Mariano giaquinta, giuseppe modica, jiří souček cartesian currents in the calculus of variations i cartesian currents (ergebnisse der mathematik und ihrer grenzgebiete 3 folge a series o

One should thinkof 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 satisfythe hom

1947-Cartesian currents in the calculus of variations

G,aqu,nta, Giuseppe Modica, Juui Soucek.

Mar, and

o 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 1 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

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

materi-al is concerned, specificmateri-ally the rights of translation, reprinting, reuse of illustrations, recitation,

broadcasting, reproduction on microfilms or in any other ways, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the

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ToCecilia and Laura, Giulia, Francesca and Sandra,

Eva and Sonia.

Non-scalar variational problems appear in different fields In geometry, for

in-stance, we encounter the basic problems of harmonic maps between Riemannian

manifolds and of minimal immersions; related questions appear in physics, forexample in the classical theory of o-models Non linear elasticity is anotherexample in continuum mechanics, while Oseen-Frank theory of liquid crystalsand Ginzburg-Landau theory of superconductivity require to treat variationalproblems 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 hasbeen growing interest, knowledge, and understanding of the general theory forthis 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 tothe scalar one - or in other words to the occurrence of singularities in vectorvalued minimizers, often related with concentration phenomena for the energydensity - and because of the particular relevance of those singularities for theproblem 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, thisamounts roughly to the question of identifying weak maps or fields, and weaklimits 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, anddifferent choices often lead to different answers concerning equilibrium points

The aim of this monograph is twofold: discussing a homological theory ofweak 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 Xand 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 stance, to that of currents and to allow vertical parts, and one is naturally led

in-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 satisfythe homological condition of having zero boundary in the cylinder X x y, andthey induce a homology map which is continuous for the weak convergence ofcurrents 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 mation and closure theorems of Federer and Fleming, that, as we shall see, play

defor-a relevdefor-ant role not only to study pdefor-ardefor-ametric but defor-also non pdefor-ardefor-ametric integrdefor-als

In the first part of our monograph, after a preliminary chapter about measuretheory and the phenomenology of weak convergence, we discuss integer rectifi-

able and normal currents, differentiability properties of the graphs of maps,

continuity of Jacobian minors, and how those notions are related in terms of proximate tangent planes, area or mass, and the homological notion of boundary

ap-We also discuss related topics as, for instance, higher integrability of

determi-nants, 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 thehomology map associated to a Cartesian current in terms of periods and cycles

In doing this we have tried to keep our treatment elementary, illustratingwith simple examples the results, their meaning and their typical use, and wealways 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 possibleindipendently of the general context, so that parts of this monograph can beeasily used separately for example for graduate courses This we hope justifiesthe size of our monograph Open questions are often mentioned and in the finalsection 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 andnon parametric formulations, and in connection with them, different notions ofellipticity (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, andVol 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 nonparametric area problem

For further information about the content of this monograph we refer thereader to the introductions to each chapter, to the detailed table of contents,and to the index

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 istero dell'Universita e della Ricerea Scientifica, Consiglio Nazionale delle Ri-cerche, EC European Research Project GADGET I, II, III, and Czech Academy

Min-of Sciences Parts Min-of this work have been written while the authors were iting different Universities We want to extend our thanks to Alexander vonHumboldt Foundation, Sonderforschungsbereich 256 of Bonn University, to the

vis-Departments of Mathematics of the Universities of Bonn, Cachan, Keio, Paris VI,

Shizuoka, and to the Forschungsinstitut ETH, Zurich, to the Center for matics and its Applications, Canberra and to the Academia Sinica, Taiwan

Giuseppe Modica

Jiri Soucek

Cartesian Currents in the Calculus

of Variations I and II

Volume I Cartesian Currents

1 General Measure Theory 1

2 Integer Multiplicity Rectifiable Currents 69

3 Cartesian Maps 175

4 Cartesian Currents in Euclidean Spaces 323

5 Cartesian Currents in Riemannian Manifolds 493

Bibliography 667

Index 697

Symbols 709

Volume II Variational Integrals 1 Regular Variational Integrals 1

2 Finite Elasticity and Weak Diffeomorphisms 137

3 The Dirichlet Integral in Sobolev Spaces 281

4 The Dirichlet Energy for Maps into S2 353

5 Some Regular and Non Regular Variational Problems 467

6 The Non Parametric Area Functional 563

Bibliography 653

Index 683

Symbols 695

Volume I Cartesian Currents

1 General Measure Theory 1

1.1 Measures and Integrals 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 Borel Regular and Radon Measures 10

(Borel functions and measures, Radon measures Caratheodory's rion in metric spaces Vitali-Caratheodory theorem Lusin's theorem) 1.3 Hausdorff Measures 12 (Hausdorff measures and Hausdorff dimension, spherical measures Iso- diametric inequality Cantor sets and Cantor-Vitali functions Caratheo- dory construction Hausdorff measure of a product)

