1, 2, then from (15) we infer (x2, y2) 0 C314 (11j, (x1, y1)), hence 3
Y2-Y11 < 4 (Ix2 - X 1
+ Iy2 - y, 12)1/2 <
3 (Ix2
4
- x1I + Iy2 - yll)
i.e.
Iy2-Y1I < 3Ix1-x21. This yields at once uniqueness, and shows that the map
(x, 0) E A -> (x, y) E E; n B((0, 0), 5/2) is Lipschitz. Extending such a map to a Lipschitz map defined on R x {0} D A we then easily get (16). From the above we conclude that E is an n-rectifiable set and
n+N \ E) < 2 /l(R n+N) This shows that
1.4 Rectifiable Sets and the Structure Theorem
97
00
MI, = MoU U.Mk k=1
with µ(Mo) = 0 and Mk C JVk, dVk being n-dimensional Cl-submanifolds of 1Rn+N Since for any x in MN, the existence of the approximate tangent plane implies that (17)
0(x) =
lo
p))
=: Bn(Il,x)
we have
for p - a.e. x E MN, . 0' (µ, x) > 0 Therefore on account of Theorem 5 in Sec. 1.1.5, we infer 7-C(Mo) = 0, hence Mµ is 7ln-measurable and countably n-rectifiable. As B*n(ii, x) is a Borel function, we also infer from (17) that 0(x) is K"-measurable on M. Finally (17) yields
µ(B(x,p))
lim
PoHn(Nk rB(x, p))
6(x)
which in turn implies µ = O7-Cn L M, by the differentiation theorem of measures in Sec. 1.1.5.
We conclude this subsection by stating without proof the important and difficult Besicovitch-Federer structure theorem for purely unrectifiable sets. Let A be an arbitrary set in 1Rn+N which can be written as countable union of sets of finite Nn-measure. Consider the collection of all countably n-rectifiable subsets of A ordered by inclusion. Since the union of families of countably nrectifiable sets in A is again a countably n-rectifiable set, by Hausdorff maximal principle there exists a maximal element R, and we have
A = RUP where R is countably n-rectifiable and P contains no countably n-rectifiable subset of positive 7-ln-measure.
Definition 6. A set P which contains no countably rectifiable subset of positive 7C'-measure is called purely n-unrectifiable.
Purely unrectifiable sets Q of Rn+N are characterized by the fact that they have 7-ln-null projection via almost all orthogonal projection onto n-dimensional subspaces of Rn+N Of course also purely unrectifiable sets Q can be written as countable union of sets of finite 7{'-measure. An example of a purely 1-unrectifiable set in R2 is given by C x C where C is the Cantor set in Sec. 1.1.3, of dimension 1/2. In fact, as we have seen, in this case,
0 < 7-ll(CxC) < oo. Therefore if M C C x C is 1-rectifiable we should have either 7{1(7r(M)) > 0 or 7{1(F(M)) > 0, being it and the projections of ]R x IR into the first and second factor. A contradiction as
98
2. Integer Multiplicity Rectifiable Currents
7 '(ir(M)) <_ x1(C) = 0 ,
x1(ir(M)) <_ x1(C)
= 0.
Theorem 3 (Besicovitch-Federer structure theorem). Let Q be a purely n-unrectifiable subset of ]n+N with 00
Q = UQk,
(Th(Qk) < 00
Vk.
k=1
Then 7-C (p(Q)) = 0 for almost (in the sense of Haar measure for 0(n + N, n)) all orthogonal projections of PJ +N onto n-dimensional subspaces of Rn+N, P E
0(n+N,n).
Notice that only the purely n-unrectifiable subsets could possibly have the null projection property stated in Theorem 3. An immediate consequence of Theorem 3 is
Theorem 4 (Rectifiability theorem). Let A be an arbitrary subset of Rn+N which can be written as countable union cc
A = U Ak
with 7-ln(Ak) < 00
Vk
.
k=1
Suppose that every subset B C A with positive 'H' -measure has the property that
fn(p(B)) > 0 for a set p E 0(n + N, n) of positive Haar measure. Then A is countably n-rectifiable.
Of course the 7-lk-density at lk-a.e. point x of a k-rectifiable set is 1 while it is zero o*k(Hk,.M,x) = 1
1
for Hk-a.e. x E M
0
for Hk-a.e. x 0 M
A natural question is to ask for which sets A of finite 71k-measure k non necessarily integer, there exists for 7-lk-a.e. x E A ok (Hk,
A, x) = ok (Hk, A, x) = B*k (7-lk A, x)
The following theorem gives a complete characterization of such sets
Theorem 5 (Preiss). Let A C Rn, with fk(A) < oo. Then Bk(7-lk, A, x)
exists for fk-a.e. x E A
if and only if k is integer and A is k-rectifiable.
1.5 The General Area and Coarea Formulas
99
1.5 The General Area and Coarea Formulas In this last subsection we would like to state the area and coarea formulas for
Lipschitz maps f : M -* RN, where M is an lu'-measurable and countably n-rectifiable set.
Recall that an n-dimensional C', r > 1, submanifold of 1Rm, m > n, is a set M c RI such that for any y c M there are open sets U and V in Rm with y E U, 0 E V and a C""-diffeomorphism 0: U-* V with 0(y) = 0 and
o(M n U) = V n ]Rn where 1Rn = {(xl, ... , xm) E Rm local representations of M
:VnR' ->Rm, (indeed
I
xn+1 =
... = x' = 0}. In this case we have
v)(VnRT)=MnU,
0(0)=y,
_ ¢ VrR-) such that
ax
(o), ... , ax (°)
are linearly independent vectors in Rm which form a basis of the tangent space
TyM of M at y. The tangent space TM of M at y is the subspace of iRm consisting of those -r E JRm such that r = 'y(0) for some C1 curve y M, y(0) = Y.
Let f : M -> RN be a Cl function on M. The directional derivative DTf E RN in Y, T E TyM is defined by
DTf
dtf('Y(t))It=o
for any Cl curve y : (-1,1) -> M with y(0) = y, ,(o) = T, and it turns out to be independent of the particular curve y. The induced linear map dfy : TM -> RN defined by
dfy(T) := DTf
,
T E TyM
yields the tangent map. For scalar functions of class C', f : M -- ]R, the directional derivatives can be represented in terms of the gradient V' f (y) defined as the element in TyM such that DTf `drETTM. If r1, ... , r,,, is an orthonormal basis of TM, the gradient is also given by the formula n
VMf(y) = E(DTkf)rk k=1
If f is a C1 function defined in a neighbourhood U of M with f = f on M, the gradient VM f is related to the ordinary gradient of f by
100
2. Integer Multiplicity Rectifiable Currents
V M.f (y) = P(V f (y)) where P : Rm --i TyM is the orthogonal projection onto TyM.
The divergence of a vector field X = (X',. .. , X"n) : M similarly
ii
is defined
rn
> ek (VMXc)
div MX
k=1
where e 1, ... , e,n is the standard basis of R. It is easily seen that n
divMX =
7-k
k=1
where r 1 , . .. , Tn
is any orthonormal basis for T. M. We have
Theorem 1 (The divergence theorem). Let M be such that the closure M is a smooth compact manifold with boundary aM = M \ M. If Xy E TyM for each y E M, i.e., X iss a tangent vector field, then
J M
div MX
dW = - JX. rl
dHn-1
am
where rl is the inward pointing unit co-normal of aM, i.e., 1771 = 1, rl is tangent to M, 77 is normal to aM and points into M. In general, i.e., without any assumption on 3M, we have
Jd1VMX = 0 M
for any tangent vector field X such that spt X n M CC M. The mean curvature of a C2 submanifold M C R'n is expressed in terms of its second fundamental form. For y E M we denote by Tl...... n an orthonormal basis of TyM and by vi, ... , V _n any orthonormal basis of (TyM)-L, the orthogonal complement of .,y, that is, vl, ... , v,,,,_n is a basis of the normal space of M at y. The second fundamental form of M at y is the bilinear form
By : TM x TyM --* (TyM)' defined by
m-n B. (-r, rl)
(rl - DTva(y)) va(y) a=1
It is not difficult to infer
Proposition 1. We have
1.5 The General Area and Coarea Formulas (i) Let -y
101
: (-1, 1) -> M be a C2 curve satisfying -y(O) = y, 'y(0) = r, TI = 1.
Then
By(rr,r) = (y(0))1 where v-1- denotes the orthogonal projection of v onto (TM)J-.
: U -+ M is a C2 map
(ii) Let U be a neighbourhood of (0, 0) E R2 and let
with 0(0, 0) = y and
a (0,0) = r'
a
Then
r
By (r, ) =
(
(°'0) - ''
a2 atlat2
(0, 0)
11 I
.
The vector (%y(0))± in (i) is the norm al component of the curvature of -y; in other words By (T, r) measures the normal curvature of M in the direction r. From (ii) it follows that By is a symmetric bilinear form with values in (TyM)1. The mean curvature vector field H of M at y is the trace of By
H(y)
By ("rk, "rk) E (TyM)1 k=1
or equivalently
m-n n
H(y) :=
E E(rk - DTkva(y)) va(y) - U=1 k=1
so that
H :_ -
m-n (div Mva) va a=1
In terms of the mean curvature vector H one can write the divergence theorem for generic vector fields X not necessarily tangent to M. Theorem 2. Let M be as in Theorem 1. Let X : M -> R m be any smooth vector field. Decompose X into its tangent and normal part m-n
X = XT+X1,
X1 = a=1
Then we ha ve
divX1 = and
Xy
f div MX d7-l" = - f X . H d7-C -
M
M
aM
d7ln-1
102
2. Integer Multiplicity Rectifiable Currents
Without any assumption on aM and provided sptX n M cc M, we have
fdivMXd7e2 = _ f X H d7-ln . M
M
Consider now an 7-1n-measurable and countably n-rectifiable set M C R", n < m. As we have seen M can be written as
M = MODUMk k=1
where H (Mo) = 0, Mk is ?in-measurable for all k, and each .Mk lies in some n-dimensional C' submanifold Nk, Mk C Nk. Suppose f : U - R be a locally Lipschitz map from a neighbourhood of M into R. Of course for any k fjvk is also locally Lipschitz, hence 'H'-a. e. differentiable (compare (A) of Sec. 2.1.2 and is differentiable Ch. 3), also it is easy to check that at all points x where affine space L := x + T 14 at the point x, we have fj L is differentiable on the we can set VNk f (x). Thus and grad f L (x) =
Definition 1. The gradient off relative to M is defined for 7fn a.e. x E Mk by
VMf(x) := VNxf(x)
.
We explicitly note that V--"f(x) exists and belongs to T,,M for 7fn-a.e. x E M, and, up to a set of 7{n-measure zero, is independent of the particular decomposition M = UkMk used in the definition. This is easily seen as TxNk = TM for 7ln a.e. x E Mk and T,M is independent of the particular decomposition. For 7-ln-a.e. point in M for which TxM and VM f (x) exist we can then define
the linear map
d'Mfx:TTM- R
by
r ETxM .
dMfx(T) = If f : M --> RN is a Lipschitz map we then set N
dMfx(r) := E(T . VMfe) ee s=1
where e1,.. . , eN is the standard basis in RN, and we define the Jacobian JM (x) for 1f -a.e. x E M as the Jacobian of the linear map dM fx : TxM RN, i.e.
JM(x)
(det (d'p'i fx)*(dM fx))1/2
JM(x)
(det (d'Afx)(dM fx)*)1/2
Often we shall also use the notation
if n < N if n > N .
2 Currents
J. (d' f (x))
JM(x)
for
103
.
Taking into account the approximation property of Lipschitz maps (compare (B) of Sec. 2.1.2) and the fact that sets of zero measure are mapped into sets of zero measure by Lipschitz maps it is not difficult to show that the area and coarea formulas hold for Lipschitz maps f : M -> 1R N, i.e.,
Theorem 3. Let M be an 7-ln-measurable and countably n-rectifiable subset of R', and let A C M. be an H'-measurable set. Then for any non-negative 7-Lnmeasurable function u : M -> R we have
fu(x)J(x)d7-C(x) =
J
ud7-l° I d7-Cn(y) (f-'(y ,I )nA /
if n < N, and
f
udl1 dNN(y) fu(x)Jr(x)dW(x) = f \ RN f-1(y)nA ifn>N. We finally notice that from the decomposition M = UkMk, together with the CI-Sard type theorem in Sec. 2.1.3 and the approximation property (B) in Sec. 2.1.2 it follows
Proposition 2. Let M be an 7-ln-measurable and countably n-rectifiable subset of R'+' and let f : M -. ISBN be a Lipschitz map. Then the slices M n f -1(y) are countably (n - N) -rectifiable subsets of JRt for 7{n -a. e. y E RN.
2 Currents In this section we deal with the general notion of currents and in particular of integer multiplicity rectifiable currents. We do not aim of course to completeness but only to discuss basic definitions and results which are relevant for the sequel. In particular not all proofs are included, and sometimes they are postponed to later chapters.
After the first two subsections on multivectors, covectors and differential forms, where we also fix some notation which will be used later on, basic definitions and results, as the important closure-compactness theorem for integer multiplicity rectifiable currents, are presented in Sec. 2.2.3 and Sec. 2.2.4. In Sec. 2.2.4 we also present some elementary examples, while further developments are discussed in Sec. 2.2.5, Sec. 2.2.6 and Sec. 2.2.7.
2. Integer Multiplicity Rectifiable Currents
104
2.1 Multivectors and Covectors Vectors of a finite dimensional vector space V may be multiplied giving rise to new objects called multivectors; for v1, ... , vk E V the result
C = vln...AVk is called a k-vector. But before describing such a procedure, let us first fix some notations. Usually we shall identify V with Rn with the standard basis e1,. .. , en; its dual V* will be again identified with IRn with dual basis e1, ... , en, so that
<ei,e-1 > = bi
.
We shall use the standard notations for ordered multi-indices
1(k, n) :_ {a = (al, ... ak)
(1)
and for convenience we set
I
ai integers, 1 < al < ... < ak < n}
I(0,n) = {0}
and
if a E I(k,n) . IaI = k If a E 1(k, n), k = 0,1, ... , n, we denote by a the element in 1(n - k, n) which complements a in {1, 2, ... , n} in the natural increasing order; of course we have
a = a, 0 = (1,... , n). If lal =n-1 and 10I = 1, we shall often write i instead of a, i = 1,... , n, and j instead of /3, j = 1,... , n, and i for 1.... ,n). Often in the sequel we shall have to consider maps from Rn into RN. We shall
denote the standard basis of RN by El, ... , EN, so that (el, ... , en, El, ... EN) is the standard basis of Rn x RN; with respect to those bases the coordinates , xn, yl, _ (x, y), and in order to stress this will be denoted by (xi, Y )
we often write 1l
x Ry instead of R x RN.
RN be a linear map, identified, with respect to the standard Let G : Rn basis with the N x n matrix
G=
i = 1,...,N , j = 1,...,n
and let (2)
n := min(n, N)
Given two ordered multi-indices with 1/3 = k, 1 < k < n, la = n - k, we shall denote by (3)
Gea
the k x k-submatrix of G with rows determinant will be denoted by (4)
M« (G)
/3k) and columns (al, ... , ak). Its
det Ga ;
we shall set (5)
MOD (G) := 1
2.1 Multivectors and Covectors
105
k-vectors. Let V be a finite dimensional vectors space over 1R, with basis el,... , en. For k, 1 < k < n, generic k vectors can be multiplied by means of the wedge product 1;
:= v1A ... /nvk
obtaining a new object. This wedge product is characterized by the following properties (i)
it is multilinear vin ... A(avi + bwi) A ... nvk = av1A ... twin... AVk + bv1A ... Awin ... nvk
(ii) it is alternating
vln ... Av2A ... AV3 A ... AVk = -VlA ... AV3 A ... AV A ... AVk
equivalently
v1 A ... nvk = 0 if vi = vi for some i 0 j or
v1A ... nvk = 0 if v1, ... , vk are linearly dependent. In terms of the basis e1,. .. , en, computation of l; yields an object of the form
= E `e« oGI (k,n)
where 6' E 118, and ea is in { ea = eat A ... Ael,_
I
a C 1(k, n) }
.
The set of all linear combinations of the e,,,, a c I(k, n), is the space of k-vectors, denoted AkV . It is a vector space of dimension (k). Every k-vector v1A... AVk defines a k-linear and alternating map with domain the product of k copies of V*, the dual space of V vin ... /nvk
:
V* x ... x V* --> 1[8
defined by
(vin...Avk)(wl,...,wk)
det(<w',vj >)
,
wi EV*
In fact it turns out that AkV can be identified with the linear space of all k-linear
and alternating maps on V*. More precisely, by considering the k-linear and alternating forms e, a E I(k,n) defined by (6)
e"'(wl, ... , w k )
det (wa) ,
O E V*
106
2. Integer Multiplicity Rectifiable Currents
we have for any Q = ()31, ... , Ok) E I (k, n), e1, ... , en denoting the dual basis of V* ea(eQ
...
SO,,
SO := det(Sai)
,
and the ea defined this way form a basis of the linear space of all k-linear alternating forms on V*. This shows that Ak V is defined intrinsically, i.e., independently of any fixed basis.
k-covectors. They are defined similarly to k-vectors replacing V by its dual V*. The linear space of k-covectors is denoted by AkV := AkV*, and is identified with the linear space of k-linear and alternating forms on V, with basis ea, a E I(k, n), given by (7)
ea(v1, ... , vk) := det (va')
,
v1,. .. , vk E V .
Exterior product of multivectors. Given two multivectors e E AkV
,
77 E
AhV, h, k > 0, h + k < n, e _ EaEI(k,n) Saea,'7 = >QEI(h,n),70eQ we define the exterior product of 6 and 77 by (8)
6A77 := E E a77QeaneQ cEI(k,n) /3E1(h,n)
where eaneQ by (i) and (ii) is given by eaAeQ
J0
if ano #0
l a(a,(3)eau0 ifaf1Q=0
where a U Q E 1(h + k, n) is the multi-index obtained reordering the multi-index (a, Q) and a(a, /3)
is the sign of the permutation which reorders (a, 0) in the natural increasing order, and v(0, 0) = 1. This way eA77 E Ah+kV. In order to give a meaning to (8) also in the case h + k > n we set eaneQ = 0
forh+k>n.
It is not difficult to see that the exterior product defined above has the following properties: Let a, 5 E AkV, /3 E AhV, -y E ASV, and c E R. Then (i) (a + S)A/3 = (aA/3) + (SAO) , c (aA/3) = (ca)n/ = aA (co) aA/3 = (-1)hk$Aa
(ii) (iii) (iv) (v)
OA(a + S) = (OAa) + (/3A5)
(aA/3)A'Y = aA(OA-y)
If v1, ... , vk E V _ A1V, and e1, ... , k E V*, then
v1A...Avk(e1,...,Ck) = det()
2.1 Multivectors and Covectors
107
Properties (i),...,(v) in fact characterize the notion of exterior product, so that it is an intrinsic notion independent of any fixed basis. Of course one can proceed similarly in order to define the exterior product of covectors. In this case (v) reads: f o r v1, ... , vk E V we have (9)
C'A...A k(U1i...,vk) = det()
.
Simple k-vectors. A k-vector
E AkV is called simple or decomposable if it can be written as a single wedge product of vectors, = Vi A ... AVk for some V 1 ,- .. , vk E V. Of course An,V - J1 and all n-vectors are simple as they are multiple of the base vector e1A ... Aen. A1V e_- V, hence all 1-vectors (vectors) are simple. The
two-vector := e1Ae2 + e3Ae4
provides an example of a non decomposable 2-vector in A2R4. Assume in fact it
is simple = v1Av2, then n = 0, i.e. e1Ae2Ae3Ae4 + e3Ae4Ae1Ae2 = 0
and, as
e1Ae2Ae3Ae4 = (_1)2 e3Ae4Ae1Ae2
we infer e1Ae2Ae3Ae4 = 0: a contradiction, since e1Ae2Ae3Ae4 is a basis of A4]R4.
Simple k-vectors play a fundamental role since they represent oriented kplanes in V, as we shall see below, and they are one of the main reason to introduce the space of k-vectors. In fact in many respect it is convenient to work with a linear space instead of working with simple k-vectors which do not form a linear space.
Duality between AkV and AkV. Covectors and multivectors are in duality. Given _ ,aEI(k,n) Sae( E AkV, W = EaEI(k,n) waea e AkV the duality between AkV and AkV is defined by (10)
<e,w> := E Eawa aEI (k,n)
It is easily seen that <, > does not depend on the fixed basis. In fact one can show that on simple n-vectors we have
= w(vl,...,vk) and a linear form on AkV is uniquely identified by its values on simple vectors. In particular, from (7) and (11) we infer that the duality product between a simple k-vector and a simple k-covector is given by
2. Integer Multiplicity Rectifiable Currents
108
.
= det() where < v2, w2 > is the duality between V and V*. The duality above of course allows to identify canonically, i.e., independently from the choice of the bases, (AkV)* and AkV* =: AkV. Finally notice that, since the k-vectors ea are simple, a k-vector e is zero if w> = 0 for all simple k-covectors w. The inner product of multivectors. Suppose that V is endowed with a inner product v w, and let e1, ... en be an orthonormal basis. We can then define an inner product in AkV (respectively in AkV) in such a way that ea (respectively ea) be an orthonormal basis of AkV (AkV) by setting
E arla
(12)
w u :_ E waOa aEI(k,n)
aEI(k,n)
where
= EaEI(k,n) Caea, 77 = EaEI(k,n) 17ae. and w = EaEI(k,n) waea, o =
EaEI(k,n) aaea
The inner product defined this way actually depends only on the scalar product in V and not on the fixed orthonormal basis 61, ... , en, and it is invariant for orthogonal changes in V. To see that we consider the duality map
R : V --* V* ,
=v-w Vv,
w -- Rw,
and we define R# AkV --* AkV by setting R#S(viA...AVk) = C(Rv1i...,Rvk)
Since Rei = e, el, ... , en being the dual base in V*, we easily see that
R#ea = ea . As by definition R# depends only on the inner product in V, (13)
C-
77 =
R#'1> ,
and the duality product is independent of the coordinate system, we see that also the inner product is independent of the coordinate system. Given now two simple k-vectors v1 A ... AVk, w1 A ... AWk from (13) we com-
pute
v1A...nvk w1A...nwk = = R# (wl A ... nwk) (vl, ... , Vk) = (W1A ... nwk) (Rvl, ...
= det () = det (v2 wj) hence (14)
(v1A... nvk) (win... nwk) = det (v2 wj)
,
RVk)
2.1 Multivectors and Covectors
109
Consequently the modulus of a simple vector is given by (15)
v1A... AVk
l
=
det (vi vj)
Let us interpret geometrically such a formula in the case V = R. We associate to the simple k-vector v1A... AVk the linear map
T:Rk -* R' ,
Tw := Ewivi.
Such a map is represented by the matrix with columns the components of the vectors v1i...,Vk
T :_ (v1 ...
I vk),
I
i.e., vi := Tei, and the matrix (vi vj) is exactly T*T. Therefore we see that
Iv1A...Avk1 = ITeln...ATekI = detT*T
(16)
One can interpret v1 A ... nvk I as the k-dimensional measure of the k-parallele-
piped generated by the vectors v1,...,vk,i.e. Iv1A... AvkI
= Hk(T[0, 1]k))
In fact from the area formula
hk(T[0, 1]k)) = .JTfk([0,1]k) = detT*T = v1A... nvk Also, if we write
v1A... AVk = E l;aea aEI(k,n)
we have
Sa = (v1A ... Avk)(eal , ... , elk) = det (< vi, ea' >) = det (v'')
and (v'') is the matrix obtained from T choosing the rows al, ... , ak. Hence (17)
detTT* = det(vivj) = IviA...Avk12 = E (ta)2 aEI (k,n)
(detva')2 aEI(k,n)
This proves Cauchy-Binet formula in Sec. 2.1.1. In fact in terms of the projections
Pa : l[8n -> Rk
Pav = (val, ... , vak )
the matrix (v'') is nothing else than Pa o T, hence (17) yields
T*T = E (det (Pa o T)) 2 aEI(k,n)
.
110
2. Integer Multiplicity Rectifiable Currents
Usually we shall work with the Euclidean norm in Rn and the induced norm )1/2
Iei = (
(S-)2 IaI=k
in the class of k-vector t; E AkRn. However in connection with the calculus of variations it is convenient and natural in many instances to consider the so-called mass and comass norms on multivectors and covectors. Here we confine ourselves
to give just the definitions and mention a property, compare Federer [226] and Vol. II Ch. 1. The comass norm of w E AkV is defined by (18)
II w II
:= sup{
E AkV, ICI < 1, e simple}
I
.
Of course
Ilwll<_Iwl. The mass norm of e E AkV is defined by (19)
IVII
sup{<1=, w> I w E AkV, II W II < 1} ,
and of course
11 0 ?
161
.
One can also show that for some c = c(n, k) one has
IIwII ? 1 IwI
,
II
II
<_
cICI
.
However in general
11wII < wl,
Iei
and we have:
II = For example for the 2-vector ICI = vr2.
wl
if and only if 1; is simple if and only if w is simple
:= eine2 + e3Ae4 in A2 R4 we have II e II = 2,
Simple k-vectors and oriented k-planes. Let us come now to one of the main motivation to introduce k-vectors. We shall show that, similarly to vectors which identify oriented one-dimensional lines, unit simple k-vectors identify the oriented k-dimensional planes in V. We shall assume V endowed with an inner product. First we observe that k vectors v1, ... , Vk are linearly dependent if and only if v1A... AVk = 0.
2.1 Multivectors and Covectors
111
In fact if V1, ... , vk are linearly dependent we know that vin ... nvk = 0. Suppose
on the contrary that vi, ... , vk are linearly independent. Then we can complete them in order to form a basis of V and consider k elements w1, ... , wk of the dual basis, so that < vi, wj > = S2 . We then find
= det() = 1 hence v1 A ...
nvk 0 0. Let P be an oriented k-dimensional subspace of V and let r1, ... , Tk be a positively oriented orthonormal basis of P. We then set bP := T1A...ATk . If v1i ... , vk is another basis of P then we have k
vi =
z=l ...,k
aijTj j=1
and it turns out that v1A...AVk = detA7-1A...ATk as one can easily verify checking on simple covectors. Hence, if vi, ... , Vk is another oriented orthonormal basis (so that det A = 1, being det A > 0) we have
P = v1 A ... hvk . This shows that SP is uniquely determined by the oriented k-plane P. In general, if v1i ... , vk is any oriented basis of P, then v1 A ... AVk
P = Iv1A ... AvkI Conversely, let
E AkV be a simple non-zero k-vector. Set
Pg := {v E V I eAv = 0} . Of course PP is a linear subspace of V. The dimension of Pg is k. In fact, being C simple, = v1 A ... nvk, and C 0 0 implies that v1, ... , vk are linearly independent; moreover, the observation above says that v E PP if and only if v is a linear combination of v1, ... , vk. It follows that
PP = span Orienting P in such a way that v1, ... , Vk be positively oriented we then see that APE _
1
2. Integer Multiplicity Rectifiable Currents
112
This proves our claim in the beginning. We would like now to read the previous characterization of simple vectors in a somewhat more algebraic way. Let C _ Eaea E AkV be a simple vector with C 74- 0. Of course a 0 for some a and by a suitable choice of the basis we may assume that a = (1, ... , k). This way we may reduce to a Cartesian situation. As such a situation is especially relevant for us in the sequel, we shall discuss it with some details.
Simple n-vectors in the Cartesian product Rn x RN. With the notation above every n-vector in A,,,Rn+N can be written as
_E
aQ eane,3
Ial+IR1
where a and Q are ordered multi-indices. We shall refer to Caa as to the components of and to Co' as to the 00-components or the first component of C.
Denoting for any k, 0 < k < n := min(n, N) by Vn,k the linear sub-
space of AnRn+N given by the linear combinations of n-vectors of the type v1A... Avn-knw1A... Awk with vi E R', wi E ]IAN, so that Vn,k := An-kin ® AkRN
(20)
,
the n-vector e can be uniquely decomposed as n
_ E (k)
(21)
k=0
where e(k) E Vn,k is given by
as eaAEg
(k) _ HI=k
in particular c(o) _ X00 e1, ... /den .
Similarly every n-covector splits as n
w,,,3 eaAE,3 _ EW(k)
W=
W(k) E Vn,k
=
An-kRn ®AkRN
k=0
la+Ql=n
and the duality <, > reads as
<W, > _ T WaQ aQ Ia+J01=n,
=
<W (k) , (k) > k
C <W(k), C> = <W, e(k) > = <w(k), e(k) >
,
2.1 Multivectors and Covectors
113
Let G := (G)21 ,...,n, i=1,...,N be the N x n-matrix associated to the linear transformation that we still denote by G from R' into RN. The vectors
i=1,...,n
ei+Gei,
yields a basis of the tangent plane to the graph of G in Rn+N. The unit simple n-vector
M(G) M(G)
(22)
where
M(G) := (el + Gel)A ... A(e,,, + Ge,,,)
(23)
identifies the plane graph of G, and in fact orients such a plane. It will be then called the tangent n-vector to the graph cG of G. Notice that if we denote by id m G the map id x G : RI -> Rn+N given by
(id >a G)(x) = (x, Gx)
(24) then
9,3 :_ (id >a G)(lRn)
.
Notice also that l: depends only on the orientation of R and not on the special fixed basis. Conversely, if e is a unit simple n-vector with positive first component, boo > 0, then 1; represents an n-plane with no vertical vector, and therefore there exists
a linear map L : Rn -> RN such that
_ M(L) M(L) More precisely we have
Proposition 1. Let V be an n-dimensional subspace of Rn+N, i.e., an n-plane in 118'+N through the origin oriented by the unitary n-vector V
V=
VaQ eaneQ
,
E (V' )z = 1 Iad+1131=n
laI+IRI=n
and let 7rv be the restriction to V of the orthogonal projection 7r to R'. We have (i) V contains vertical vectors, i.e., non zero vectors z such that 7r(z) = 0, if and only if V00 = 0. (ii) If V00 > 0, then there exists a linear map L : R' -- RN such that V = range (id >a L) and
V
I M(L) I
,
M(L)
(ei + Lel)A ... A(en + Len)
Moreover -7rv is the inverse of id >a L : R' --+ V.
2. Integer Multiplicity Rectifiable Currents
114
(iii) If J., denotes the Jacobian determinant of lrv, then
J,1 =
Voo
Proof. Suppose that V contains at least a vertical vector z1. Let (z1, ... , zn) be a basis for V. We split each zi as vi + wi, where vi E R and wi E RN. Since zl is vertical we have vi = 0, and, as z1A... nzn is a multiple of V
V = kzjA...Azn = k(vl+wl)A...A(vn+wn) Being V00 = k det (v1, ... , vn), we therefore deduce V°° = 0. Suppose V does not contain vertical vectors. Choose a basis z1, . . . , zn in V and split as previously each zi as zi = vi+wi. Clearly the n vectors vi are linearly independent, otherwise we could find Ai such that E Aivi = 0 and therefore > Aizi = > Aiwi would be
vertical. Thus V00 := kdet (v1, ... ,v) # 0. This proves (i). Define the linear map L : R' --, RN by Lvi = wi, where zi = vi + wi is a basis for V. Clearly, vi +Lvi = zi thus the range of id x L is V, moreover irv is the inverse of id x L. Finally, M(L) _ z1A ... nzn zln ... /nzn M(L) I
zjA...Azn Iz1A...AznI
) = 1 Moo( L
oo
>0
we therefore conclude that V = M(L)/IM(L)1. This proves (ii). If V00 > 0, being lrv the inverse of id x L, the Jacobian determinant of 7rV is the inverse of the Jacobian determinant of id < L, thus
_
1
1
IM(L)I
Jidr
=
Voo
If V00 = 0, then 7rv(V) is a lower dimensional subspace of F", thus J,( = 0. In both cases V00 > 0 and V°° = 0, we therefore have J, = V°°. This proves (iii) and concludes the proof of the proposition. In coordinates it is not difficult to compute, compare (31) (33) below, (25)
M(G) _
a) Ma (G) eaAEO
where recall M9 (G) = det Ga denotes the determinant of the (/, a)-minor of G Ga = JEa and MO'(G) = 1. In particular we have M(o) (G) = e1A ... Aen (26)
M(l)(G) _ E(-1)n-ZG2eine, M(k) (G) _
Q(a, a)Ma (G)eaAe'R InI+I
2.1 Multivectors and Covectors
115
and in terms of the components l;aa of := M(G)/CM(G)(
boo _
1
M(G)j (27)
X19
=
CU O
o (a, a)Ma
X00 =
(-1
boo
In conclusion we can then state E Anon+IV with
Proposition 2. An n-vector
non-zero first component is
simple if and only if
/
X00
v(a, a)Ma I
((-1)"-Z X00)
We also see that the map A ,mn+Iv
i ces} M : { N x nmatr -
is injective, and that the map
M(i) : {N x n-matrices} --> Vn,l :=
An-lRn ®Allf81`'
yields an isometry of linear spaces. This allows us to say that El :_ {l= E A,Rn+N
I
t; = M(G) for some N x n-matrix G}
is the graph of an algebraic function over Vn,1.
M(Gg
Fig. 2.3. The map M.
2. Integer Multiplicity Rectifiable Currents
116
For each t; E A.,,R,+N with t; 00 > 0, we can associate to C the N x n-matrix Ge defined by (28)
Ge
l
M(1)(\X00) .
Then t; -- Gg is "linear" and GCI El = M-1, i.e.,
GM(G) = G for all N x n-matrices G and l; = M(Ge) if and only if a is simple and
E Al where COO
Al := {l; E AI TRn+N
= 1} .
Also
GC = 0 if and only if e(1) = 0 .
In terms of Gg Proposition 2 above reads: E fl j n+N with 600 > 0 is simple if and only if M(Ge) (29) Finally we notice that, as dim An]IPn+N = and simple n-vectors can and by 660, simple n-vectors form a "small" be parametrized by e(1) nN) -nN-1 manifold of dimension nN+1; in other words, there are (n conditions for simplicity.
Induced linear transformations. With any linear map L : V -> W between two finite dimensional vector spaces V and W, we can associate the linear maps AkW defined by AkL : AkV
AkL(V1A... Avk) = Lv1A... ALvk
(30)
.
One can show that the family of linear maps AkL is characterized by the following two properties (i) A1L = L (ii)
Ah+kL(aAQ) = AhLaAAkL/.3 ,
a E AhV, 9 E AkV
With respect to bases e1,... , en in V and E1, ... , EN in W, the map L can be Accordingly we then find represented by a matrix (aj') so that Lei = >Z AkLea
ca 6,3 0
where (31)
c« = det (aa')
i.e. the matrix associated to AkL has as entries the k-order minors of the matrix A associated to L. In fact, denoting by e', ... , EN the dual basis of E1, ... , EN we have
2.1 Multivectors and Covectors cO = (AkL)(ea)(en'..., eOk) _ (Leaf A ... ALeak) (EQl
117
... 16 Yk )
= (EQ' A ... AeOk) . (Leaf A ... ALeak ) = det (<eQ=, Lea, >) = det (<ea=, aa, ei >) = det (aaj ) and, by (16)
AkL = det T*T. It is also readily seen that, if = v1 A. - - Avk is a simple vector and P the
(32)
subspace generated by V1, ... , vk, then AkL(e) is again a simple vector, and, in fact it is the k-vector associated to the plane L(P), if L has maximal rank or 0. Returning to n-vectors in A,,Rn+N and linear maps G : Rn --4 RN, we can then write
M(G) = (e1 + Gel)A... A(e, + Gen) = An-(id x G)(eiA ... Aen)
.
The computations to prove (31), then yield a proof of the decomposition (33)
M(G) _ E u(a, a) Ma (G) eanep
.
laI+IAI=n
In fact computing M(G) we get terms of the type
... vein... AGejA... Aehn ... AGekn ... reordering such terms then we get
M(G) = Ea(a,a)ean(A1 G)(e5)
(34)
a
that by (31) is just (33). Similarly, there is a linear induced map A k L: AkW --> AkV characterized by the properties
(i) A1L=L*, L*:W-+V (ii)
Ah+kL(WAS) = AhL(w)AAkL(b) , w E AhW, S E AkV
and again defined by AkL(W1A...AWk)
= L*wln...AL*wk
.
One easily checks that AkL is the dual map of AkL, i.e., (35)
<e,AkLw>
V 6 E AkV, w E AkW
.
118
2. Integer Multiplicity Rectifiable Currents
and (36)
jjAkLjJ = IAkLHH
If G : W --+ Z is another linear map, we have
Ak(G o L) = AkG o AkL .
(37)
Equalities (35) and (37) express the so-called Binet formulas , compare Vol. II Ch. 2, which in the case V = W = Z and k = dim V amount respectively to det L = det LT
det GL = det G det L
,
as one can see using (31).
2.2 Differential Forms Let X be an n-dimensional manifold. A k-vector field C on X is a map
I;:xEX -+ e(x)EAkTXM. Similarly, a k-covector field on X is a map
w:xEX -+ w(x) EAkTXMf-- Ak(TXM)* . A k-covector field is also called a (differential) k-form. In fact it can be generated by applying the exterior differential d to functions and using the wedge product. For the sake of simplicity here we shall only consider k-forms on an open set
U of R', so that w is a map
w:UCRn -+ AkR')we shall discuss the general situation in Ch. 5. Exterior differentiation. As usual, let e1, ... , e,z be the standard basis in Rn. We denote by dxi the 1-form on Rn defined as the constant map (Jn)* dx2:xERn --+ e'EA ell ... , en being the dual basis of Rn, so that dxi(x) = xi. For any smooth function f : U -+ R its differential is defined as the 1-form df : U --+ A'R' n
df : x E U
E ff+ (x)dxi E A i=1
Of course df does not depend of the fixed basis, in fact for any smooth vector
field :xEU--+ 1;(x)
we have
< (x),df,> = =
. dtf(x+tC(x))jt=o
2.2 Differential Forms
119
Notice also that the definition of dxi is consistent with the definition of df ; in fact if f (x) := xk, then n
dfy = > ffi (x) dx2 = bkie2 = ek i=1
For any ordered multi-index a E I(k, n) the k-form
dxa:xEU -f eaEAkRn can be written as
dxa = dxa' A ... Adxak
,
and every k-form w : U C IEFn -+ Ak1n as w(x) = Eal_k wa(x) dxa; so that
w = E wa dxa lal=k
where the product of a k-form w by a function is defined by
(fw)(x) := f(x)w(x), and more generally the exterior product of two forms by
(wAS)(x) := w(x)A5(x)
.
The exterior derivative of a k-form, k > 1, w = k I=k wa dxa with smooth coefficients wa (x) is defined by
dwandxa _
dw
wax= dx''Adxa Ial=k i=1,...,n
lal=k
This way (i) df agrees with the ordinary differential for C' function f (ii) d(w + 77) = dw + dry for all k -forms w and 77 (iii) d(whr7) = dwA'i + (-1)kwndi] if w has degree k (iv) d(dw) = 0, for any form of class C2.
Notice that (iii) generalizes the ordinary differentiation rule for the product of functions, and (iv) expresses in case w is a function the equality of mixed derivatives
DiDj f = DjDi f . Conversely one can easily show that the exterior derivative operator d is characterized by properties (i)...(iv). This in particular implies that dw is intrinsically defined, independently of the coordinate system.
2. Integer Multiplicity Rectifiable Currents
120
The space of infinitely differentiable and compactly supported k-forms in U is defined by wadxa wa E C,°(U)} D"(U) I
lal=k
Similarly we define wadxa
£k(U)
I
wa E Coo(U)}
lal=k
and we speak of rn-differentiable k-forms. We may then regard the exterior derivative as an operator
d : £k (U) -> £k+l(U)
.
Of course it increases of one the degree of a form and decreases of one its regularity property.
Pull-back. Let U C R and V C RN, and let f : U -* V be a smooth map. To every k-form w = > wQ dyQ E Dk (V) we can associate the form f #w E £k (U), called the pull-back of w and defined by
f#W = E w,o of d fa
(1)
1131=k
where
dfQ :=
dfQ'A...Adf,Qk
dfQh =
n
fR! dxi
.
It is not difficult to show that for any x E U we have (2)
f#w(x) = Akdff(w(f(x))
and
f#(dyalA... Ady3k) = df01A...
Adf,ok
so that pulling back is an intrinsic operation, and f:# enjoys the following properties (i)
(ii) (iii) (iv)
f#(w+77) = f#(w)+f#(77), f#(wAr7) = f#(w)nf#(,7),
f#(dg) = d(g o f ), if g : V --> 1R, d(f#w) = f # (dw) if f E C2.