crite-1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems 23

(Lebesgue's decomposition theorem, Radon-Nikodym differentiation orem Vector valued measures and Riesz representation theorem)

the-1.5 Covering Theorems, Differentiation and Densities 29

(Vitali's and Besicovitch's covering theorems Symmetric differentiation and Radon-Nikodym theorem Densities Approximate limits and mea- surability Densities and Hausdorff measure)

2 Weak Convergence 36

2.1 Weak Convergence of Vector Valued Measures 36

(Definitions Banach-Steinhaus theorem and the compactness theorem Convergence as measures and L1-weak convergence Lebesgue's theorem

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)

(Weak convergence in L' and weak and weak" convergence on Banach

spaces Riemann-Lebesgue's lemma Radon-Riesz theorem and variants)

(The proof of Lebesgue theorem on weak convergence in L' Weak vergence of the product)

con-2.5 Concentration: Weak Convergence of Measures 55

(The universal character of the concentration-distribution phenomenon The concentration-compactness lemma)

2.6 Oscillations: Young Measures 58

(Equivalent definitions of Young measures Examples)

2.7 More on Weak Convergence in Ll 64

(Convergence in the sense of biting and convergence of the absolutely

continuous parts)

Integer Multiplicity Rectifiable Currents 69

1 Area and Coarea Countably n-Rectifiable Sets 69

1.1 Area and Coarea Formulas for Linear Maps 69(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 the- orem)

1.4 Rectifiable Sets and the Structure Theorem 90

(Countably n-rectifiable and n-rectifiable sets The approximate gent space of sets and measures The rectifiability theorem for Radon

tan-measures Besicovitch-Federer structure theorem)

1.5 The General Area and Coarea Formulas 99(Area and coarea formulas for sets on manifolds and for rectifiable sets The divergence theorem)

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 SimSim-ple 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

Inte-gration 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 n- dimensional 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

(Currents carried by smooth graphs Rectifiable and integer ity rectifiable currents The closure theorem for integer multiplicity 0-

multiplic-dimensional and 1-multiplic-dimensional rectifiable currents Examples Image of

a rectifiable current under a Lipschitz map The Cartesian product of

rectifiable currents)

(Slices of codimension one Slices of codimension larger than one)

2.6 The Deformation Theorem and Approximations 157(The deformation theorem Isoperimetric inequality Weak and strong

polyhedral approximations The strong approximation theorem for mal currents)

nor-2.7 The Closure Theorem 161

(The classical proof: slicing lemma, the boundary rectifiability theorem, the rectifiability theorem White's proof)

3 Notes 173

Cartesian Maps 175

1 Differentiability of Non Smooth Functions 179

1.1 The Maximal Function and Lebesgue's Differentiation

Calderon-Zyg-1.2 Differentiability Properties of W 1,P Functions 192

(Differentiability in the LP sense Calderon-Zygmund theorem Sobolev theorem)

Morrey-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 Approximate Differential and Lusin Type Properties 210

(Approximate continuity and approximate differentiability Lusin type properties are equivalent to the approximate differentiability)

1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps 218(Area formula, graphs and degree for non smooth maps Rado'-Reichel- derfer theorem)

2 Maps with Jacobian Minors in L' 228

2.1 The Class AI (.(l, RN),Graphs and Boundaries 229(The class A'(0, 1RN ) The current integration over the graph Conver- gence of graphs and minors)

(The map x/Ixl Homogeneous maps: boundary and degree)

2.3 Boundaries and Integration by Parts 238

(Approximate differential and distributional derivatives Analytic

for-mulas for boundaries Boundary and pull-back )

2.4 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

the sense of Sobolev Weak and strong anchorage)

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 the- orem)

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

4.1 The Transfinite Inductive Process 299

(Ordinal numbers The transfinite inductive process and the weak

se-quential 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 The Join of Cartesian Maps 313

(Composition and join of Cartesian maps Weak continuity of the join)

4 Cartesian Currents in Euclidean Spaces 323

1 Functions of Bounded Variation 327

smooth open sets)

1.3 De Giorgi's Rectifiability Theorem 346

(Reduced boundary De Giorgi's rectifiability theorem Measure

theo-retic boundary Federer's characterization of Caccioppoli sets)

1.4 The Structure Theorem for BV Functions 354

(Jump points and jump sets Regular and singular points The ture of the measure total variation Lebesgue's points and approximate

struc-differentiability of BV-functions)

1.5 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

2.1 Limit Currents of Smooth Graphs 380

(Toward the definition of Cartesian currents)

2.2 The Classes cart (,fl x IRN) and graph(.f2 x ISBN) 384

(Cartesian currents and graph-currents)

2.3 The Structure Theorem 391

(The structure theorem The map associated to a Cartesian current.