2.2 Differential Forms
121
Differential forms in a Cartesian product. Later we shall be interested in
differential r-forms in U :_ 0 x RN C 112' x RN. As for r-vectors and r-covectors, we can write every r-form in Dr(U) as
w=
Wap dX'Ady" I+1O1=r
I
or
min(r,N) W(k) 1: k=max(O,r-n)
w = where
E
W(k) :_
w,,o dx' Ady 3
.
I-I+IR1=*
PI=k
Integration of differential forms. Let M be a smooth, say C1, k-dimensional manifold that for the sake of simplicity we assume to be embedded in R. Let (Ui, Oi) define local charts in M and cpi : V C Rk --p m f1 Ui, oi = ?Pi 1, be a local parametrization of M with parameter variables u. The tangent space of M at cpi (u) is given by
Tw,(u)M := span {D1cc (u), ... , Dkco (u)} and the unit k-vector
S
D1cpi(u)A... ADkcpi(u) Dlcpi(u)A... ADkco (u) I
!L E Ui
Recall that two parametrizations yield the
defines an orientation of
same orientation if the Jacobian determinant of the transformation which reduces one chart to the other is positive. In this case we have
Dico (u)A... ADkwi(u) Dicpi(u)A... ADk
D1cpj (u)A... ADkcpj(u) I Di.Wj (u)A... ADkWj(u)
for cpi(u) = cOj(u)-
If one can choose charts or parametrizations in such a way that 1; (cpi (u)) defines
a smooth unitary tangent vector field on M, one says that M is an oriented manifold with orientation l;. Suppose now w is a smooth k-form on M (or in a neighbourhood of M in ), and suppose spt w C Ui, then classically the integral of w on M is defined as the integral of the k-form cp#w on Rk
Jw :=
(3)
M given by
Jco w
V
2. Integer Multiplicity Rectifiable Currents
122
f
f<w(u)eiA.. . Aek > dfk(u)
w
M
.
V
One easily computes
=w(co (u))(Dco (u)e1h...ADcpi(u)ek) = = ADkWi(u) (u)) > JW, (u) = (u)),
(4)
In the general case, that is if sptw is not contained in a chart Ui, and M is oriented by 6 E A,, TM, one shows by means of a decomposition of unity {i
that
f w = E ftiw) ,
}
cPi#(eiA ... /yen)
a'2 V
M
is well defined. According to the computation in (4) and taking into account the area formula in Sec. 2.1, J. (u) d7-lk (u) = d7-lk L M, we then find
fw
(5)
=
M
f<e(x)w(x)>d(x). M
This shows that a k-form is a natural object to integrate over an oriented, kdimensional rectifiable set, because it is sensitive only to both the location x E M
and the tangent plane to M at x. Given a k-rectifiable set M C R'ti, we say that M is oriented by Ak]R
(i) (ii)
M ->
if
is Hk-measurable, for 7 1k-a.e. x 1e(x) I = 1 and fi(x) is associated to TIM, in the sense that is the wedge product of k vectors of a basis of TIM.
In this case we define the integral of the k-form w on the oriented rectifiable set M simply by (5). Of special relevance for us will be the case in which M is the graph say of a smooth map u : (1 C R' --, RN. In this case we easily infer from the above (6)
f w = J(idu)#w = f <w(x, u(x)), M(Du(x)) > dx . S2
0
2.3 Currents: Basic Facts Let U C R' be an open set, 0 < k _< n. We denote by Dk (U) the space of all infinitely differentiable and compactly supported k-forms in U topologized by the usual topology which is characterized by the assertion that
2.3 Currents: Basic Facts
w' dxa -p W
wa dxa ,
123
i --* 00
aEI(k,n)
aEI(k,n)
if there is a fixed compact K C U such that (i) (ii)
sptw"CKVaEI(k,n) and V i lim DRw' = DQwa `d a E I(k, n) and every multi-index 0.
Definition 1. A k-dimensional current in U is a continuous linear functional on Dk(U). The space of k-dimensional currents in U is denoted by Dk(U). Given T E Dk(U), we can define for any a E I(k, n) the components of T (with respect to standard basis e1, .
, en in Rn) by setting for any cp E C°O (U)
. .
T'(co) := T(Vdxa) Clearly with a certain abuse of language we have
T = E Taea IaI=k
where we regard ea as the operator
ea (E wQ dxO ) := Wa ea(dxo) = ba = aa . 5a2 ... yak or equivalently
Ta(wa)
T(w)
w=
if
wa dxa . aEI(k,n)
aEI(k,n)
From this point of view a current is just a collection of ordinary distributions indexed by a E I(k, n). Accordingly we have
Definition 2. Let Th,T E Dk(U). We say that the sequence {Th} converges weakly to T, Th - T, if Th(W) -i T(w) for all w E Dk(U). The first notion which distinguish currents from ordinary distributions is that of boundary. It is simply defined as the dual operation of the exterior derivative.
Definition 3. The boundary of T E Dk(U) is defined as the (k - 1) -current (1)
We set
8T(77) := T(dr7)
8T = 0
if
V
rl E Dk-1(U) .
T E Do (U)
.
2. Integer Multiplicity Rectifiable Currents
124
We shall see that the boundary operator
: Dk(U) -f Dk-1(U)
8
comprises two apparently different concepts
(a) that of derivative of a function (b) that of boundary of submanifolds. Before discussing the important subclass of currents with components which are measures in U, let us make a few general remarks. As for distributions the support of a current T E Dk(U) is defined by
sptT = n{K C U
I
K relatively closed in U, T(w) = 0 for all w E Dk (U) with spt w c U \ K} .
One can define the Cartesian product of currents. Let Ti E Dki (Ui), Ui C Rni,
i = 1,2. The current T := T1 X T2 E Dk1+k2 (Ul X U2) is defined as follows. If w E Dk1+k2 (U) has the form
w = E wap (x, y) dxa Adya ICI=k1
=k2
then we set
T(w)
(2)
= Ti I
If Ial
k1i
J(3(
w«R(x, y) dy3
T2 (
\ jal=k1
Ial=k2
k2, IaI + IQI = kl + k2, and
I
/
dxa)
E C°°(U), we set
T(cpdxandy,3) = 0.
(3)
One easily shows that
O(Ti x T2) _ &T1 x T2 + (-1)k1T1 x 0T2
(4)
.
As a special case consider T E Dk (U), U C Rn, and the 1-current Q (0,1) l defined as integration of 1-forms on the oriented segment (0, 1) of R. Let w E Dk+1([0,1] X U). Split w into its horizontal and vertical parts, w = WH +wv where
WH(t,x) = E wHa(t,x)dtndxa Ial=k
wv (t, x) =
wva(t,x)dxa Iaj=k+1
and denote by [w](0,1) the k-form in U obtained by averaging wH with respect to the t variable
2.3 Currents: Basic Facts
125
1
U wxa(t, x) dt dx".
[w](o,l) (x)
0
JJ
From the definition of product of currents we infer, compare (3), that Q (0, 1) x T(wv) = 0, while from (2)
\
1
(5)
(0,1) 11 x T(w) =
JT( o
wx(tx) dxI dt //
Ial=k
T([w](0.1))-
Also (6)
a([(0,1)lxT) _ (81-8o)xT-Q(0,1)}jxaT =61xT-8oxT-Q(0,1)1xaT
where 8°, 81 E Do (R1) are the Dirac masses at 0 and 1, 8o(cP) = W(0)
,
81((P) = W (1)
V c0
E D°(R1)
Currents which are representable by integration. Those are currents with components which are measures with locally finite total variations instead of mere distributions.
Definition 4. Let U C R and V C U be open sets, and let T E Dk(U). The mass of T in V is defined by MV(T) := sup{T(w) I W E Dk(U), sptw C V, Iw(x)I < 1 V X E U} . Actually there are very good reasons for replacing in the previous definition the Euclidean norm I w (x) I by the comass norm defined in Sec. 2.2.1, compare Vol. II Ch. 1.
Definition 5. Let U C 1 and V C U be an open set, and let T E Dk (U) . The mass of T in V is defined by Mv(T) := sup {T(w) I w E D"(U), spt w C V,
IIHII<1VXEU}. The reason for such a definition will be apparent in Vol. II Ch. 1 where dealing with approximation theorems, compare also Sec. 2.2.6. The Euclidean and mass norms are equivalent, in fact
MV(T) < Mv(T) < cMv(T) where c is an absolute positive constant, see Sec. 2.2.1; but in general we have
MV(T) < Mv(T).
2. Integer Multiplicity Rectifiable Currents
126
If V = U we shall simply write M(T) instead of Mv(T), and we set
M(T) < oo} Mk,lo,:(U) :_ {T E Dk(U) Mv(T) < oo d V CC U1.
Mk(U) :_ {T E Dk(U)
I
I
Of course T E Mk,loc(U) (respectively Mk (U)) if and only if the Ta's are Radon measures (respectively Radon measures with finite total variation in U).
Introducing the total variation of T with respect to the comass and the Euclidean norm
:= sup {T (w) := sup{T(w)
II T II (w)
ITIO
II w(x) II <_ o(x)}
I
Iw(x)I <_ W(x)},
I
cp > 0, an immediate consequence of Riesz theorem for measures is
Theorem 1. Let T E Mk,1oc(U). We have (i)
There exists a ITI-measurable function f with 1 < f < c such that IIT 1= ITI L f.
(ii)
There exists a unique I T II -measurable map I
1
U ---* Ak1n with I ' I I =
II T II-a.e. such that for all w E Dk(U)
T(w)=f d1ITII.
(7)
U
Moreover
VVCCU, V open
IITII(V)=Mv(T)
and T is the Radon-Nikodym derivative of T with respect to II T II dT
T
dIITII
(iii) Also
T(w) =
fdITI U
TI(V) = Mv(T)
VVCCU, V open
and I fTI = 1 ITI-a.e.
Notice that conversely, given a non-negative Radon measure µ and a µmeasurable and locally summable map f : U -+ AkRn the current T = p L f defined by T(w)
:
fd u
2.3 Currents: Basic Facts
127
belongs to Mk,loc(U) and we have
IITII = ILLIIf1I
µ a.e.
Let T E Mk,loc(U), i.e., T(w)
= f< T, w> d I i T11. Though in general
MV (T) 54 MV(T), if T is simple, we infer from Theorem 1
1 = f ITI = f II T II = f
ITI-a.e., equivalently I ITI I-a.e..
Therefore we conclude in this case Mv(T) = MV(T). In the sequel we shall always work with the mass and comass norms, but we understood equalities like IITII = ITI,
IITII = ITI
II T II-a.e. or ITI-a.e.
in the case that T is simple. From the definition of mass we readily infer
Proposition 1 (Lower semicontinuity of the mass). Let Tj,T E Vk(U). If Tj T then for any V C U, V open, MV (T) < lim inf j,oo MV (Tj) . and from the compactness theorem for Radon measures
Proposition 2 (Compactness-closure theorem). Let {Tj} C Mk,lo0(U) be a sequence satisfying supj MV (Tj) < cc d V CC U. Then there exists a subsequence {Tj,} and T E Mk,loc(U) such that Tj, - T. Moreover
M(T) < lim inf M(Ty) < cc j -moo
if the masses of Tj, are equibounded.
By the monotone convergence theorem any current with finite mass, T E Mk (U), can be extended to all k-forms with Borel bounded coefficients, and again we have
T(w) := f<,w>d(IT((. U
A similar extension works also for T E Mk,loc(U) and for compactly supported k-forms w in U with bounded Borel coefficients. In particular, if T E Mk(U) and A C U is any Borel set we can define the restriction of T on A, T L A, by (8)
TLA(w) := fdMTM. A
More generally, we can define for any bounded Borel function
128
2. Integer Multiplicity Rectifiable Currents
T L f (w) :=
(9)
f<w>fdIlTIl.
However a certain care is needed when dealing with weak convergence. From {Tj} C ivtk(U), sups M(T?) < oo, Tj - T, i.e.,
Tj (w) -+ T(w)
VwEDk(U)
one can easily infer that TT (w) --* T (w) on k-forms with continuous coefficients with compact support, but in general the Tj (w) do not converge to T (w) if w has only bounded Borel coefficients (even with support contained in U). The reason for that is of course that a sequence of functions in L' ((1) which converges as measures need not converge in L1, compare Ch. 1.
For future purposes we note that for any T E Mk,lo.(U), T := II T II L T, the
current [ (0, 1) x T has locally finite mass and actually II
(0,1) J1 x T II =
G1
Q (0,1). x
x I ITI I
= e1AT.
For the reader's convenience we include the simple proof. With the same notation as above let w E Dk+1(U), W = WH + WV. As clearly II WH II < II w II, we have II [w](0,1) II (x)
f II w(t, x) II dt, hence 0
(0,1)1 x T(w)I
=
IT([w](0,l))1 < II T II(II [W](0,1) II)
IITII
JIIwt,x )I Idt)
C1xIITII(liw(t,x)II),
0
from which we infer II [ (0,1)1 x T II < Ll x II T 11. To prove the opposite inequality, for e > 0 and f E CO ([0, 1] x U), let w E Dk ([0, 1] x U) be such that for any
t
T(w(t,x)) > II T Integrating with respect to t in (0, 1) we get 1
x T II. To conclude the proof of the II claim it suffices to notice the following equality II
1
1
(0,1)1xT(w) = f T(wH)dt= f dt f <WH(t,x),T(x)>d1ITII 0
=
0
f<w(t,x),eiAf>d' x 11 T II.
2.3 Currents: Basic Facts
129
Normal currents. The class of normal currents is defined as Nk(U)
M(T) +M(aT) < oo}
{T E 1)k(U)
aT E NIk-1(U)} ,
{T E Mk(U) and the class of locally normal currents by
Nk,loc(U) := {T E Vk(U) J Mv(T) +Mv(3T) < oc V V CC U}
If T E Nk(U) one sets
Nv(T) := M v(T) +Mv(0T). Of course also for normal currents a closure-compactness theorem holds: If Tj E Nk,l°°(U) with
sup{Mv(Ti) +Mv(aTi)} < oo
d V CC U ,
then there exists a subsequence {Tj, } of {Tj } and T E Nk,lo,(U) such that
Tj, -S T
.
An interesting result for normal currents, which will be proved in Theorem 3
in Sec. 4.3.1, states that they cannot concentrate on small closed sets; more precisely we have
Theorem 2. Let T E Nk,loc(U), U C Rn, and let p,, denotes for any a E I(k, n) the orthogonal projection 1
n
Ck'
ak
Then, for any closed set E C U with Nk(pa(E)) = 0 `d a c 1(k, n), we have
ITII(E) = 0
i.e. TAE=0. n-dimensional currents in Rn. Those currents will be discussed with some details in Ch. 4. Here we shall confine ourselves to state for the reader's convenience a few facts. Let U be an open set in R'. Of course
-D- (U) = {bdx'A... Adxn
I
b E D°(U) = C,° (U)}
.
Thus to any n-current T it corresponds the distribution
b --> T(bdx'A...hdxn) and, conversely, to any distribution S E Do(U) it corresponds the n- current
130
2. Integer Multiplicity Rectifiable Currents
bdx1A... Adxn
S(b)
.
In particular, to any function u E L o, (U) we can associate the n-current
T,(bdx'A...Adx'):= fbudx.
Tv,:D'(U)->R,
U
Obviously
Mv(T,,) =
fluldx
V V CC U.
V
As every w E D'-'(U) can be written as w = Ej' 1(-1)j-'a; dx1A... Adx3A ... ndxn, we compute 09T.)(W)
=
fudivadx
,
a = (ai, ... , a.) .
We therefore see that every function u E BV0,,(U), i.e. every L1-function whose distributional derivatives are Radon measures defines the n-dimensional locally
for any V cc U, we have normal current T, moreoverfudx, MV 09T.) =
MV (T.) =
fdDu. v
V
Conversely we shall see in Ch. 4
Theorem 3. For any locally normal current T E Nn,lo,;(R'n) there is a function
u E BV(U) such that T = T. As for any n-current in U C Rn we have T = T°e1A ... Ae,, and aT = 0 implies DiT° = 0, i = 1, . . . , n, in the sense of distributions, i.e., To =const, we also infer the following characterization of n-currents in U without boundary in U.
Theorem 4 (Constancy theorem). Suppose U is a connected open set in Rn, T E Dn(U) is such that aT = 0, then T is a real multiple of the current integration on U, i.e.,
T = rl[U
for some r E R. 1 For the reader's convenience we compute here the boundary of k-currents in Rn, n = 1, 2, 3.This will show in which sense a means derivative.
1. In R1 we have 0-dimensional currents S which are simply distributions, and for which 8S = 0, and 1-currents. Clearly 1-currents have the form
T = T1e1
2.3 Currents: Basic Facts
131
and according to d27 = 27' dx for 27 E D°(R), we compute
8T(r7) = T(di2) = Tl(7]) = -DT'(r7)
i.e., 8T = -DT'. 2. In 1R2, apart from 0-currents, we have currents of the form S = S' el +S2 e2 E Dl (1R2).
T = To e,Ae2 E D2(R2)
Since for 77 = 97, dx1 + 772 dx2 E D' (IR2) we have d77
771,x2dx2Adx1 +772,xidx1Adx2 = (772,x1 - 77i x2) dx'Adx2
we can compute 8T (77)
= T (d7?) = T° (772,x1 - r71,x2) _ = -DxiT°(r72) + Dx2T°(771) _ -(DxiT°el - Dx2T°el)(77)
i.e.
8T = Dx2T°el - Dx1T°e2 and aT may be regarded as minus the gradient of T°. For 77 E D° (R2) we have d77 = 77x 1 dxl + 77x2 dx2 hence aS(77) = S(d77) = S1 (77x1) + S2 (77x2) = -Dx1 S1(77)
-
Dx2 S2 (77)
i.e.
aS = -div (Sl, S2) E Do (]R2)
.
3. In R3, a part for 0-dimensional currents which are distributions, currents have
the form
T = T°e,Ae2Ae3 E D3(]R3) S = S1e2Ae3 +S2 e3Aej +S3 ejAe2 E D2(1 3) R = R'el + R2e2 + R3e3 E Dl (R3) As in ]R2 we easily computes
8T - - gradient of To
,
aR - -div (R1, R2, R3)
For 77 = r7idxl +772dx2 +773dx3 E D1 (R3) we have d77 = (772,x1
- r7i,x2)dx1Adx2 + (773,x2 - r72,x3)dx2Adx3 + (773,x1 - 71i,x3)dxlhdx3
we therefore see that aS(77) = -curl (S1, S2, S3)(771, 772, 773)
2. Integer Multiplicity Rectifiable Currents
132
Image of a current. For its relevance in the sequel we conclude this subsectioi
by discussing the notion of image or pushing forward of a normal current f#T by a smooth map f . Under mild conditions one can define the pushing forward of a current T E Dk (U), U C R1. Let f : U ---p V, V C R', be a C°° map such that f lspt T is proper, i.e., f -1(K) n spt T is compact in U for any compact set K. Then the current f#T E Dk(V) is defined by (10)
f#T(w)
T ((f#w)
w E Dk(V)
where C E Co (U) with c = 1 in a neighbourhood of spt T n spt f #w. One easily checks that such a definition is independent of ( and (11)
a f# = f#a ,
spt f#T C f (spt T) .
Suppose now that T has finite mass, i.e. T E Mk(U). As we have seen T extends to all differential forms rl with Borel coefficients and f Inj d1J T 11 < oo. In particular for any map f : U -+ V of class Cl with bounded derivatives in U, supu I D f I < oc, the form f # (w) has bounded and continuous coefficients, hence T is well defined on f # (w). Therefore we can conclude that for any f of class C1 and T in Mk(U) f#T is well defined again by
f#T(w)
(12)
T (.f #w) .
In particular if U = V x RN and Tr : U -> V is the orthogonal projection onto the first factor, lr(x, y) = x, ?r# is well defined for any T E Mk(U). However in general f# is not continuous with respect to the weak convergence of currents. For example if U = (0, 1) x R, V = (0, 1) the sequence of currents Tk defined as integration over the line segment from the point (0, k) to (1, k), we have M(aTk) = 0,
M(Tk) = 1,
7r#(Tk) = To
but Tk - 0 as k -s oo which says that 7r# is not continuous. We explicitly note that (12) implies that f#T has finite mass if T has finite mass and actually f#T I I = f#11 T 1 1 ,
f -!'(f (x)) = AkDf(x)(T(x))
Similarly, assuming that M(T), M(8T) < oo and f E C2(U) has bounded second derivatives the pushing forward of aT
f#8T(w) = aT(f#(w)) = T(df#(w)) is well defined, but in general
f#aT 0 of#T. Let us consider now mappings f : U --* V which are just Lipschitz continuous. In this case for any w with smooth coefficients the pull-back f #w is a form with
2.3 Currents: Basic Facts
133
bounded measurable, but not necessarily Borel coefficients. Consequently we can apply T to f #w only if II T II is absolutely continuous with respect to the Lebesgue measure, and in general T (f #w) as in (12) is meaningless. The following example illustrates the situation. Take U = V = 1 and let f : Iii -> lib be defined by
f (x) :=
x+1 forx<-1 for -1<x<1 0 x-1 forx>1.
If w = cp(y) dy E D' (R),
f#w = co(f(x)) f'(x) dx =: zb(x) dx
where O(x) is defined and smooth for x # f1. But b(1) and 0(-1) are not defined. By taking T := 81 E No (IR) we then see that
T(.f#w) = 81(') = 0(1) is meaningless.
However it turns out that if T is a locally normal current in Dk(U) and f is a Lipschitz map which is proper on spt T, then f#T can be well defined. But before doing that let us show why we need to assume f spt T proper.
Let f : U := (0, +oo) y V := ]IP be just the inclusion map, and let T [ (0, b) D, b > 0. We have aT = 8b so that T E N1(U), consequently f#aT = 8b. The push-forward of Q (0, b) by f is instead the current Q (0, b) E Dl (IR), f#T = Q (0, b) , hence of#T = Sb - 8o. Therefore we have f#aT # of#T. As one would like to have the equality f#aT = af#T, what is wrong is the fact that f is not proper f-1([-1, 1]) = (0,1] and (0,1] is not compact in U = (0, +oo). Suppose now f Lipschitz and proper on sptT, T E Nk,loc(U). Let 0 E C° ° (Rn) be a standard mollifier and let r-'O(r-1x)
'f'r(x)
r>0
and
fT
f *0,
be a smoothing of f. We define the image of T by f as
f#T(w) = limo fT#T(w)
w E Dn(V)
i.e. as the weak limit of the well defined currents fT#T, compare Sec. 5.1.2. In fact we have
Proposition 3. If T E
f : U -> V is Lipschitz and fIsptT proper,
then
f#T(w) := o fT#T (w) 11
2. Integer Multiplicity Rectifiable Currents
134
exists and is independent of the mollifier 0 for all w E Dk(V). It agrees with f#T defined by (12) if f is of class C'. Moreover spt f#T C f (spt T) and k
sup JDfj) Mf-1(w)(T) f-1(W)
Mw(f#T) <
for any open set W CC V. Finally
f#8T = of#T . The proof of Proposition 3 is based on the important homotopy formula for currents. Consider two smooth maps f, g : U -> V and a smooth homotopy
h : [0,1] x U -> V,
h(0, ) = f,
h(1, ) = g.
Suppose h is proper in [0, 1] x spt T. Then h#(1(0,1)1 x T) is well defined and we have
8h#(1(0,1)1 x T) = h#a([ (0,1) x T) = h#(51 xT - So x T- Q(0,1) x 8T) = g#T- f#T - h#(Q(0, 1) 1 x9T) . Therefore we can state
Proposition 4 (Homotopy formula). Let U, V be open sets in R', U --+ V be smooth maps, and let h : [0, 11 x U -- V be a smooth homotopy between f and g; finally let T E Dk (U) . Assume that h-1(K) n spt T is compact in [0, 1] x U for any compact set K C V, then we have (13)
g#T - f#T = ah#(1(0,1) ] x T) + h#([ (0,1) fl x aT)
where h#(Q (0,1) l x aT) := 0 if k = 0. Recalling (5) Q (0, 1) JI x T(w) = T([w](o,l))
we can write (13) as (14)
T(g#w
- f#w) =T([h#dw](o,l) +d[h#w](o,l))
where the last summand has to be taken equal to zero if k = 0. It is easy to see that (13) and (14) are also valid even if h is not proper provided T has compact support in U, T E Ek(U). By applying (14) to currents of the type
w -.
J
wnrl
rl E Dn-k (U)
2.3 Currents: Basic Facts
135
we then recover the usual homotopy formula for forms g#w
(15)
- f#w = [h#dw](o,1) + d[h#w](o,l).
In the special case of the affine homotopy
h(t, x) := (1 - t) f (x) + tg(x) we have
Ak+1Dh(t, x)(elAT(x))
h# (t,x)
= (9(x)-f(x))AAkDxh(t,x)(T(x)) hence
h#(Q(0,1)1 x T)(w) =
f <w(h(-)),(9-f)AAkDxh(T)>d(L' x IIT II)
which implies
M(h#(Q(0,1)]1 x T)) :5 cIITII(I9-fI(IDfIk+IDyIk)) and therefore (16)
M(h#Q (0, 1) ] x T) < sup If - 9I sup (ID.f I + IDgI)k M(T) spt T
spt T
Returning to Proposition 3, we consider the affine homotopy
h(t, x) := tf, (x) + (1- t) ff (x) corresponding to two parameters T, Q > 0. The homotopy formula yields for any O
w with spt w c K, K CC V, f,-#T(w) - fv#T(w) = h# (If (0,1) x T)(dw) + h# (E (0, 1) x 0T)(w) therefore, if moreover IwI, Idwl < 1, we infer from (16)
I.fT#T(w) - ff#T(w)I < cM(T)
sup f-' (K) nspt T
IfT -
f01 (Lip f)k
The proof of Proposition 3 can be now easily concluded. We conclude this subsection by stating one more simple but useful consequence of the homotopy formula.
Let T E Dk(U) with spt T C U. Suppose that U is star-shaped relative to the origin, and consider the affine homotopy which joins the null map to the identity map
h(t, x) = tx
.
We define the cone over T as the current in Dk+1 (U) given by
136
2. Integer Multiplicity Rectifiable Currents
OxT := h#(Q(0,1)JJxT). From the homotopy formula we infer
8(OxT) = T-(Oxcxc T) which shows that every boundaryless current T is a boundary, i.e., if T E Dk(Rn)
and aT = 0, then there exists R E Dk+1 such that T = 8R. Another immediate consequence of the homotopy formula is the following
Proposition 5 (Poincare lemma). Let U be a starshaped domain in IEBn and let w be a closed form in U. Then w is exact. In fact, assuming U starshaped with respect to the origin and considering the affine homotopy between 0 and the identity, h(t, x) := tx, (15) yields w = g#w = d[h#w](o,l).
2.4 Integer Multiplicity Rectifiable Currents In the class of k-dimensional currents in some open set U of R' we may identify two typical situations which in some sense are extremal from the point of view of smoothness. (i)
Given a Lebesgue measurable k-vector field k-dimensional current
T := ,Cn L C =
Gn
L:
in U,
Cc'ell the
L COea
oI=k
is a current "distributed" on the n-dimensional set U. Actually by considering a non-negative measure y and a 4-measurable k-vector field C we can realize k-dimensional current
T := 1JLC ii)
which are "distributed" in sets of any "dimension" less than n. Given a smooth k-dimensional oriented submanifold of IPn, M, M C U, the current integration of infinitely differentiable k-forms with compact support in U, Q M , defines a k-current which lives on the k-dimensional set M. Moreover, as we have seen in Sec. 2.2.2, the current
QMI(w) =
J
M
w
w E Dk(U)
2.4 Integer Multiplicity Rectifiable Currents
137
can be represented as
I M I (w) = f<(x),w(x)>d7-Lk M
where fi(x) orients for H' -a-e. x c M the tangent plane TAM of M at x. One also refers to Q M I as to the current carried by M.
Integer multiplicity rectifiable currents arise as the natural extension of the currents of the type Q M 1, in which we allow M to be xk-measurable and countably rectifiable and we also allow integer multiplicity. As we shall see in Sec. 2.2.6
such a class turns out to characterize the class of the weak limits of the kdimensional currents carried by smooth submanifolds. But before turning to precise definitions, let us return to the k-dimensional
currents which are integration over smooth graphs, as they shall be specially important for us in the sequel.
Currents carried by smooth graphs. Let ,(2 be an open set in 1Rn and let u: f2 C 1Rn _+ RN be a smooth map. Its graph is defined by
9u,o := {(x, u(x))
E IRn+N
I
x E .(2}
and the Cartesian current carried by the graph of u is given by the n-dimensional current
G. i.e., by
Cu (w) =
f (id x u)#w = f<w(xu(x)),M(Du(x))>dx
d w E Dn (U)
n
S2
where U := .f2 x 1R`v. By (33) in Sec. 2.2.1 we have
(a, a) f waQ(x, u(x)) Ma (Du(x)) dx
Gu(w) IaI+IQI=n
D
for any n-form w
w =
E waQ(x, y) dxahdya Iat+-IQI=n
where a(a, a) is the sign of the permutation which reorders the multi-index (a, a) in its natural order, and Ma (Du) denotes the determinant of the minor of Du with rows din) and columns (al, ... , an). Alternatively this can be seen as follows. We have
duo' A...hdu8k =
up ;k dxjlA...Adxj'k
and those terms will contribute to waQ dxandyQ only if the differentials
138
2. Integer Multiplicity Rectifiable Currents
and dx31,...,dx31-
dx«l,...,dx«>.-,.
are different, so that (ii,... , jk) must be a permutation of 5. Reordering (ji, ... , jk) to d, and then (a, a) we then get dx«1 A ... Adx«n-k AduQ1 A... Adupk = Mg (Du) dx«ndxa = c- (al &) Ma (Du) dx'A ... Adxn . {E Ma (DU) 211/2 is the element of area
It is worth noticing that I M(Du) on cu,.fl and the n-vector
M(Du) _
a(a, a) Ma (Du)
l
I«I+IQI=n
_ (el + Dlu''ei)A ... A(e,, + D,luiei) or more precisely M(Du)/jM(Du)j is the unitary tangent n-vector orienting gu,,?. Thus we can write Gu (w) _
f <w, M(Du)
> d7-ln L Gu,,f2
compare Sec. 3.1.5 and Sec. 3.2.1. For future purposes we would like to discuss some special cases. This in particular will show in which sense the boundary operator describes the geometric boundary.
0
Dimension and codimension 1. Let n = N = 1, Q = (0, 1), U = (0,1) x R.
As a 1-forms in U have the form w = cpl (x, y) dx + cp2 (x, y) dy, with cpl, W2 E C,c°(.fl x R), we get
G.(w) = J[i(x,u(x)) + cp2(x,u(x))u'(x)] dx . 0
For any 0-form 77 = cp(x, y), we have d77 = cps dx + cpy dy so that
if[cox(x,u)+co(x,u)uu]dx aGu(97) = G(dr7) = 0
dx(x, u(x)) dx = cp(b) - cp(a) 0
where b = (1, u(1)), a = (0, u(0)). From the previous computations we see that
aGu=0
2.4 Integer Multiplicity Rectifiable Currents
139
if we regard Gu as a current in U, as 77 E Do(U), and 8Gu = Sb - Sa E Do (IR x IR)
if we regard G,,, as a current in R x R. Notice that 8b - dd E Mo(R x IR) and is the current carried by the graph of ul arz, so that, regarding Gu as a current in IR x IR
8Gu = Nan,uiej the graph of ul an being oriented in the induced way.
2 Codimension 1. Suppose n > 1, N = 1. Then Gu,f2 has dimension n and an n-form in IRn+1 can be written as w = wo (x, y) dxl A ... Adxn +
Then Gu(w)
f
=
wi (x, y) dx' A ... Adxzn ... Adxn ndy . n.
[wo(x, u(x)) +
j:(-1)n_zwz(x, u(x)) ux, I dx . i=1
By Gauss-Green theorem we can compute Gu (dri) = Q agu,.? (77) = Q guy an J 0/)
so that 8Gu = 0 if we regard Gu as an element of Dn (.fl x ]R) 8G,, = Q 9u1637 1 if we regard Gu as an element of Dn (Rn x IR)
This can be easily seen, for instance, if n = 2 and Q = {(x1,x2)
1
x1 < 0}. If
77 = cot dx1 + 02 dx2 + cp3 dx3 E Dl (IR2 x IR)
one easily computes
(id x u)#d7) _ (-colx2 + 2x1 - Wlx3ux2 + (Q3xiux2 -cp3x2ux1)dx'Adx2 { - D2 [c1(x, u(x))] + D1
l
(x, u(x))] + Dl [co3 (x, u(x) )]ux2
- D2[co3(x, u(x))]uxl } dxlndx2.
Denoting by v = (1, 0) the exterior normal to 812, we then get
J
(id x u)* d77
f[c02(o,x2, u(0, x2)) + c3(0, x2, u(0, x2))ux2] dx2 9uian
(77)
Similarly one proceeds in the general case, compare Sec. 3.2.3.
2. Integer Multiplicity Rectifiable Currents
140
0
R2
Mapping from R2 into R2. Let u E C1(.(l, R2), 7 C I[F2. Any 2-form in R2 X can be written as 2
wij (x, y) dxindyj
w = wo (x, y) dxl ndx2 +
+ w2 (x, y) dy' Ady2
i,j=1
Thus G,,(w) is given by 2
[Ou (x)) + E (-1)z_lwi n
X, u(x))uxi + W2 (X, u(x))det Du(x) I dx.
i,j=1 11
Rectifiable and integer multiplicity rectifiable currents. Let U be an open set in R'. Given (i) an lk-measurable and countably k-rectifiable set M C U (ii) an lk-measurable map 9 : M -> IEi; which is locally 71k L.M-summable (iii) an 71k-measurable map 1; : M - AkR' with I ICI I = 1 xk L M-a.e.
we say that a current T E Dk (U) is of the type 'r(M, 0, ), T = ir(M, 9, 1;) if
T(w) =
T= By changing l; --* l sign 0, 9 -> 9 sign 0, we can and do assume that 0 > 0, and we call 9 the multiplicity of r(.M, 9,1;). Sometime one refers to
set (T) := {x E IR'
9k(kk T 1 1
,
x) > 0}
as to the set of T. Of course T determines set (T), while possible M's are determined only up to sets of Nk-measure zero.
Definition 1. We say that a current T E Dk(U) is rectifiable if and only if T = -r(M, 9, C) for some M, 0, C as previously and moreover l; (x) is a k-vector associated to the tangent plane TTM for 71k L M-a.e. x. If moreover the density 0 is integer-valued, then we say that T = r(Nl, 0, C) is integer multiplicity rectifiable, briefly i.m. rectifiable. The class of integer multiplicity rectifiable k-currents in U will be denoted by Rk(U). For currents of type rr(.M, 9, l;) we have
1ITHH = 97,ckLM
2.4 Integer Multiplicity Rectifiable Currents
141
while, according to our discussion after the definition of the mass and the Euclidean mass in Sec. 2.2.3, for rectifiable currents we have and
IC11=ICI=1
IITII=ITI=BfkL.M.
Of course the class of rectifiable currents in U form a linear space, also
Ti +T2 E 7Zk(U) ,
if
T1, T2 E Rk(U)
but in general AT E Rk(U) only if T E Rk(U) and A is an integer. The following two theorems of Federer and Fleming make the class of i.m. rectifiable k-currents Rk (U), very natural and important especially in connection with the calculus of variations.
Theorem 1 (Closure theorem). Let {T3} C Rk(U) be a sequence of i.m. rectifiable k-currents in some open set U C 118' satisfying
sup[MV(Tj) +MV(aTj)] < oo
(1)
VV open, V CC U
j
and weakly converging to some current T E Dk (U), Tj -s T. Then T E Rk (U)
.
This theorem together with the compactness theorem for general currents yields
Theorem 2 (Compactness theorem). Let {T3} C Rk(U) be a sequence satisfying (1). Then there exists a subsequence {Tj, } of {T3 } and T E Rk (U) such
that Tj' - T. Defining the weak convergence in Nk,ioc(U) as
Tj - T if
weakly in Nk,1oc(U)
Tj - T
in Dk(U)
sup (MV (Tj) + My (aTj)) < 0o
j
bVCCU
we can state Theorem 1 as: Rk (U) n Nk,l0C (U) is (sequentially) weakly closed in Nk,l0C (U)
Notice that not every current in Rk (R') belongs to Nk (JRI) . For instance a collection of 2-dimensional disjoint disks in R3 with radius rj satisfying 00
00
rJ2 j=1
< 00 ,
T3 = 00 j=1
clearly yields a current T in 7 2(IR3) for which M(aT) = oo.
2. Integer Multiplicity Rectifiable Currents
142
(a)
Fig. 2.4. (a) A sequence of density 1 currents converging to a current of density 2, (b) A rectifiable current with boundary of infinite mass.
(b)
We shall return to Theorem 1 later in Sec. 2.2.6 and Sec. 2.2.7. But first we would like to present a few examples with the aim of illustrating Definition 1 and Theorem 1 and Theorem 2. Of course limits of smooth manifolds Mj, identified with the rectifiable currents 7(M (MjQAi ) with density 1, can be in principle rectifiable currents with integer density not necessarily 1, therefore it is natural to allow integer densities in order to identify such limits. As for distributions we cannot expect any regularity of the limits in the sense of currents of smooth manifolds Mj, if there is no control on the mass of the currents Q Mj ][, i.e.,
supM(QMj f) = +oo .
Assuming only Q Mj - T in Dk(U), the components of T need not even be measures. 4
Consider the sequence of smooth maps
uj
:
[0,1] --> R,
uj(x) := sin 2 j7rx
.
For the associated i.m. rectifiable currents Gui carried by the graphs of uj we trivially have
M(Gu,,) -) oo
,
asj ->oc.
Also i
Gu, (cp(x, y)dx) = fca(x,uj)dx 0 1
Gu, (z/)(x, y)dy) = f(x,uj)jdx 0
2.4 Integer Multiplicity Rectifiable Currents
143
Fig. 2.5. Young measure and distributions as zero and first components of the limit of i.m. currents. 1r
J 0
d J V1 (X, T) dr - f(xr) i
(171
[ dx
0
dx
0
f
Ui
1f
- J dx 0
'0. (x, T) d-r
0
Therefore, taking into account Proposition 2 in Sec. 1.2.6 we see that the zero component of Gu,, converge to the Young measure in (0, 1) x 1[P given by
G1L(0,1)xA,
r
J
dA
)n2dx
,
o 1
Gui (co(x, y)dx)
-- f(x,sin2x)dx 0
while 1
Gu; (0(x, y)dy)
sin/27rx
- f dx J 0
Ox (x, T) dr
0
which is not a measure, but just a distribution in R x R.
Next we want to show that the closure theorem does not hold for realmultiplicity rectifiable currents.
5 The closure theorem for i.m. rectifiable 0-currents. Let k = 0, U = R'n. It is readily seen that T E 7Zo(U) if and only if
2. Integer Multiplicity Rectifiable Currents
144
T=
aisz;,
zi E R', ai E Z .
,
i=1
Let {T3 } be a sequence in Ro (JRY) with
N(T) := M(T) +M(aT)
sup N(T3) < oo ,
.
j
Since the coefficients aii) are integers, we can choose a common s and write each Teas
Tj = i=1
with supj,i Ja(') < oc. Therefore passing to a subsequence we have
P)
-+ zi
or
z2j) I
-+ oo ,
and
ai
as j -* oo, from which we infer the (elementary) closure theorem
T=
Tj
aiazt E Ro(U) . i=1
Let us allow now real densities. Assume for instance k = 0, n = 1 and U :_ (0, 1), and consider the sequence of rectifiable currents with real multiplicity 1
Tj
bile
1
3
i=1
We then have, compare Sec. 1.2.5, that T. converge weakly to T = ,C1 L (0,1) which belongs to Mo (1k) but is not rectifiable. Of course similar examples occurs
in any dimension. For instance the sequence of real-multiplicity rectifiable 1dimensional currents in R2
Tj := 1EQ(0,1)x{i/j} 3i=1 with sups N(T3) < oc converges to the 1-dimensional normal, but non rectifiable current T := C2 L (0, 1) x (0, 1) e1. 6 The closure theorem for i.m. rectifiable 1-currents in R. and let T = -r (M, 9,T) R, (R). Of course we can assume
= oe1,
Let k = n = 1,
v:M -+ {tl}.