Weak convergence of Cartesian currents)

2.4 Cartesian Currents in Codimension One 403

(BV functions as Cartesian currents and representation formulas tor-Vitali functions SBV functions)

Can-2.5 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

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)

3.2 Degree Mapping and Degree of Cartesian Currents 460

(Definition and properties of the degree for Cartesian currents and maps)

3.3 The Degree of Continuous Maps 471

(The degree of continuous Cartesian maps agrees with the classical

de-gree for continuous maps)

3.4 h-Connected Components and the Degree 474(Homologically connected components of a Caccioppoli set)

4 Notes 479

5 Cartesian Currents in Riemannian Manifolds 493

1.1 The Deformation Theorem 494

(Proof of Federer and Fleming deformation theorem and of the strong

Fed-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 Hodge Operator 531

(Interior multiplication of vectors and covectors Hodge operator The

L2 inner product for forms)

2.3 Sobolev Spaces of Forms 536

(The classes Lr2,(X) and WP'2(X))

erator The lemma of Gaffney at the boundary Hodge-Morrey

decom-position Hodge representation theorem of relative cohomology classes)

2.7 Weitzenbock Formula 559

(Connections and covariant derivatives Levi-Civita connection Second

covariant derivatives Curvature tensor and Laplace-Beltrami operator

on forms)

2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology 565(De Rham cohomology groups Poincare duality Relative cohomol- ogy groups on manifolds with boundary Cohomology long sequence.

Poincare Lefschetz duality)

3 Currents and Real Homology of Compact Manifolds 570

3.1 Currents on Manifolds 572

(Currents in X Flatness and constancy theorems)

3.2 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' In- tersection of cycles is the wedge product of Poincare duals Kronecker

index Intersection index)

3.5 Relative Homology and Cohomology Groups 608

(Homology and cohomology in the Lipschitz category Closure of cosets Generalized de Rham theorem) 4 Integral Homology 615

4.1 Integral Homology Groups 615

(Integral homology groups Integral relative homology groups Isoperi-metric inequalities and weak closure Torsion groups Integral and real homology) 4.2 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) 5 Maps Between Manifolds 631

5.1 Sobolev Classes of Maps Between Riemannian Manifolds 632 (Density results of Schoen-Uhlenbeck and Bethuel d-homotopy White's results) 5.2 Cartesian Currents Between Manifolds 640

(Approximate differentiability Area formula Graphs Cartesian cur-rents The class carta,l(!2 x y)) 5.3 Homology Induced Maps: Manifolds Without Boundary 648

(Homology and cohomology maps associated to a Cartesian current) 5.4 Homology Induced Maps: Manifolds with Boundary 658

(Homology and cohomology maps associated to flat chains) 6 Notes 663

Bibliography 667

Index 697

Symbols 709

Volume II Variational Integrals

1.1 The Abstract Setting 3

(Lower semicontinuous and coercive functionals Weierstrass theorem.

r-relaxed functional) 1.2 Some Classical Lower Semicontinuity Theorems 10

(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 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)

2.2 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)

2.3 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 The Polyconvex l.s.c Extension of Some Lagrangians 45

(Area of graphs The total variation of the gradient The Dirichlet gral The p-energy functional The liquid crystal integrand)

inte-3 Regular Integrals in the Class of Cartesian Currents 74

3.1 Parametric Integrands and Lower Semicontinuity 75

(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

(The relaxed functional in classes of Cartesian currents and maps)

3.4 Relaxed Energies in the Parametric Case 90

(The approximation problem for parametric integrals A theorem of Reshetnyak Flat integrands and Federer's approximation theorem In-

teger multiplicity rectifiable and real minimizing currents)

4 Regular Integrals and Quasiconvexity 106

4.1 Quasiconvexity 107

(Quasiconvexity as necessary condition for semicontinuity Rank-one

convexity and Legendre-Hadamard condition)

4.2 Quasiconvexity and Lower Semicontinuity 117

(Quasiconvexity as sufficient condition for semicontinuity in classes of Sobolev maps or Cartesian currents)

4.3 Ellipticity and Quasiconvexity 127

(Ellipticity, quasiconvexity and lower semicontinuity) 5 Notes 131

2. Finite Elasticity and Weak Diffeomorphisms 137

1 State Space and Stored Energies in Elasticity 139

1.1 Fields and Transformations 139

(Non local and non linear structure of transformations) 1.2 Kinematics 140

(Bodies, states, and deformations of a body Deformations as 3-surfaces in R6, or as graphs of diffeomorphisms) 1.3 Local Deformations 143

(Infinitesimal deformations as simple tangent vectors to the deformation surface) 1.4 Perfectly Elastic Bodies: Stored Energy, Convexity and Coercivity 147

(Stored energy: different forms and constitutive conditions Polyconvex-ity and coercivPolyconvex-ity) 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

2.1 Kinematical Principles in Elasticity: Weak Deformations 156 (Material body and its parts Impenetrability of matter Weakly invert-ible maps Weak one-to-one transformations Existence of local defor-mations Weak defordefor-mations 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 Convexity-like Conditions 170

(Convexity is not compatible with elasticity Noll's condition Polycon-vexity and Diff-quasiconPolycon-vexity) 2.4 Coercivity Conditions 175

(A discussion of the coercivity conditions) 2.5 Examples of Stored Energies 179

(Ogden-type stored energies for isotropic materials) 3 Weak Diffeomorphisms :; 182

3.1 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 diffeo-morphism) p , 4 3.2 The Classes dif (0, Rn) 191

(Weak diffeomorphisms with non-prescribed range Closure and com-pactness properties Elastic deformations as weak diffeomorphisms) 3.3 Convergence Theorems for the Inverse Maps 199

(Convergence of the ranges and of the inverse maps) 3.4 General Weak Diffeomorphisms 204

(Weak diffeomorphisms with vertical and horizontal parts: structure, closure, and compactness theorems) 3.5 The Dif-classes 213

(The approximation problem for weak diffeomorphisms) 3.6 Volume Preserving Diffeomorphisms 214

(The Jacobian determinant of weak one-to-one maps and of weak

de-formations)

4 Connectivity Properties of the Range of Weak Diffeomorphisms 216

4.1 Connectivity of the Range of Sobolev Maps 217

(Connected sets and d,_1-connected sets sets are "mapped" into connected sets by Sobolev maps) 4.2 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 diffeomor-phisms 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 An a.e Open Map Theorem 243

(A.e open sets and a.e continuous maps Weak diffeomorphisms in W1' ', p > n - 1, "are" a.e open maps) 5 Composition 247

5.1 Composition of weak deformations 248

(Composition of one-to-one maps and of weak diffeomorphisms) 5.2 5.3 On the Summability of Compositions 250

(Binet's formula and the summability of the composition) Composition of Weak Diffeomorphisms 254

(The action of weak diffeomorphisms on Cartesian maps and the pseu-dogroup structure of weak diffeomorphisms Weak convergence of com-positions) 6 Existence of Equilibrium Configurations 260

6.1 Existence Theorems 261

(The displacement pressure problem Deformations with fractures) 6.2 Equilibrium and Conservation Equations 264

(Energy-momentum conservation law and Cauchy's equilibrium equa-tion) 6.3 The Cavitation Problem 268

(Elastic deformations do not cavitate) 7 Notes 278

3. 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 Re-verse Holder inequalities with increasing supports) 2.2 Classical Regularity Results 303

(Morrey's regularity theorem for 2-dimensional weak harmonic maps) 2.3 An Optimal Regularity Theorem 307

(A partial regularity theorem and the existence and regularity of en-ergy minimizing harmonic maps with range in a regular ball: results by Hildebrandt, Kaul, Widman, and Giaquinta, Giusti.) 2.4 The Partial Regularity Theorem 319

(The partial regularity theorem for energy minimizing weak harmonic

maps: Schoen-Uhlenbeck result)

3 Harmonic Maps in Homotopy Classes 333

3.1 The Action of Wl'2-maps on Loops 334

(Courant-Lebesgue lemma In the two-dimensional case the action on

loops is well defined for maps inW122)

3.2 Minimizing Energy with Homotopic Constraints 336

(Energy minimizing maps with prescribed action on loops Schoen-Yau, Saks-Uhlenbeck, Lemaire, Eells-Sampson and Hamilton theorems)

3.3 Local Replacement by Harmonic Mappings: Bubbling 337(Jost's replacement method and existence of minimal immersions of S2)

4 Weak and Stationary Harmonic Maps with Values into S2 339

4.1 The Partial Regularity Theory 339

(An alternative proof More on the singular set)