From the decomposition theorem for normal n-currents in IR', compare Sec. 4.3.1, or by a simpler argument in this case, we see that each T E Ri (J) n Ni (IR) has the form
2.4 Integer Multiplicity Rectifiable Currents
T=
145
giQ(ai,bi)
0. Since aT = Ei=1 qi (Sb - Say) and
where fail f1 {bi} = 0, qi E Z, qi M(aT) = 2 E8=1 IgiI > 2s we get
s < N(T)
M(T) + M(aT)
.
If {Tj } C Rl (R), sups N(Tj) < oo we can therefore write S
Zj
qi
Q (ai
bi
)
i=1
for some fixed s independent of j. If moreover we assume that spt TT in contained in some fixed compact set we then find as previously a(j)
-, ai,
qz.7)
b(j) ""' bi,
--, qi,
Tj -s
qi Q (ai, bi)
i=1
We would like to show now that the assumption
sup M(aTj) < 00 i
is essentially necessary in the closure theorem. We shall see that if the previous bound does not hold, and Tj - T, we can have
Concentration: the support of T may be a lower dimensional set Distribution: T = T(M, 0, T) where T orients M, but 0 is not integer (iii) T = T(M, 0, C), where 0 is integer but 6 is not tangent to M. (iv) combination of the previous phenomena, compare Ch. 1. (i) (ii)
7 Concentration. Consider the sequence of 1-dimensional currents
Tj := j Q (0,1/j) ]j E 7Z1(R) fl N1(IE8) Of course M(TV) = 1, M(aTj) = 2j -, oc, and we have T,j
-s boe1
which belongs to M1([8), but is not rectifiable, its support is a zero-dimensional set. 8
Distribution. Consider the sequence of 1-dimensional currents
i-A i
T, i=1
E 71(R)nN1(R)
2. Integer Multiplicity Rectifiable Currents
146
where 0 < a < 1. Of course supj M(Tj) < oo, M(0Tj) = 2j -- oo. One easily sees that
Tj - T := A
(0, 1)
and, trivially, T is not i.m. rectifiable. 9
Consider the sequence of i.m. rectifiable 1-currents in U = R X R = R2
j (0, 1/j) X {i/j}l E R1(R2) n N1(11 2) .
Tj
i=1
Then Tj = T(Mj,1, e1), Mj = Ui=1(0,1/j) x {i/j}, and M(Tj) = 1, M(aT3)
E Fig. 2.6. One dimensional i.m. rectifiable currents converging to a non rectifiable current.
2j -- oo. Clearly 711
Tj
L Nl3
H' L M,
T = 7-(.A4, 1, el)
M := {0} x (0,1) .
Since the tangent vector to M is M := e2 # e1, the limit current T is not rectifiable. 10 We remark that M(aT) = +oo in the previous example. More generally, if T = 'r (M, 0, C) E Mk (RI) where M is a smooth k-dimensional submanifold, 0 > 0, and C is a smooth vector field which is not tangent to M, then we have
M(aT) = +oo. Let us prove it in the special case T = Q (0, 1) x {0} e2 E .M1(R2) , as the same argument works in general, compare also Lemma 1 in Sec. 3.3.2. We have
T=T1e1+T2e2, consider the 0-form
T1-0,
T2=711 LM
,
.M=(0,1)x{0}.
2.4 Integer Multiplicity Rectifiable Currents
147
77(x1,772)
where
if xI < 2
1
3-xI if2<xj <3 ifxI>3
0
and, for a fixed E > 0,
-1
if x2 < -E
(1/E)x2
if Ix21 < E
1
ifx2>E.
One computes 1
8T(77) = T2(7l2) = fz(x'o)dx =
1 E
0
Being 1771 < 1, we therefore infer M(aT) > 8T(77) = E, i.e., M(aT) = oo. 11 Concentration-distribution. The different phenomena corresponding to (i) (ii) (iii) may appear in a combined form. Let us see two simple cases. Set for j = 2, 3, .. .
Mj:=U i=1
z
\j
z
3
C(0 1).
Clearly Tj E R1 (]R) n Ni (EP); as xi (Mj) = 1/j2, we have
M(Tj) = 1,
while
M(aTj) = 2j3 -* oo.
The hypotheses of the closure theorem are not satisfied, but
Tj - T := r((0,1),1, e1) = Q (0,1)11 E 7Z, (R) n Ni The concentration behaviour is seen from the fact that
1'(Mj) -' 0 ,
>
11(Mj) < 00
3
and for any positive e, we can find a Borel set E with x1 (E) < E such that for large j we have
IITj11((0,1)\E) = 0, i.e., M3 C E. Such a concentration however produces the i.m. rectifiable current Q (0, 1) 1 which is distributed in (0, 1). Consider now the following variant. Set again
148
2. Integer Multiplicity Rectifiable Currents
Fig. 2.7. One dimensional i.m. rectifiable currents converging to a 1-dimensional diffused current. Tj
T(Mj,j2,el) E 7Z1(R2)
where this time
i_
Mj :=
1
,
7-12LQ
,
i) x f h
C R2
i,h=1
Then
j2 7 1LMj
Q= (0,1) x (0,1)
and
Tj
T:= T(Q, 1, el) = 7-12 L Q e1
.
Again we have concentration, with this time a non rectifiable current distributed on the 2-dimensional set Q. Of course in this case we again have M(aTj) = 2j4 -+ 00. 12 Finally we would like to remark that the vanishing of OT in some sense fixes the orientation of T. Let B(0, 1) be the unit ball in 1R2, trivially T =
T(B(0,1), 1, eine2) defines an i.m. rectifiable current in R2 without boundary in B(0,1). Also T1 +T2, where T(B(0, 1/2), 1, eiAe2)
T1 T2
:=
T(B(0,1) \ B(0,1/2),1, -eine2)
,
is an i.m. rectifiable current in R2 ; however we have
a(T1 +T2) L B(0,1) = 2QOB(0,1/2) ]
0.
2.4 Integer Multiplicity Rectifiable Currents
149
Image of a rectifiable current under a Lipschitz map. Let T be an i.m. rectifiable k-current in U C R', T = -r (M, 0, C) E 7Zk(U), U C 1R71, and let f : U C R'n -+ V C RN be a Lipschitz map such that f, spt T is proper.
Then the push-forward or the image of T under f we have already defined in the general context of currents turns out to be an i.m. rectifiable k-current which can be explicitly written as
f#T(w) = f
(2)
<w(f(x)), (AkdMfx)S(x)>0(x) d7-(k(x) .
M
This can be easily seen in the case in which T is the current integration over a smooth submanifold and f of class C1, and actually in the same way in our present situation, as we can write M apart from an 7{k-zero set as union of 7-lk-measurable subset of k-dimensional C'-submanifolds. Recall in fact that
I
an orientation of the rectifiable set f (M). Therefore from the general area formula we have
f#T(w) = f <w(y),
(3)
f(M)
xEf-'(y)rm+ 0(x)
> d7lk(y) I
I
where (4)
Nl+ = {x E M I JM(x) > 0}
.
In fact choosing any orthonormal basis (Tl (y), ... , Tk (y)) in Ty f (M) in a 7-lk-measurable way for 7-lk-a.e. y E f (M), setting 71(y)
Ti
(y)A... ATk(y)
and for Nk-a.e. y E f (M)
N(y) :=
9(x)E(x) xEf -' (y)nM+
where e(x) = ±1 is defined by
(ilkdMfx)e(x) (AkdMfx)t(x)
= 6(x ) 77(f (x))
2. Integer Multiplicity Rectifiable Currents
150
we can write
f#T(w) =
J
Nd7-lk
f(M)
f#T = r(f(M), INS, sign Nil) and clearly
E
IN(y)I =
0(x)
xEf -1(y) nM+
_I
0
I
Fig. 2.8. (a) 7r# Q S' J = 0, 7r# I I Q flI I = 2-L' L (-1,1); (b) -#Q AB -J = Q (0,1) ], =AA.C' L(0,1).
Later we shall be interested to the case in which f is the projection it
:
(x, y) E Rn+N -+ x E 1n and T = -r (M, 0, ) E Rn(R n+N) As a trivial example consider the case n = N = 1 {(x, kx)
Nt and
I
x E (0,1)} , e + kE
T = T Mill 1+k2
k> 0
E R1(R2) .
Then we have 7r#T = T((0,1), 1, e) = Q (0, 1) 1
.
Notice that
T - H1LM 1+k2+711LM
1+k
and, as measures, we have
7r(7-I1L.M) =
l+k2H'L(0,1)
k2e
2.5 Slicing
151
while -ir#e = e, lr#E = 0 so that 7r#T=
1+k2H1L(0,1)
1+k2
e=f1L(0,1) e.
Remark the difference between projecting a measure and projecting a current!
The Cartesian product of rectifiable currents. It is also not difficult to show that the Cartesian product of two i.m. rectifiable currents we have defined in the general context of currents is also a rectifiable current. More precisely, if Ti = T (Mi, 8i, Si) E Rki (Ui) ,
U i E ]E8'
,
i=1,2
then
Ti X T2 = 7-(Ml x M2, 8102, eiA 2) E Rki+k2 (Ul X U2) Notice however that T1 and T1 x T2 rectifiable do not imply that T2 must be rectifiable. For instance take
Ti = kQ(0,1)l , so that
T2 = k Q(0,1)
T1xT2 = [(0,1)x(0,1)1.
2.5 Slicing Let T be a k-dimensional current in Rn and let f : lI , R' be a Lipschitz map. Corresponding to y E R' one often needs to consider the part of T in f -1(y) as a (k - m)-dimensional current k > m, usually denoted by . There is a measure theoretic method to define the slice of a current T by f at least if T has certain regularity property. It has very interesting applications for instance to intersection theory in algebraic geometry and to complex analysis. Later we shall use some elementary properties of the slices. Therefore we shall collect here a few facts concerning slicing. First let us consider codimension-one-slices, i.e. the case m = 1. In order to
define the slice of T by f at y we need T to have some regularity property. We shall now see that it can be defined for locally normal or i.m. rectifiable k-currents, and of course the two definitions agree if T E Rk(Rn) f1 Nk,1oc Il$ n
Definition 1. Let T E Nk,ioc (U) where U is an open set in R', i.e., let
Nv(T) := Mv(T) +Mv(aT) < oo and let f : U
IR be a Lipschitz function. We set
d V cc U
,
2. Integer Multiplicity Rectifiable Currents
152
(b)
(a)
(c)
Fig. 2.9. (a) Q B(0,1)1 n if < t}, (b) -8Q B(0,1) L if < t}, (c) < QB(0,1)
(1)
a(T L { f < t}) - 8T L { f < t}
-a(TL{f >t})+aTL{f >t}
t >.
where
If > t} { f < t}
{x E U I f (x) > t} {x E U f (x) < t} I
Of course we have
:= = if
M(TL{f =t})+M(aTL{f =t}) = 0,
and this happens for all but countably many values of t. Therefore is well defined for 7-11-a.e. value t.
Consider now the case of i.m. rectifiable currents. As a consequence of the theory of rectifiable sets and of the general coarea formula, in particular taking into account Said type lemma in Sec. 2.1.4, we infer Lemma 1. Let .M be an 7-lk-measurable and countably rectifiable k-dimensional
set in R', f : R -+ I[8 be a Lipschitz function, and
M+ := {x E M I Jm(x)
:_ I (AkdMfx)eI > 0}
where 1; (x) is a k-vector orienting TIM . Then for 7-1' -a. e. t E R we have (i) ,Mt := f-1(t) n M+ is countably rectifiable and 7-1k-1-measurable. (ii) For 7 1k-1-a.e. x E .Mt TM t and TxM both exist, TxMt is a (k - 1)dimensional subspace of TxM and
TxM = {T + AVM f (x)
I
T E TxMt, A E IR}
where V- "f (x) is the gradient off projected on TxM.
2.5 Slicing Moreover
fdtfgd71_1
(2)
R
153
= JpvMfIgd7k M
Mt
for all non-negative 7-lk-measurable functions g.
Applying (2) to g X{f
f
r
tt
I V f lg d7-lk =
Mn{f
J -00
ds
J M..,
g
dHk-1
so that the left hand side is an absolutely continuous function of t and for a.e.
tE]l. (3)
dt
f
IVMf Ig dRk
= fgdh/c_1 Mt
Mn{f
Using the scalar product <, > in AkTTM, define now for l; E AkTTM and w E TAM the (k - 1)-vector 6 L w by (4)
:= <e,wA97>
drlEAk-1TM.
Then one sees that for 7-L1-a.e. t E ]l8 and 7-fk-1-a.e. x c Mt
Fig. 2.10. Slice
VMf (x) L I VMf (x) is a unit simple (k - 1)-vector.
E
Ak-1TTMt
154
2. Integer Multiplicity Rectifiable Currents
Definition 2. Let T = -r(M, 0,1;) E Rk(U). For a.e. t E R we define the slice of T by f at t
as the rectifiable current in 7Zk_1(U) given by
:= 7-(JV1t, et, fit) where for 7-1k-1-a.e. x E Mt
et(x)
fi(x)L
et(x)
0+,&t
Vmf(x) I v f (x) I ,
0+
0XM+
We have
Proposition 1. Let T = -r (M, 0, 1;) E Rk(U), U C RT. Then (i) For any open set V C U
f Mv()dt= f IvMflBdxk<Jsup IoMfIMV(T). MnV
Ilt
(ii) If moreoverT E Nk,lOC(U), i.e, Mv(aT) < oo for all V c c U, then for a.e.
tER
= a(TL{f
aTL{f < t}
.
(iii) If also aT is rectifiable, then for a.e. t E R
= -a
.
For locally normal currents in Nk,lo,:(U) we finally have
Proposition 2. Let T E Nk,lo,(U), U C Rn. Then
(i) spt C sptTn{f=t}
.
(ii) For all open sets V C U
Mv() <supIDfI limio hMv(TL{t < f < t+h}) v
Mv() <supjDfl limia Mv(TL{t - h < f
fMv(TL{f
a
Thus bb
fMv()dt< supIDfIMv(TL{a < f
V
2.5 Slicing
155
We shall not prove Proposition 1 and Proposition 2, but we would like to make a few remarks on their proofs. Proposition 1 (i) is a consequence of (2). The proof of Proposition 1 (ii) is done by approximating f with its smoothings fem. The proof of Proposition 1 (iii) is simple. One applies (ii) of Proposition 1 to OT
<3T, f, t> = a(3T L {f < t})
,
while applying a to the identity in (ii)
a< T, f, t > = -a(aT L If < t})
.
In order to prove (2) one first consider the case f E C', and then one gets the result by approximation. For f E C' and -y : IR --* [0, +oo) smooth and increasing we get
[a(TLyo f)(3TLyo f)] (W) =(TL7of)(dW)-aT(yo fW)=T(-yo fdw)-T(d(yofw)) =T(yofdw-d(-yofw)) = -T(-y' c, fdfAw)
.
Choosing -y such that -y = 0 for t < a, -y = 1 for t > b, and
0 < y' <
b+¢ E
fora0,
we infer
a(T Ly o f)(W) - (aT L-y o f)(W) --> if b 1 a and this implies (i). Also
IT(y'ofdfAW)I
SsupIDfI-±-Mv(TL{a< f
which implies (ii) and (iii). Quite more complicated is to deal with slices of codimension larger that 1. To
motivate the procedure let us consider the special case T = r(M,1, ) where M is a smooth k-dimensional submanifold in U C Ilk' oriented by the k-vectorfield 6, with ?-l!` (M) < oo, and let
f:R' -- R',
k>m
be smooth. For x e M we factor
e(x) _ r7(x)AC(x) so that 77(x) is a simple m-vector of TIM, 17(x) = 1
= Jf (x) ((x) is a simple (k - m)-vector of TxM, C(x)I = 1, and
,
156
2. Integer Multiplicity Rectifiable Currents ker d1M f (x) is associated with C(x)
in case J fm (x) > 0. Using the coarea formula one finds that if w E Dk-m (U) and cp is any real valued Borel function on ]gym, then
T L f#(cpdyln...ndym)(w) =T(f#(cpdy1n...ndym)nw)
f
=
f#(cpdy1A...Adym)Aw(x)>d7-lk(x)
M
= Jpj(x))Jf (x)<((x),w(x)>dHk(x) M
= li
=
f f
f
cc(y)
<((x),w(x)>d7-lk-m(x)d7-Cm(y)
Mn f-1(y)
W(y),r (Mn f(y),1,()(w)dxm(y)
R-
It follows from the differentiation theory that for 71 -a.e. y TL
f#(XB(y,p)dyiA...
Adym)/(wnpn) -, T(M n f-1(y),1, C)
in Dk-.,,,.(U) as p - 0. When T E Nk(U) is an arbitrary normal current (or even a flat chain), one introduces for y E U and p > 0 the (k - m)-currents TP :=
1
wmP,m,
TL
f#(XB(y,P)(t)dt1n...ndtm)
E Dk-m(Rn)
and defines the slice of T at y as the weak limit of Tp
:= li oTp whenever it exists. It turns out that exists for 7-lm-a.e. y. In fact for a E Dk-m(U) we have f# (T L a) E N,,,, (Rm), hence, compare Theorem 2 in Sec. 4.3.1,
f#(TLa)(w) = fhaWdX,
w E Dm(Rm)
for some ha E Ll(]Rm). Consequently TT(a) = (_l)m(k-m) f ha(t) d7-Lm B(y,p)
and limp_,o Tp (a) exists for each fixed a and 7-lm-a.e. y by the Lebesgue differentiation theorem. If we choose a denumerable subset C of Dk-m (U) which is dense
2.6 The Deformation Theorem and Approximations
157
with respect to the uniform topology we infer that for 7-lm'-a.e. y limp-o Tp(a) exists for all at E C, and, as
IT( a)1 <M(f#(TLa))
Dk-,,,,(U).
Also one has (i) (ii)
spt c f -1(y) n spt T,
a = (-1)"' ,
k > m.
Moreover (iii) (iv)
E Nk_m(U) for fm'-a.e. y (T L f#dy'A... ndym)(w) = f (w) dxm(y), w E Dk-m(U)
(v) M(TL f#dy'A...Adym) = fM()d?-lm(y), and finally (vi)
is i.m. rectifiable for 7..( a.e. y if T is i.m. rectifiable.
Finally we remark that there is no problem in extending the theory of slicing to the case in which f takes values in an oriented m-dimensional submanifold of class C1.
2.6 The Deformation Theorem and Approximations In this section we shall state without proofs the important deformation theorem and some of its consequences: the isoperimetric inequality and the approximation theorem.
Consider that e-grid or E-skeleton, E > 0, of Rn+N Ln,e
{x E In
I
1x E Zn+N} E
and denote by Ln,E the n-dimensional faces of Ln,E, or equivalently the family of all n-dimensional cubes in lRn+N with side E and vertices in Ln,e
Theorem 1 (The deformation theorem). We have (i) Let T E Nn(Rn+N) and let E be any positive real number. Then there exist a real polyhedral chain P made of n faces in Ln,E, i.e.
P = E aj j Fj I
aj E R
Fj E.Cn,c
and
R E Nn+l(1E2n1N)
such that
S E Mn(In+N)
T = P+aR+S
.
158
2. Integer Multiplicity Rectifiable Currents
Moreover P is controlled as follows
M(P) < cM(T) ,
M(aP) < cM(aT)
and the rest aR + S is small in the sense that
M(R) < ceM(T)
M(S) < cEM(aT)
and
M(3R) < c{M(T) +EM(aT)} , where c is an absolute constant. Finally spit P, spt R C U5 (spt T)
spt aP, spt aR C Ub (spt aT )
where b = b(E) = 2 n _+ NE and U5(A) :_ {x
I
dist (x, A) < b}.
(ii) If moreover T E Rn (Rn+N) is i.m. rectifiable, we can choose P and R i. m.
rectifiable, i.e. R E Rn+1(iR +N) and
P = E aj [ Fj
with aj E 7G
,
FjEG,,,E
and, if also aT is i.m. rectifiable, we can choose S i.m. rectifiable, too.
Such a result is obtained by deforming T into Ln,e, so that the result is automatically a polyhedral chain. The main error term is aR, where R is the surface through which T is deformed; S is a secondary error term due to moving aT into the skeleton Ln,E. It is not difficult to see that there is no real problem to perform such a procedure by projecting from points inside the (n + 1)-cubes defined by Ln,E, if T does not wind about the center projection and has no autointersections. In the general case the key point is choosing the center of projection suitably by observing that in the average the distortion is still controlled. We shall postpone the proof to Sec. 5.1.1. A first consequence of Theorem 1 is the following
Theorem 2 (Isoperimetric inequality). Let T be a boundaryless i.m. rectifiable (n - 1)-dimensional current in Rn+N with compact support. Then there exists an i.m. rectifiable n-current R with compact support such that aR = T and (1)
M(R) < c
M(T)n/(n-1)
where c = c(n, N).
Proof. From the deformation theorem we can write in correspondence of E > 0
T = P+aR+S . The result follows by choosing e, but strangely, E large. From
2.6 The Deformation Theorem and Approximations
159
M(S) < c_-M(OT) = 0 we infer S = 0. Next we observe that if
P= E aj[F,
O,
F3 EGA,E
we have
N(e) integer M(P) = N(e) en-1 , as a. E Z and M(QF, ]]) > en-1. From M(P) < cM(T), and the previous inequality we therefore infer by choosing (2cM(T))1/(n-1)
e that
N(e)en-1 = M(P) < cM(T)
=Zn-1
,
i.e. N(e) = 0. Then P = 0 and T = 8R with spt R compact and
M(R) < csM(T) = c'
(M(T))n/(n-1)
As in the classical case it turns out that the best constant c in (1) is that corresponding to equality in
M(vT) <
cM(T)n/(n-1)
which arises when T is a k-dimensional ball.
The next consequence of the deformation theorem we want to state is the following
Theorem 3 (Weak polyhedral approximation). Let T E Rn(U), U open in Rn+N, be such that aT E Mn_1(U). Then there is a sequence of i.m. polyhedral chains of the form Pj
a(i) QF50) ]I
a(j) E Z, FS(j) E Ln,j,
s
S
such that (2)
sups (M(P,) + M(8PM < oo
PP - T
weakly in Dn(U)
and consequently (3)
aPj -1 aT
weakly in Dn_1(U)
.
-- 0
160
2. Integer Multiplicity Rectifiable Currents
Proof. First assume U = Rn. We apply the deformation theorem with e = ej to obtain
T = P3 +&R;+S3
with
sup(M(Pj) +M(aPj)) < c(M(T) +M(aT)) < oo M(RS) < cep M(T) ,
M(SS) < cep M(aT)
.
Hence S? - 0, Rj - 0, consequently ORS - 0, and we get Pj - T. In the general case we take cp : 118'+N _, ]18 with cp > 0 in U, cp - 0 in n+N \ U such that {x v(x) > Al CC U V A > 0. For x1-a.e. A > 0 we infer from Sec. 2.2.5 that TA := T L {x W(x) > A} is such that M(aTA) < oo. Since spt TA CC U, we can apply the previous argument and conclude the proof by choosing a suitable sequence A, 10. I
I
Notice that in particular Theorem 3 says that every i.m. rectifiable ndimensional current is the weak limit in the sense of currents of a sequence of smooth n-dimensional submanifolds of I18'+N One can also prove, compare Sec. 5.1.1
Theorem 4 (Strong approximation). Let T be an i.m. rectifiable current in Rn+N, T E Rn(Rn+N) with compact support and OT E Rn_1(118'). Let e > 0. Then there exist an i.m. rectifiable polyhedral chain P in Rn+N and a C1diffeomorphism of ][8'+N into ll8'+N such that (4)
M(P - f#T) + M(aP - of#T) < e
.
Moreover
spt P C {x dist (x, spt T) < e} =: Ue (spt T) I
(5)
Lip f-1 < 1+e Lip f < 1+e , f(x) - xI < edxEI[8'+N and
We emphasize that inequality (4) means that P and f#T, as well as aP and f#8T coincide up to a piece of small mass. Usually the class of i.m. polyhedral chains of dimension n in Rn+N is denoted by P,(R,+N) It is defined as the class of currents of the form
where a, . E Z and Zj are n-dimensional simplexes in R +N Finally we would like to state a similar approximation theorem with mass for normal currents. Here it is essential to use the mass norm of n-vectors instead of the Euclidean norm, compare Sec. 2.2.1 and Vol. II Ch. 1. We denote by Pn,(I8'+N) the class of polyhedral chains with real coefficients. We have, compare Vol. II Sec. 1.3.4 and Federer [226] for a complete proof
2.7 The Closure Theorem
161
Theorem 5 (Strong approximation theorem for normal currents). Let T belong to N,,(R'+N) with sptT compact. Then there exists a sequence of polyhedral chains P; with real coefficients, P3 E Pn(Rn+N), such that
T
Pj
weakly in Dn(Rn+N)
aPj - aT weakly in Dn_1(Rn+N) and
M(T)
M(PG)
and
M(aPj) -+M(8T).
2.7 The Closure Theorem In this final subsection we would like to indicate the main steps in the proof of the closure theorem for i.m. rectifiable currents, Theorem 1 in Sec. 2.2.4.
Let U C Rn+N be an open set and let T E Nn,loc(U). For a E U and r < ra, := dist (a, aU) we know from Sec. 2.2.5 that the slice of T over 9B(a, r) is well defined as
:= 8(TLB(a,r))-(8T)LB(a,r) and is an i.m. rectifiable (n-1)-current for ?1-a.e. r < ra, if moreover T E Rn(U). We have
Lemma 1 (Slicing lemma). Let Tk,T E Nn,loc(U) be such that sup N v (Tk) < no
bVCCU
k
Tk - T
weakly in V,(U)
Then, given a e U, for.Cl-a.e. r < ra there exists a subsequence, depending on r, {Tk} of {Tk} such that
supNv() < no
(1)
(2)
VVcc U.
k
If moreover for some Vo CC U we have Nv0 (Tk) -+ 0 then {Tk} can be chosen in such a way that also Nvo () -+ 0.
Proof. Passing to a subsequence we can assume that 11Tk11+I
aTkH - Y
where p is a positive Radon measure on U. If p(OB(a, r)) = 0, (notice that this holds for all but countably many r at most), we then infer
TkLB(a,r)
TLB(a,r)
,
9TkLB(a,r)
aTLB(a,r)
,
162
2. Integer Multiplicity Rectifiable Currents
hence
a(TkLB(a,r)) -s 8(TLB(a,r))
,
consequently (1) holds. For 0 < rl < r2 < rd and V CC U, we also know from Sec. 2.2.5 that r2
f Mv()dr < cMv(TkLB(a,r2)). rl
The same holds for
a = -a(aTk L B(a, r)) _ - hence
r2
f fk(r)dr < c Nv (Tk L B (a, r2)) . rl
where
fk(r) := Nv() > 0.
Fatou's lemma and the equiboundedness of Nv (Tk) then yield r2
r2
lkm inf fk(r) dr < lim of j fk(r) dr < oc
.
rl
rl
Therefore for a.e. r E (rl, r2) we have liminfk_. fk(r) < oo. Choosing those is we can finally find a subsequence such that (2) holds. The rest of the claim is trivial.
A part from the simple remark in Lemma 1, the first key ingredient in the proof of the closure theorem is the following theorem which says that if almost all slices of T by spheres, i.e., by f (x) := Ix - al, are rectifiable then T is rectifiable, the second ingredient is the boundary rectifiability theorem stated below.
Theorem 1. Let T E Mn,lar(U). Suppose OT = 0 and that for all a E U and almost all r < ra, the slices
:= a(T L B(a, r)) are i.m. rectifiable. Then T is i.m. rectifiable.
Let us postpone the proof of Theorem 1 and let us give the Proof of the Closure theorem. We shall prove it by induction on the dimension n of the currents (keeping fixed n + N and U).
Let Tk E Rn(U), supk Nv(Tk) < oo V V CC U, Tk -L T in Dn(U). By the same argument in Theorem 3 in Sec. 2.2.6 we can assume U = Rn+N and supk N(Tk) < oo.
2.7 The Closure Theorem
163
The theorem clearly holds for 0-dimensional currents, compare Sec. 2.2.4. Suppose that it holds for (n - 1)-dimensional currents and let us prove it for n-currents. From the proof of the boundary rectifiability theorem below (where one uses the closure theorem for (n - 1)- currents) we infer that
aTk E Rn_1(U) and of course N(aTk) < oo. Using again the closure theorem for (n-1)-currents we therefore infer aT E Rn-1(U). By the isoperimetric theorem, for instance, we can then find S E Rn_1(U) such that OS = aT. By considering
T := T-S
TK := Tk - S,
we see that it suffices to prove the theorem in the case aT = 0. Fix a. By the slicing lemma, for a.e. r < ra, there is a subsequence {Tk} of {Tk} such that
,
sup N() < oo. k
Again by the proof of the boundary rectifiability theorem / E R,-,(R n +N )
and by the inductive assumption we conclude
E Rn_1(Rn+N) The conclusion now follows from Theorem 1.
Theorem 2 (Boundary rectifiability theorem). Let T E R ,,,(U), U open in I[Bn+N Suppose
VVccU.
MS (T) < oo
Then OT is an i.m. rectifiable current in U, aT E Rn_1(U). Proof. By the polyhedral weak approximation theorem in Sec. 2.2.6 we find a sequence {Pk} of polyhedral chains with integer coefficients with
supN(Pk) < oo
Pk - T .
,
k
Trivially
aPk E Rn-1(U)
,
sup N(aPk) < oc . k
Therefore applying the closure theorem (note: for (n -1)-currents) we infer that aT is i.m. rectifiable, as aPk - aT.
2. Integer Multiplicity Rectifiable Currents
164
Remark 1. We have seen that the closure theorem for boundaryless i.m. rectifiable (k - 1)-currents in lRn implies the boundary rectifiability of i.m. rectifiable k-currents in R. Let {Ti} be a sequence in Rn_1(Rn) satisfying spt T2 C B(0,1) ,
5Ti = 0 ,
Ti - T in Dn,_1(Rn)
We can write
Ri E Rn(ln) ,
Ti = aRi ,
spt Ri C B(0,1) ,
sup N(Ri) < oo . k
Trivially, since the Ri's are n-currents in ll8n, then Ri - R, R E Rn(R'), and
Ti - T=8R By the boundary rectifiability theorem we then conclude T E Rn,_1(R' ). In other words: The closure theorem for boundaryless i.m. rectifiable currents of codimension 1 is equivalent to the boundary rectifiability theorem for i.m. rectifiable currents of maximal dimension.
Let us discuss now the proof of Theorem 1. It relies on the important and difficult Besicovitch-Federer structure theorem, compare Theorem 3 in Sec. 2.1.4. The first step consists in proving the following lower density lemma.
Theorem 3 (A lower density lemma). Let T E V (U), U C Rn+N with
MS(T)+Mv(aT) < oc
VVCCU.
Then
(i) For II T II-a.e. x
A(x, r)
_
T OIITII(B(x,r)) - 1 where
A(x, r) := inf{M(S) 18S = a(T L B(x, r)), S E Dn(U)} . (ii)
If moreover aT = 0 and if a(T L B(x, r)) is i.m. rectifiable for every x and almost every r, then there is S > 0 such that
B*(IIT II,x) > d for II T
II-a.e. x E U.
Proof. Clearly .(x, r) < II T II (B(x, r)), so if (i) is false there is an e > 0 and a set X such that I I T I I (X) > 0 and for each x E X there exist arbitrarily small
balls B (x, r) with
A(x, r) < (1 - e) M(T L B(x, r)) . We assume
2.7 The Closure Theorem
165
XcVccU. By Besicovitch's covering theorem, for every p > 0 there exists a disjoint collection B1, B2 ... C V of such balls covering II T II-almost all of X and having radii < p. Let Si E V,,,(U) be a current such that
M(Si) < (1 - e) M(T L Bi)
9Si = 3(T L Bi) , and set
T. := T->TLBi+I:Si. i
i
Then, denoting by xi the center of the ball Bi, we have
(T-Tp)(w) =1: (TLBi -Si)(w) i
E ,9(5.i x (T L Bi - Si)) (w)
< 1: < hence Tp
(8yi x (T L Bi - Si)) (dw)
pM(TLB2-Si)supIdwl
M(Ox(TLBi-Si))supIdwl
2pM(T L Bi) sup dwl < 2pMv (T) sup Idwl
,
T as p --+ 0 and
Mv(T) < liminfMv(Tp) p-0
(3)
.
On the other hand Mv(TT)
<
Mv(T->TLBi)+>Mv(Si) i
i
Mv(T-TLBi)+(1-E)Mv(TLBi) i
i
Mv(T)-EEMv(TLBi) i
< Mv(T) - EII T 11(X) which contradicts (3). To prove (ii) consider a point x at which (i) holds, and let
f (r) := M(T L B(x, r))
.
Then for r small, r < R,
f (r) < 2A (x, r) By the slicing theory in Sec. 2.2.5, as 8T = 0,
M(8(T L B(x, r))) < f' (r)
for a.e. r
2. Integer Multiplicity Rectifiable Currents
166
and by the isoperimetric inequality A(x
c f'(r)
r)(n-1)/n <
for a.e. r
7
since the slice a(T L B(x, r)) is rectifiable. Thus f(r)(n-1)/n <
c f'(r)
or
dr
f (r)1/n > c'
.
Therefore
f(r)
>_
(c'r)n
for r < R.
Next step is the following rectifiability theorem
Theorem 4 (Rectifiability theorem). Let T E M,(U), U C Rn+N, with aT c M,-,(U). Set
M :_ {x E U I and suppose that
0' (11 T II , x) > 0}
IITII(U\M) = 0
i. e. that
e*n(IITII,x)>0
for IITII
- a.e. x EU.
Then T is a real-multiplicity rectifiable current.
Proof. First we show that Nl is a countable union of sets of finite fn-measure. In fact, setting Mk :_ {x 19*n (II T II, x) > 1/k} by Theorem 5 in Sec. 1.1.5, we obtain
k Hn(Mk) < II T II (U) < oc
M = UMk
fn (.Mk) < 0C.
k
Let us prove now that l;(x) orients TIM, for H'-a.e. x E M. Write 00
M = Mo U U M j=1
with Hn(M0) = 0, M3 pairwise disjoint, and Mj C N N; a C1 submanifold. From
2.7 The Closure Theorem (4)
0*n(JI
0*n(1I T 1j, U Mr, x) = 0,
167
T II,JVj \ Mj, x) = 0
r,-Ej
for Hn-a.e. x E M j, writing as usual 77,,,a (y) := x + A-1(y - x) we have for any w E V(Rn+N)all x E .Mj for which (4) holds, and all A sufficiently small d < ?-ln , #aw+>B6f
yx,a#T (w) = T (rl#Aw) =
(A)
N.
where e(A) -- 0 as A 10, i.e. 77y,A#T(w) =
I
<6(x + Az), w(z) >B(x + Az) dH'(z) + e(A) .
7=a (Nj )
Since Nj is of class C1, this gives limr7x,a#T(w) = 0(x)
r
JP
w(z) > d7-ln(z)
for 'H'-a.e. x E Mj, where
P=TXNj =TxMj On the other hand from M(aT) < oo we get 1ar7y,a#T(w)I_ 1?7x,a#8T(w)I = 18T(71#aw)j
IWlPWn0*n(HHaTHH,x)
A
for H'-a.e. x E Mj. Thus for such x
lim M(aix,a#T) = 0. a-o Finally we clearly have
lim M(r7X,a#T) < oo .
a-.o
Thus we conclude that for fn-a.e. x E M we can find aZ 10 such that
97z,a,#T - Sx
,
where (5)
S. (W) = 0(x)
SS E Dn(Rn+N)
aSx = 0
r <e(x),w(z),>d?-ln(z) J
.
T. M
We finally claim that (5) and OS,, = 0 imply that fi(x) orients TIM. Without loss of generality we can assume
T 0 M= Rn x {0} C Rn+N
2. Integer Multiplicity Rectifiable Currents
168
Choose
w :=
x'cp(x)dx"A...Adx2°-1
where j > n+ 1, cp E and 5Sxo = 0
0 =
C°O(Rn+N).
E
Dn-1(Rn+N)
Since xi = 0 on Rn x {0} we deduce from (5)
asxo(w) = Sxo(dW) _
= B(xo) f w(x)<e(xo), dx'AdxilA...
Adxin-1 > d7{n(x)
ToM 0(xo)
f
e,Aej1A... / \eZn-1 dH'(x)
ToM As W is arbitrary we then get
e(xo) ejAei1A... Aei,,_1 = 0
whenever j > n + 1
This yields at once C = ±e1 A ... Aen as required, since
1.
Proof of Theorem 1. By the lower density lemma, Theorem 3, there is 8 > 0 such that o (MT
'X) > 8
for 11 T 11-a.e. x
.
Thus if we let
M = {x E 1[$n+N
0'(11 T 11, x) >Jj
we have 11 T
(6)
I(Rn+N
\ M) = 0,
and from Theorem 5 in Sec. 1.1.5
Hn(M) < b-1HH T 11 (M) < oc
(7)
.
On the other hand Theorem 3 in Sec. 4.3.1 and (6) imply that
11 T
absolutely continuous with respect to xn L M, thus we can write
11TI1(f):=J fOd7"(n and
(8)
T(w) = f<w(x),T(x)>8(x)d(x). M
One can now apply the rectifiability theorem Theorem 4 and infer that T is a real multiplicity rectifiable current. Therefore it remains to prove that O(x) E Z
7-Lna.e.xEM.
2.7 The Closure Theorem
169
With the notations in the proof of Theorem 4 we have
Si := rlx,a,#T - 0(x) If TXM
for xn-a.e x E M
where QTXM]I is oriented by e(x), and N(S,) < oo. Let d(y) := IyI, y E ]lBn+N Applying the slicing lemma we infer that for a.e. r and for a subsequence Sn3 , that we again denote by S we have
<Sj,d,r> - 9(x)IfTMf18B(0,r)If with sups N(< Si, d, r>) < oc. Choosing now r in such a way that is rectifiable, we infer that all the < S? , d, r > are rectifiable too and the closure theorem for (n - 1)-currents then yields that 9(x)QTTM fl 8B(0, r) If
is rectifiable, i.e. 0(x) E Z.
A direct proof of the closure theorem.. We close this section giving an alternative proof of the closure theorem, in fact of Theorem 1, which does not use the structure theorem nor the rectifiability theorem, but the rectifiability criterion for measures Theorem 2 in Sec. 2.1.4. A key point is again the lower density lemma Theorem 3
The proof proceeds by induction, and similarly to the previous proof it is enough to prove the following claim:
Theorem 5. If T E Dn(lRn+N), 8T = 0, M(T) < oo, and for every Lipschitz function f , the slice = a(T L { f < r}) is i. m. rectifiable, then T is i.m. rectifiable.
In the proof one can and do freely use the closure theorem for (n - 1)-currents in R,+N which is available because of the inductive hypotheses. Proof of Theorem 5. As in the proof of Theorem 4 we infer using the lower density lemma that if we let
M = {x E Rn+N
I
B; (11 T 11, x) > d}
then (9)
(10)
I T 11 (IRn+N \ .M)
= 0,
xn(M) < 5-'j 1TII(.M) < co.
Moreover from Theorem 3 in Sec. 4.3.1 and (9), we then infer that absolutely continuous with respect to xn L M, thus we can write
T(w) = or
T II is
170
2. Integer Multiplicity Rectifiable Currents
f<w(x),r(x)>dm(x)
T(w) =
T(x) := O(x)T(x)
M
and we need again to show that M is rectifiable, for 7-ln-a.e. x E M, T(x) is a
simple vector associated to TAM, and 8(x) is an integer. We then prove the claim using the rectifiability criterion of measures, Theorem 2 in Sec. 2.1.4 instead of the Besicovitch-Federer structure theorem. From general measure theory we know that for Hn almost every a E M we have (11)
0'* (fn, M, a) < 1
(12)
lim
f
7-ln(.M n B(a, r))
I F(x) - F(a) I d?ln(x) = 0
MnB(a,r)
and we fix one such a point a E M. By (12) and the definition of M we have
O (Hn,M,a) = F(a)I-le*(IITII,a) > 0
(13) and
limsup7-l' L77a,A(M)(B(0, R)) = lim sup A-n7{n(M n B(a, AR)) A-0
A-0 wnRn8n* (7 (, M, a) < oc
where as usual
,
A-1(x - a). It easily follows passing to subsequences
7-ln L ?7(M)
(14)
p
a Radon measure, and
J
f<wr(a)>ddb =: sa(w)
<w, (a) >dn L a.\, (M)
for any n-form w E V
,n+N ). Setting
(
Tak = 'la,Ak Yf'T from
(M)) _
MB(o,1) (T)1k - F(a)Rn L
nMB(a,akR) (T - , (a)7-1n L M)
J
.\k n
A
1F(x) - f(a) I d7-ln (x) --'. 0
MnB(a,Akr) we also infer (15)
TAk Tak 11
- Sa
lL L F(a) = S.
2.7 The Closure Theorem
171
Since for every Lipschitz f, almost all slices are rectifiable, from (15) the slicing lemma, and the (n - 1)-dimensional case of the closure theorem we infer that almost all slices <µLT(a), f,r> are i.m. rectifiable. Thus by Theorem 3 we have the lower density bound
j
0*(µ,x) -! T(a)I > 0
(16)
for µ-a.e. x.
We now claim that p is translation invariant in at least n-directions, and, since the set of point where (16) holds has locally finite 7-ln-measure, in exactly
n-directions. Denoting by V the subspace of jn+N of vectors v such that T is invariant under translation in the direction of v, we shall in fact prove that F(a) E AnV. This implies that dim V > n, and by the previous remark, dim V = n and 7(a) is simple. By making an orthogonal change of coordinates, we can assume that V has a basis consisting of vectors of the standard basis of Rn+N, e1, ... , en+N. Of course, in order to prove the claim, it then suffices to show that if el V, then F(a) has no-component of the form e1Aei2A ... Aei,,. By mollification we can also assume that II µ LT(a) jI = 7{,+N L 6 for a smooth 6. Then for all f E (Rn+N) 0
=
LT(a))(fdx2A...Adxn) = (pLT(a))( ax dxindx2A...Adxn)
(0(tZ
ai-xi0(x)dx= f>aif i where
ai := the eiAe2A... nen component of T(a) . Hence
i.e. 0, consequently FL L r(a) is invariant under translation in the direction E aiei. Since el V we conclude that a1 = 0 which means that 7(a) has no component of the type e1Ae2A ... nen. The same argument can be used to prove that T(a) has no component of the type elsei2A ... Aei,,. Returning to the proof of the theorem, we then find a collection PI, ... , Pp of n-planes parallel to the n-dimensional subspace determined by T(a) such that p
jU -
j=1
aj(xn LPj)
By (11) and (14) p
Eai < on*(7{n, M a) < 1 j=1
172
2. Integer Multiplicity Rectifiable Currents
and by the lower bound (16) aj > 8/IT(a)I, sop must be finite. We now claim p = 1. Suppose not and let
3E < min{dist (Pi, Pj)
I
Pj I.
Pi
Fix j and let f (x) := dist (x, Pj ). Also assume without loss of generality that Pi is parallel to ]I8n x {0} C Rn+N and set U:= B(0, R) C I2n and W := U x I[8N C fgn+N By the second of (15) lim Mw (T.\, L {E < f < 2E}) < ?(a)Iµ(W n {E < f < 2e})
koo
so by the slicing lemma we can find a further subsequence and t c (E, 2e) such that
kL{f
(17) Tl
and such that
Mw(8(Ta,, L { f < t})) --> 0. In particular
Mu(87r#(TAkLIf
(18)
Also 7r#Ta, L If < t} is an n-dimensional current in IR with bounded mass and boundary mass, hence it is represented by a BV function Uk
7r#T,\k L If < t}(W) =
JU
UkW
with
MU(87r#(Ta,, L{f < t})) = fIDuk. U
By Poincare inequality for suitable /3k we infer
f IUk - akI dx < C U
JU
IDUkI.
From (17) (18) it follows that (3k
Q = ajIT(a)I,
fInk -pI dx
0.
U
But
QkCn'({XEU
I
Uk(X)=0}) < J *k-ak(dx < c U
so
f U
IDUki
3 Notes
Gn({xEU
I
uk(X)=0})
173
0
Thus using also (14)
lim G'({x E U I G'(U) = k-.oo
uk(x)
0})
urn " (7f(fla,ak (M) n if < t}) n u)
<
lim7{CL7]a,Ak(M)({f
k--.w
µ({ f < t} n w) = aj7-C(U) . It follows that each aj > 1, and by (11), (13) and (14) 0 < 0 (7l", M, a) < E aj < 0'n* (f', M, a) < 1
j
Hence p = 1 and al = 1. Furthermore P1 must pass through the origin since for every r > 0 ,u(B(O, r))
> lim ff'L r)a,ak (M)(B(0, r)) k-oa liminf)\- H'(M n B(a,)\r)) a-.o
r-"0 (HjT j], a) > 0 by (13). Finally note that P1 is determined by T(a) and therefore does not depend on particular subsequences.
This shows that M has an approximate tangent plane for fl -a.e. a E M. Hence M is n-rectifiable by Theorem 2 in Sec. 2.1.4. As in the previous proof, see the proof of Theorem 1, or observing that by the (n -1)-dimensional closure theorem µ LT(a) has integer multiplicity slices and µ = 7{n L P1, we finally get that 0(a) = ft(a) is an integer.
3 Notes 1 The classical references for geometric measure theory are Federer's treatise [226], and Simon's book [592]. In our presentation we mostly followed Simon [592] and its terminology and notation, which slightly differ from those of Federer [226]. The reader will surely find also interesting consulting Morgan [485] Hardt and Simon [357]. We also mention Whitney [674], the foundational paper of Federer and Fleming [230], and De Rham [189].
2 The rectifiability theorem for Radon measures Theorem 2 in Sec. 2.1.4 is essentially due to De Giorgi [177]. In our presentation we have follow Simon [592]. The structure theorem is due to Besicovitch [83] for k = 1 and to Federer [223]. A presentation of Besicovitch's original proof can be found in Falconer [218], the general case is discussed in Federer [226].
174
2. Integer Multiplicity Rectifiable Currents
Concerning rectifiable sets, we would like to mention the following interesting criterion, see e.g. Alberti, Ambrosio, and Cannarsa [8] and Simon [596]. Given any set S C R' and any point x E R', define T(S, x)
J{rp j
r > 0, p = li .
x - xh X
- xh
l xh I x}t xh E S \f{x},
and the tangent space of S at x, Tan(S, x) as the smallest subspace of R' which contains T(S, x). We then have
Theorem 1. If
dimTan(S,x) < k
d x E S,
then S is fk-measurable and 7-lk-countably rectifiable.
3 The content of Sec. 2.2.1 and Sec. 2.2.2 is quite standard. The reader may consult e.g. Federer [226], and for an elementary presentation Fleming [238]. In Sec. 2.2.3, Sec. 2.2.4 we presented basic definitions and facts, as well as some simple examples. The reader is referred to the books quoted above. There one also finds the proof of the general slicing theorem, of the deformation theorem, and of its consequences. For the isoperimetric inequality the reader is referred to Almgren [22], where the optimal inequality in any codimension is proved, and to Almgren [21]. For further information on isoperimetric inequalities the reader is referred to Burago and Zalgaller [117] and to its bibliography.
4 The closure theorem was first proved for codimension one currents, i.e. (n - 1)dimensional currents in R' by De Giorgi [177]. We shall present it in Ch. 4 in the context of Caccioppoli sets. The ideas introduced in De Giorgi [177] play an important role in the general case. The first proof of the closure theorem in Sec. 2.2.7 is essentially the classical Federer-Fleming proof in Federer and Fleming [230] as modified by Simon [592], except for one point where we have used the lower density lemma, Theorem 3 in Sec. 2.2.7. Theorem 3 in Sec. 2.2.7 is taken from White [669], compare also Fleming [240] and Solomon [604]. The second proof, which develops in the same spirit as De Giorgi's proof in codimension 1, is due to White [669].
3. Cartesian Maps
In this chapter, we shall be concerned with the notions of graph, tangent plane to a graph, and weak convergence of graphs for non-smooth maps from IR' into RN, n, N > 1.
We begin in Sec. 3.1 by investigating the differentiability properties of Sobolev maps. As it is well known in this context the basic notion, at least for scalar maps, is that of distributional derivative, and Sobolev maps do not possess smoothness properties in the usual sense. In particular Sobolev maps are not even almost everywhere differentiable in the classical sense, except in the case that they have gradient summable with a power larger than the dimension n. However a theorem of Calderdn and Zygmund states that they are almost everywhere differentiable in the so-called LP-sense, compare Sec. 3.1.2. Essentially as consequence of this result we shall infer, compare Sec. 3.1.3,
that Sobolev maps, u E Wl,r(,f2,IRN), fl C IR', p > 1 possess the following Lusin type properties (i) (ii)
there is a denumerable disjoint family of measurable sets {Fk}, such that ulF, is Lipschitz continuous and meas (( \ UkFk) = 0. except on arbitrarily small open sets, u is of class C'.
It turns out, compare Sec. 3.1.4, that those Lusin type properties are equivalent to a sort of measure-theoretic pointwise differentiability notion called approximate differentiability almost everywhere. Approximate differentiability provides the right way to express local approximability by linear transformations and, just using the equivalence with the Lusin type properties mentioned above, the classical formulas for area and coarea and for the degree can be extended to almost everywhere approximately differentiable maps. All that may be regarded as a special case of the rectifiability theory of Besicovitch, De Giorgi, Federer and Fleming, we have partly described in Ch. 2. For us the important outcoming is that the graph of a map which is almost everywhere approximately differentiable is well defined, and is a countably nrectifiable set. Our definition of graph, CJu,s-i in Sec. 3.1.5, more than to the usual notion of graph of a continuous function defined in 0, is modeled after a kind of 1-graph in which each points z E Gu,SQ carries the tangent plane to CJu,n at z = (x, u(x)). An important consequence of such a definition is Lusin's property (N), i.e., that null sets in .f2 are mapped into 7i -null sets in fix RN, a
176
3. Cartesian Maps
property which is typical of Lipschitz maps, but which does not hold in general for continuous maps. This way we see that the graph and its tangent planes are not really related to the distributional derivative, but to the in some cases weaker and in general different notion of almost everywhere approximate differentiability of the map, which in fact will be the starting point for us in this chapter. As a simple example the real map x/Ixj : R -* R has a rectifiable graph with horizontal tangent plane everywhere except at zero, corresponding to the naive, and correct in the sense of approximate differentials, idea that it has zero (approximate) differential everywhere except at zero. A more complicate example which shows a lot of fractures in the graph is Cantor-Vitali function for which the approximate differential is zero almost everywhere and the graph in the previous sense consists of infinitely many horizontal segments, compare next chapter. Usually the approximate differential is denoted by apDu. As here it is our main notion of derivative, in the sequel we shall simply write Du for apDu if no confusion with the distributional derivative may arise. Approximately differentiable maps provide essentially the natural and correct class of maps whose graphs are ?f''-measurable and countably n-rectifiable
sets in the product Q x RN. Their tangent planes, at points of approximate differentiability, are described by the minors of the Jacobian matrix in terms of the approximate differential exactly as in the classical case. Consequently the n-dimensional area of the graph of an almost everywhere approximately differentiable map is finite if and only if all the minors of the Jacobian matrix are summable.
In Sec. 3.2 we introduce and discuss the class of approximately differentiable maps with summable Jacobian minors, denoted by A' (,(l, RN). Each map u E A'(17,RN) turns out to be identified completely by the integer multiplicity rectifiable n-dimensional current Gu defined as integration of compactly supported differentiable n-forms in 1 x RN over the graph GJu,.fl of u in .fl naturally oriented by the independent variables. Being interested in weak limits of sequences {uk}, there are at least three important elements which turn out to be relevant for a satisfactory geometric description of such weak limits, compare Ch. 1 and Ch. 2. (i)
the tangent plane, that is the existence almost everywhere of the tangent plane to the graphs of the {uk}'s or equivalently the rectifiability of the graphs of the {uk}'s.
(ii)
(iii)
the area of the graph or better the equiboundedness of the areas of the approximating graphs {uk}. In fact limits of graphs may be measures in the product RI x ISBN already in the case n = N = 1, if they do not have equibounded masses. the boundary in ,(2 x RN. The rectifiability of the graphs and the equiboundedness of of their areas are not sufficient to provide a good geometric descrip-
tion of limits of smooth graphs. In fact such limits may again be measures
3. Cartesian Maps
177
distributed in the product .f2 x 1RN, if the control of the size of boundaries of the approximating sequence is lost.
With respect to the first two points, the class Al (f2, RN) yields a satisfactory setting. In the sequel of Sec. 3.2 we discuss the boundary question. In the scalar case, N = 1, the summability of the distributional derivative turns out to be equivalent to the coincidence of the approximate differential and the distributional derivative and to the non existence of boundaries of the current Gu inside the cylinder (2 x R
aGuLQxIR = 0. This is no longer true in the vector valued case, N > 1. It will be true for maps u E Wl,?`(0,1RN), with n = min (n, N), that
3GULQXRN = o.
(1)
However (1) does not hold in general for maps u E Wl,p(.f2, RN) if p < n. For instance the map x/jxj from the unit ball B3 of ]R3 into JR3 belongs to W1'p for all p < 3; as det D(x/I x1) = 0, its graph has finite area, but one has aG
L B3 XR3 = -Jo x a{ B3
Condition (1) is obviously satisfied by smooth maps, and is also closed with respect to the weak convergence of graphs. In fact (1) can be equivalently stated as
G,,,(&,)) = 0
for all (n - 1)-forms w supported in ,f2 x 1R N. Thus it is natural to require (1) also for reasonable weak maps. This way we are naturally led to introduce the class of Cartesian maps as cart l (0, IRN) = {u E Al (.f2, JRN) aGu L !2 x ]RN=01 In Vol. II Ch. 2 we shall see that (1) expresses in the context of finite elasticity
the physical requirement that a perfectly elastic body does not fracture if we deform it. In Sec. 3.2.3 of this chapter we shall see that condition (1) amounts not only to the integration by parts formula for the derivatives (or equivalently to the existence in Ll of the distributional derivatives) but to all formulas of integration by parts, actually in the product space ,f2 x IRN, for all Jacobian minors, corresponding to the classical differential identities which state that all minors are null Lagrangians or divergence free fields. In Sec. 3.2.5 we shall then briefly compare traces in the sense of Sobolev spaces and boundaries. A few significant examples will be discussed in Sec. 3.2.2. In Sec. 3.2.4 we shall introduce special subclasses, denoted Ap,q, p > n - 1, q > n/(n - 1) whose
elements are Cartesian maps and discuss higher integrability of the Jacobian determinant.
178
3. Cartesian Maps
Having always in mind the question of identifying weak limits of smooth maps, we shall discuss in Sec. 3.3 the weak continuity of the minors. More pre-
cisely we ask under which conditions it happens that for a sequence {uk} in A' (fl, RN) such that strongly in L1 weakly in Ll Ma (Duk) - va weakly in Ll Uk --> u
v
Duk
we have
v = Du,
va
= Ma (Du)
.
After discussing a few examples, we shall show that essentially this happens
if aGuk LQ x RN = 0, i.e. if the {uk}'s are Cartesian maps. For the class cart' (Q, RN) we shall provide in Sec. 3.3.2 a closure theorem which essentially relies on Federer-Fleming closure theorem for integer multiplicity rectifiable currents. This way we are able to formulate quite general results concerning the weak continuity of minors which extend results as Reshetnyak's theorem on the weak continuity of determinants in the Sobolev setting. Of course cart' (Q, RN) misses nice compactness properties, it just inherits the compactness property of the L' space. This makes that class not suitable for the calculus of variations, and in fact we postpone to next chapter introducing and discussing the important class of Cartesian currents. Better compactness properties are of course enjoyed by the classes
cartP(fl, RN) :_ {u E AP(.fl, RN)
I
aGu L ,fl x RN = 0}
when p is strictly larger than 1.This will be discussed in Sec. 3.3.3. Starting from a sequence of smooth maps {uk}, for which we clearly have (2)
aGukL.0XIRN = O,
by Stokes's theorem, and assuming for instance that all Jacobian minors of {uk} are equibounded in some LP-norm with p > 1, we may then infer passing to a subsequence that
Guk - Gu where u E cartP(f2, RN). Therefore the non validity of (1) is a homological obstruction for the approximability of a map in AP(1?,RN). However it turns out that this is not the only obstruction. There are also homotopic obstructions to the approximability of maps in carte (0, RN). This follows from a result concerning the strong and the weak approximability of minors in LP. In this respect the known results seem to be still at an initial stage and better understanding seems to be needed. We refer the reader to Sec. 3.4 for precise statements and open questions.
1 Differentiability of Non Smooth Functions
179
1 Differentiability of Non Smooth Functions In this section we shall investigate pointwise differentiability properties of functions in Sobolev spaces. Let £2 be a domain in R', and let p be a real number in [1, oo). The Sobolev space W1'p(Q,118), simply W1"p(Q), we recall, can be defined as the class of all LP-functions in £ with derivatives in the sense of distributions in LP (fl)
W1'p(,fl) := {u E L2(Q)
(1)
j
Du E Lp(,f2,R")}
normed by 1/p
1/p
r (2)
II U II
J n
wl,P(n)
/
J IDuIp dx n
dx
Juj
It turns out that the linear space W1,p(,f2) is complete, i.e., is a Banach space with respect to the norm in (2), and even a Hilbert space for p = 2, with scalar product r / (u, v)w1,2(S?) := (3) u v dx + J Du Dv dx .
JQ
.n
One also shows that W1'p(Q) can be characterized as the closure of smooth functions in Wl,p, i.e., as the closure in W',P(,fl) of
{u
I
u E Cl (f2) f1 W 1'p(f2)} .
In general one interprets u in W1,P as a class of equivalence of functions in LP; in this section we shall instead work with a suitable representative of u and we are interested in its pointwise behaviour. Of course we cannot expect that Sobolev functions possess smoothness properties in the usual classical sense, but we shall show that a Sobolev function u is almost everywhere differentiable in the LP-sense, which in turn implies a Lusin type property, that is Sobolev functions are of class C1 except an arbitrarily small open sets. Of course the situation is
much easier for p > n. In this case it turns out that u is "almost everywhere differentiable" in the classical sense, and in fact we can find a representative of u which is continuous in ,f2 and almost everywhere differentiable: this is the content of the classical Morrey-Sobolev theorem. From our point of view the important outcoming of this will be the possibility of defining the graph 9Ju,.f2 of a Sobolev function u E W1,P(0, RN) and of showing that ccu,A is a countably n-rectifiable set for all p > 1. Actually both Lusin type property and the rectifiability of gu, fl turn out to be equivalent to the almost everywhere approximate differentiability, Sec. 2.1.4. In fact the notion of approximate differentiability, which is no more related to the distributional derivatives, is the right notion in order to define the graphs of a map and its tangent planes and will be one of the basic notions in this book.
180
3. Cartesian Maps
In Sec. 3.1.1 we present a few relevant facts about Hardy-Littlewood maximal operator which will be used later. As a consequence we shall then prove Lebesgue's theorem about the convergence of averages of a function on balls and an estimate of the Hausdorff dimension of the set of points which are not Lebesgue points of Sobolev functions. We conclude Sec. 3.1.1 proving the classical Calderon-Zygmund decomposition argument and giving a characterization of Zygmund class L log L. In Sec. 3.1.2 we shall prove the classical Calderon-Zygmund LP-differentiability theorem, some of its consequences, and Morrey-Sobolev theorem.
In Sec. 3.1.3, after recalling the classical Whitney extension theorem and Rademacher theorem, we shall present Lusin type properties for Sobolev functions. In Sec. 3.1.4 we shall discuss the notions of approximate limits, approximate differentials, and we shall prove equivalence of approximate differentiability and Lusin type properties. In Sec. 3.1.5, after discussing the validity of the area formula, we shall intro-
duce the notions of graph and tangent plane for vector valued maps which are almost everywhere approximately differentiable and we shall see that the area formula holds for such maps. This allows us to define the degree for maps with Jacobian determinant in LI. However we shall see that in such a general case the degree loses homological invariance and it is not stable by weak convergence.
1.1 The Maximal Function and Lebesgue's Differentiation Theorem In order to study the differentiability properties of Sobolev functions it turns out to be very useful to work with Hardy-Littlewood maximal operator which is defined, for every u E L o, (1R) and for all x E I[8"' as (1)
f
M(u) (x) = sup
r>O
I u(y) I dy
,
B(x,r)
where
1 ff()d.y
I f(y) dy
JAI
A
A
We shall now prove a few properties of M(u) which will be used in the sequel. Given a measurable function u(x) in R', the function (2)
X(t) := meas {x E R
u(x)I > t}
I
t E R+
is called the distribution function of u, and, if u E LP(]Wn), p > 1, one has
fudx Rn
= p f 00tP-1X(t) dt . 0
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
181
Moreover, for any u E L' (1R') we have
a(t) <
(3)
t
Estimate (3) is usually referred as a 1-1-weak estimate for u. In fact (3) is weaker than the summability of u, as it is shown for instance by the extension to zero of the function 1/x in (0, 1) which trivially satisfies (3) but is not in Li. where A = II U
11L1.
One can easily see that the maximal function M(u) of a non zero u E Ll (1R.' ) is not in Ll (R'y), in other words the operator u -> M(u) is not bounded in L' ('). Indeed if I u I > 0 on a set of positive measure, then M(u) decreases at infinity no faster than c IxHowever M (u) verifies a weak (1-1)-estimate which makes it very useful.
Proposition 1 (Maximal theorem). Let u E LP(R"), with p > 1. Then for any t > 0 (4)
meas {x E Rn
M(u)(x) > t} <
I
f
s
Iu(y)I dy
.
u >t/2
In particular M(u)(x) is finite almost everywhere.
Proof. Let u E Ll (Rn) and let
At := {x E Rn
M(u)(x) > t}
I
.
For every x E At there exists a ball B(x, r(x)) such that
Iu(y)I dy > t
T
B(x,r(x))
or equivalently
measB(x,r(x)) <
t
f
Iu(y)I dy
.
B(x,r(x))
Using Besicovitch's covering theorem we can choose a denumerable set of points {xi} in At such that UiB(xi, r(xi)) A and the balls B(xi, r(xi)) do not overlap
more than 3' times. Thus measAt
< >measB(xi,r(xi)) i
(5)
<
t
f a
u(y) I dy <
f
8
B(x;,r(x;))
IIfn
Suppose now that u c Lr(R'), with p > 1. The function ii (X)
{ u(x) if I u(x) I > t/2 0
if I u(x) I < t/2
I u(y) I dy
3. Cartesian Maps
182
is easily seen to be in L' (RI). Moreover
M(u)(x) < M(v,)(x) + 2
V x E Rn
hence
{x E lE$n
I
M(u) (x) > t} C {x E Rn
M(u)(x) > t/2} .
I
Inequality (5) applied to u then yields
meas{xEIR
I
M(u)(x)>t}
< 2 t3' 1 I u(y) I dy =
2
tan f
I u(y) I dy
juI>t/2
Len
The second part of the theorem follows easily from (1).
Remark 1. If u E LP(Rn), p > 1, Holder inequality trivially yields
M(u)(x)
[M(UP)(x)I1/P
Therefore we find
meas {x E R' I M(u) (x) > t} < meas {x E Rn I M(uP) (x) > tP}
f
c
f
IuIPdy
dy,
It
ju>t/2
ItLIP>tP/2
i.e., the following (p-p)-weak estimate meas {x E
(6)
R
I
M(u)(x) > t} <
IuVPdy.
Actually we have
Proposition 2. Suppose 1 < p:5 +oo and u E LP. Then M(u) E LP and IIM(u)IILP(R') < A(p) IIuIILP(Jn)
(7)
where A(p) is a constant which depends only on p and on n, A(p) -> 00 as p -+ 1.
Proof. Using Proposition 1 we infer 00
f IM(u) I P dx = p f tP-1meas{ x I M(u) (x) > t} dt Rn
o 2ju(x)P
00
= 6np f
tP-2 ( f
Iu(x)I dx) dt = 6np
Iu>t/2
0
p
=6n2P-1
p
-
f ( f tP-2dt) Iu(x)I dx
JR"
0
IJu(x)IPdx
1
Rn F-I
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
183
Remark 2. In fact we also have u E LP if M(u) E LP, p > 1, as consequence of Lebesgue's differentiation theorem below.
Proposition 1 is a useful tool in many questions which we are not going to touch. For our purposes, we only point out a trivial consequence of (4) contained in the following proposition, and we derive the fundamental differentiation theorem of Lebesgue.
Proposition 3. Let Uk, u E LP (R'), with p > 1. (i) If Uk --i u in LP, then M(Iuk - uIP) -> 0 in measure, i.e., for any e > 0
meas {x E Rn (ii)
I
M(luk - u1P)(x) > s} -4 0 as k --} no
We have tPmeas {x E l[8n
M(u) (x) > t}
I
----->
0
as t - oc
Proof. The weak estimate (6) yields meas {x E
IR
I
M(l uk
- uI P) (x) > E} <
6p II uk - u 111P
I
thus (i) follows; while (6) and the absolute continuity of the integral yield (ii).
0 Theorem 1 (Lebesgue's differentiation theorem). Let u E LP(Rn), p > 1. Then for almost every x we have
lu(x)I < M(u)(x)
(8)
and (9)
to
f
l u(y) - u(x) I P dy = 0;
B(x,r)
in particular lim
(10)
r-.o
-f
u(y) dy = u(x)
11
B(x,r)
holds for almost every x.
Let {uk} be a sequence in C0 (l ) which converges to u in LP(lRn), Passing to a subsequence we can assume that Proof.
(11)
uk(x) -> u(x)
for a.e. x
and, since M(Iuk - uIP)(x) -- 0 in measure, that
184
3. Cartesian Maps
M(luk - uIP)(x) --> 0
(12)
for a.e. x
.
Denote by E the set where (11) and (12) hold
E := {x E R'
(13)
(11) and (12) hold}
1
Clearly meas (llU \ E) = 0. Let us prove that (8) holds for all x E E. Since Uk is smooth, we have
uk (x) = lira f uk (y) dy B(x,P) thus
uk(x)I C O t
IUk(y)I dy :5 M(Uk)(x)
B(x,p)
from this inequality (8) follows taking into account (11) and (12). In order to prove (9) define now VP(u)(x)
:= sup f lu(y) - u(x) 11 dy p
For p < r we have 1/P
Cf
l u(y) - u(x) 11 dy
< oscB(x,P)uk
+lt
dy}1/P
T
[(u(y) - uk(y)) - (u(x) - uk(x))]P
B(x,P)
< oSCB(x,P)uk + [M(Iu - ukVP)(x)]1/P + lu(x) - uk(x)I .
Thus, taking into account (8) for x E E, we find that for all k and x E E (VP(u)(x))1/P < oSCB(x,r)uk + 2[M(lu - ukl')(x)]1/P
(14)
Now for every k we choose bk so that osCB(x,bk)uk < 2-k uniformly in x, we then deduce from (14) that for any x E E
lim (Vak(u)(x))1/P < lim 2-k +2 lim
k-00
k--.oo
k-,oo
ukIP)(x)]1/P = 0
and this implies (9). Finally (10) is a consequence of (9). Remark S. For future uses, we explicitly state that (8), (9) and (10) of Theorem 1 hold for all x E E, where E is the set defined in (13).
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
185
Actually the previous considerations apply to any representative u of a func-
tion in L'(R') since we have used, compare Theorem 1, pointwise values of U.
Definition 1. Let u E
The Lebesgue set Cu of u is the set of points x E R" for which there exists A E IR such that (15)
f
l u(y) - A dy
as r --> 0 .
0
B(x,r)
For x E Lu, the number A = Au(x) is uniquely defined by (15) and it is named the Lebesgue value of u at x. It is clear from the definition that Lu and Au(x) are uniquely defined by the class of equivalence of u E Ll(IRn) and by Theorem 1 meas (W1 \ Lu) = 0 .
(16)
Any function u(x) such that u(x) = Au(x)
dx E Lu
is referred to as a Lebesgue representative of u. For instance we can set
u(x) - f 0Au(x)
if x 0 Lu if x E Lu
,
It is easy to check that Vx E Lu
Iu(x)I < M(u)(x) If moreover u E L oc(R") we also have (17)
-/
u(y) - u(x) Ir dy ---- 0
as r -> 0 .
B(x,r)
for almost every x E W'. We shall denote by Lu,p the set of points where (17) holds.
Points in ,f2\Lu are often called exceptional points, and Q\Lu the exceptional set of u. We shall now prove that the Hausdorff dimension of the exceptional set of a function u E W11 P(Q), 12 open in ll8u, is not greater then n - p. In doing that the next lemma is the key point.
Lemma 1. Let .f2 be an open set of IIRA, v be a function in Ll r(f2) and 0
EaxES-l I limsupp ' p-0
J B (x,p)
Then we have
'H-(E,,) = 0
jv(y)jdy>0
}.
3. Cartesian Maps
186
Proof. It suffices to show that for each compact subset K C 12 we have 7-((F) _ 0 where F = Ea f1 K. Set
P-' f I v(y) I dy > s-1 }.
x E F I lim sup
F(s)
B(x,p)
Obviously F = Us_1F(S), hence it suffices to show that ?-('(F(s))
0 for
all s. Let Q be a bounded open set with K CC Q CC .(1 and let min(1, dist (K, 8Q)). For all e > 0, 0 < e < d, and for all x E F(s)
d :_ there
exists r(x), 0 < r(x) < e, such that
f
r(x)-«
Iv(y)I dy > 2s
B(x,r(x))
Applying Besicovitch's covering theorem we can then choose a denumerable set
of points {xi} in F(s) such that U,B(xi,r(xi)) J F(s) and the balls B(xi,ri), ri = r(xi), do not overlap more than 3' times. Therefore we infer (18)
Eri < 2sE i
i
v(y)I dy = 23ns
I
f
Iv(y)I dy
UiB(xi,ri)
B(xi,ri)
Since a < n, this inequality implies (19)
ri <
meas(UiB(xi,ri)) C w,,,
i
i
< 23'se'-" J Iv(y)I dy
w,,,e'''-aEr"
wn, = meas (B(0, 1))
Q
From (19) and the absolute continuity theorem, applied to (18), it follows that the right hand side of (18) goes to zero as e -+ 0, and therefore that 1-(" (F(s)) = 0.
We can now state
Theorem 2. Let ,(l be an open set in ]Rh and let u E Wlo'cp'(fl), 1 < p < n. Set
El :_ {x E .(L
I
lim sup
pp-n f
P-0
I DuIp dy > 0}
B(x,p)
E2
Then we have
{x E (I lim o
u(y) dy does not exist finite} B(x,p)
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
187
=0
(20)
Wn-P (E1)
(21)
7 fn-P+e (E2) = 0
b e> 0.
Moreover the exceptional set of u has Hausdorff dimension not greater than
n - p, dimH(.fl\Gu,)
(22)
de > 0.
Proof. Lemma 1 yields at once (20). To prove (21), for a fixed xo we consider the function
u(x) dx =
O(P) :=
u(xo + Py) dy
-}
B(0,1)
The function 0(p) is differentiable in (0, oo) and
0' (P) = f Y Diu (xo + py) dy B(o,1)
hence 10' (P) I
1/P
(f
<-
I Du (x) I P dx)
.
B(xo,p)
Suppose now that xo 0 En-P+E, i.e.,
p-n+P-e
f
I Du(x) I P dx -+ 0
B(xo,p)
then
W(P)I <
cp-1+;
,
and therefore P
10(P) -O(s)1 <
which implies that xo
0'(t)I dtj <
IPA/P -SEIPI
E'2. Therefore we conclude that
E2 C E.-p+,
Vc
.
This implies (21), because of Lemma 1. Finally Poincare inequality
f B(x,p)
I u(y) - u.o,P 1 P dy < c pP
f
I Du(y) I P dy
B(x,p)
yields at once that every point in .(1 \ (E1 U E2) is a Lebesgue point for u and consequently (22) holds, if we take into account (20) and (21).
188
3. Cartesian Maps
The function
u(x) = loglog
lxj-1
which belongs to W1"2(B(0,1)), B(0,1) C R2, shows that Theorem 2 is optimal
for p > 1. Later we shall see that the case p = 1 is special. We shall in fact see in Sec. 4.1.4, Remark 3 in Sec. 4.1.4, that the exceptional set of a function u E Wl"l(f2) has zero (n - 1) -dimensional Hausdorf measure. We conclude this subsection by proving a fundamental decomposition theorem which is very useful in many instances from real and complex analysis to partial differential equations. Here we shall use it in order to give a characterization of the class of functions with maximal function in L'. Let Q be a cube in Rn. Dividing Q into 21 congruent cubes and iterating the process one obtain the so-called dyadic decomposition of Q.
Lemma 2 (Calderon-Zygmund decomposition argument). Assume that u is a non negative function in L'(Q) and let t be a positive number such that
I u(x) dx < t
.
Q
Then there exists a countably family {Qi} of cubes in the dyadic decomposition of Q such that -(i)
t< f u(x)dx<2'1t Qi
(ii)
u(x) < t
for a.e. x E Q \ UiQi
Proof. By bisection of the sides of Q, we subdivide Q into 2n congruent and equal subcubes. Those subcubes P which satisfy -1- u(x) dx < t P
are similarly subdivided and the process is repeated indefinitely (or finitely if there is no such cube). Let Q = {Qi} denote the family of subcubes so obtained which satisfy
f u(x) dx > t Q.
and for each Qi denote by Qi the subcube whose subdivision gives Qi. Since
jQil = 2" QjI we get immediately (i) as
f u(x) dx < t . Qi
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
189
If x E Q \ UiQi and is not on the boundary of some Qi then clearly it belongs to infinitely many cubes P of the successive subdivision in 2n subcubes with API --+ 0. Since, on the other hand, at every Lebesgue point x we have i
liim
t
Ju(y) - u(x) I dy < c(n) b(1limo
EP P
Ju(y) - u(x) I dy = 0 ,
B(x,,8(P))
J(P) being the diameter of P, in particular
u(x) = PIlimo + u(y) dy i
,EP JP
we infer (ii) at once.
Remark 4. Actually Calderon-Zygmund decomposition Q = {Qt} depends on the parameter t. Let {Q(t)} and {Q(s)} be the families corresponding to the parameters t and s, t < s. Then each Q(3) is contained in some Qjt). This is clear from the construction. In fact we have
f u(x)dx>s>t, Q2")
therefore in the chain of cubes {P} which are successively divided into 21 equal cubes and produce Q,8) the first cube P for which
f u(x) dx > t P
belongs to Q(t).
We shall now prove that the distribution function of the maximal function M(u)(x), the function
if
t -> t lu(x)I dx uI>t
,
and the measure of the union of the cubes Q,t) of the Calderon-Zygmund decomposition relative to Jul and t are equivalent for large values of t. In fact we have
Proposition 4. Let Q be a cube in W1, let u E L' (Q) and let t be such that
t>
f Q
3. Cartesian Maps
190
Denote by Q(t) the family of subcubes of Q in the Calderon-Zygmund decomposition relative to Jul and t. We have (23)
2_n t-1
J IuI>t
Iu(x)I dx < E
f
2 t-1
i
lu(x)I dx
IuI>t/2
and for a constant -y = 'y(n) (24)
t J
Iu(x) l dx < cl (n) meal { x E Q l M(u) (x) > y(n)t }
IuI>t
<
f J
C2 (n) t
Iu(x)I dx
IuI>7(n)t/2
Proof. Calderon-Zygmund lemma yields
f
f
Iu(x)I dx <
IuI>t
Iu(x) I dx < 2n t
IQzt) I
Q(t)
and
f
t IQzt)I << f Iu(x)I dx =
Iu(x)I dx +
Q,t)
f
Iu(x)I dx
Q;t)n{IuI>t/2} IQit)I
2
f
+
Iu(x)I dx .
Q(t)n{IuI>t/2}
from which (23) follows at once.
If x E Q(t), then
Iu(x)I dx > t
.
Q;
By considering the smallest ball B with center x containing Q?t), we then get M(u) (x) ? f Iu(y)I dy > c(n) B
l u(y) I dy ? y(n) t
.
Qit)
This proves (24) taking also into account (23) and (4).
As first consequence we shall now give a characterization of the class of functions with maximal function in L' ([8n)
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
191
Definition 2. We say that u belongs to L log L in Q if u is measurable and
JQ
Jul log(2 + Jul) dx < oc
.
Notice that
LlogL(R') C L'(IR)
.
Proposition 5. We have (i)
If u E LlogL(R7) then M(u) E Ll c(1R ).
ii)
Let Q a cube in 1R', u a measurable function with M(u) E L'(Q). Then u E LlogL(Q).
Usually Proposition 5(ii) is proved under the additional hypotheses that u = 0 outside of Q. This is not necessary and actually the two claims are equivalent. Proof. Let K be a compact in W. Using Proposition 1 we get
fM(v)(x)dx<2IKH K
J M(u)>2
M(u) (x) dx 00
00
=2IKI+J I{M(u)>t}Idt < 2JKJ +cJ (J t
2
lu(x) I dx) dt I
CI>e
Iu(x)I
= 2 1KI+ c
=2IKI +c
J IiI>1 J
Iu(x)I (
J
t dt) dx
1
Jul log Juldx.
UD>1
This proves (i). Because of Lebesgue's differentiation theorem, lu(x)l < M(u)(x)
(25)
for a.e. x in Q, hence u E L'(Q). Since (26)
J
< log 3
Q
/
J
Q
r Jul dx + J IuI>1
it suffices to prove that the last integral is finite. Set
tQ :_ Q
Jul log Jul dx
3. Cartesian Maps
192
We have
f (27)
J
lu(x) I dx + J
Jul log Jul dx < log tQ J
,
ltI>tQ
lul
lu(x)I>1
Jul log Jul dx
Using Proposition 4 we infer u(x)1
(28)
J ul>tQ
Jul log Jul dx = J lul>tQ
00
f
dt
J
lu(x)l
t
tQ
J
( f dt) dx 1
lu(x)Jdx < c1(n)lJ M(u) IL1(Q)
lul>t
We therefore conclude from (25) (26) (27) (28)
1
ullog luldx < c(n)(log(3tQ) + 1)II M(u) llL1(Q)
which proves the claim (ii).
1.2 Differentiability Properties of W1,P Functions Let u E Wlo'P(R' ), with p > 1. In this subsection we shall be concerned with pointwise differentiability properties of u, thus it is not restrictive to assume that in fact u E W1'1(Rn), or even u E W1,P(lR') and we shall think of u as of a representative of the equivalence class of u. We denote by Let u E Wlo'P
Ru := IunLDun{xlu(x)=A.u(x)} the set of Lebesgue points for both u and the weak derivatives Du where u(x) is the Lebesgue value of u at x. By Lebesgue differentiation theorem,
meas (Rn \ Ru) = 0 . For u
W' (l]), x E R and 8> 0 we define Va (Du) (x) :=
sup J
O
I
Du(y) - Du(x) IP dy
B(x,r)
Notice that, for almost all x E Ru, we have Ua (Du) (x) ---p 0
as 8 -> 0
The following proposition contains the basic estimates in order to study the differentiability properties of functions in W,1'P(IIYn)
1.2 Differentiability Properties of W" P Functions
193
Proposition 1. Let u E Wio (R ), p > 1, x E A. and u(x) be the Lebesgue value of u at x. Then for all r > 0 we have 1
(1)
B(x,r)
1u(y) -u(x)Ip dy y - xIp
dt f IDu(y)IPdy . B(x,tr)
0
In particular Morrey's inequality 1f
I u(y) - u(x) I P dy < r '
(2)
B(x,r)
J0
f IDu(y) I P dy
dt
.
B(x,tr)
holds. If moreover p > n, then I u(y) - u(x) I dy < p
(3)
1/p l n r( f I Du(y) I P dyl l \ B(x,r)
p
B(x,r)
Proof. Of course we may assume x = 0 and u(x) = 0, thus we have to prove
f f 1
IyIrpdy,
(4)
<
dt
I
B(0,r)
IDu(y)IPdy
B(0,tr)
0
provided u E Wlo' (RI) and f u(y) I dy -> 0 as r --+ 0. B(0,r)
Step 1. (4) holds for u E C1(Rn) such that u(0) = 0. In this case in fact
Du(ty) y dt
u(y) 0
so that
f
u)IP IyI
f
dy
B(0,r)
B(0,r)
f
P
1
dy
IDu(ty)I dt
0
1 11
dy f B(0,r)
0
IDu(ty)IPdt =
Jdt f IDu(y)Ipdy 0
B(0,tr)
Step 2. We now prove (4) when u E C1(IIPn \ {0}) and
f
B(O,r)
r -> 0. Assume
I u(y) I dy
0 as
3. Cartesian Maps
194
J0
dt
IDu(y)IP dy < +oo
-f B(O,tr)
otherwise there is nothing to prove. By changing the order of integration we get f1f
dzJ IDu(tz)dt = B(0,r)
fdt
f
JDu(y)j1 dy < 00
B(0,tr)
0
0
from which we deduce that for a.e. z E B (0, r) the function u, ,(t) := u(tz), t E (0, 1) is absolutely continuous. In particular there exists
,\(z) := limu(tz) t