4.2 Stationary Harmonic Maps 345

(Partial regularity results for stationary harmonic maps)

4 The Dirichlet Energy for Maps into S2 353

1 Variational Problems for Maps from a Domain of R2 into S2 3541.1 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 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:

Brezis-Coron and Jost result A global regularity result)

2 Variational Problems from a Domain of JR3 into S2 3832.1 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 Dipoles and Gap Phenomenon 400

(Dipoles and the approximate dipoles Lavrentiev or gap phenomenon)

2.4 The Structure Theorem in cart2'1(.f2 x S2), Q C J 3 409

(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 con- nection 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 onan)

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 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

per-turbed functionals of the type of Ginzburg-Landau)

5. Some Regular and Non Regular Variational Problems 467

(Existence and regularity of equilibrium configuration)

1.2 The Relaxed Energy 470(Existence of equilibrium configurations for the relaxed energy The dipole problem Relaxed energies in Sobolev spaces and Cartesian cur-

rents Equilibrium configurations with fractures)

(Approximation in energy: irrotational and solenoidal dipoles The eral dipole problem)

gen-2 The Dirichlet Integral in the Regular Case: Maps into S2 4852.1 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)

(Degree with respect to a (n - 3)-curve and the dipole problem)

(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 The Dirichlet Integral and Minimizers 506

4 The Dirichlet Integral in the Non Regular Case: a Homological

Theory 508

4.1 (n,p)-Currents 509

((n, graphs and the classes A(P),' rectifiable (r, currents, (r, mass, (r, p)-boundary Vertical (r, p)-currents and cohomology Integer

p)-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 The Dirichlet Integral 534

(Representation Minimizers and homological minimizers of the Dirichlet integral)

4.5 Prescribing Homological Singularities 543(s-degree Lower bounds for the dipole energy)

6 The Non Parametric Area Functional 563

1 Area Minimizing Hypersurfaces 564

1.1 Parametric Surfaces of Least Area 564

(Hypersurfaces as Caccioppoli's boundaries: De Giorgi,s regularity orem, monotonicity, Federer's regularity theorem Surfaces as rectifiable

the-currents: Almgren's regularity theorem Minimal surfaces as stationary varifolds: a survey of Allard theory Boundary regularity: Allard's and

Hardt Simon's results) 1.2 Non Parametric Minimal Surfaces of Codimension One 579(Solvability of the Dirichlet problem Bombieri-De Giorgi-Miranda a pri-

ori estimate The variational approach Removable singularities

Liou-ville type theorems Bernstein theorem Bombieri-De Giorgi-Giusti orem on minimal cones)

the-2 Problems for Maps of Bounded Variation with Values in Sl 5902.1 Preliminaries 594

(Forms and currents in fl x S1 BV(fl,E): a survey of results) 2.2 The Class cart(,fl x Sl) 600

(The structure and the approximation theorems)

2.3 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)

3 Two Dimensional Minimal Surfaces 619

3.1 Plateau's Problem 619(Morrey's e-conformality theorem Douglas-Rado existence theorem Hildebrandt's boundary regularity theorem Branch points and embed-

ded minimal surfaces: Fleming, Meeks-Yau, and Chang results) 3.2 Existence of Two Dimensional Non Parametric MinimalSurfaces 625

(Rado's theorem and existence, uniqueness, and regularity of

two-di-mensional graphs of any codimension) 3.3 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 Hilde- brandt, Jost, and Widman)

4 Least Area Mappings and Least Mass Currents 632

4.1 Topological Results 633

(Representation and homology of Lipschitz chains)

(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 The Mass of Cartesian Currents and the Relaxed Area 641(Graphs of finite mass which cannot be approximated in area by smooth graphs)

5.2 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 definitionsand results from general measure theory that will be freely used later on; inSec 1.2, due to its relevance for the sequel, we shall discuss in more details weakconvergence 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 setsand 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 ortheir proofs are especially relevant for the sequel; sometimes, proofs are post-poned 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 asreference 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,

(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

(ii) 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,

µ* is monotone, i.e., if A C B then µ*(A) < µ*(B),

p* is countably sub-additive, i.e., if {Ek} is a countable collection of subsets

of X, then

00

(iv) µ* is an extension of µ, i.e., p* (E) = p(E) whenever E E F,

(v) µ* can also be computed as

Notice that in general µ* is not countably additive

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

sub-additive

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 istrue 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

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 surable sets

mea-(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 tially due to Caratheodory and it is known as Caratheodory's method; for thisreason outer measures are often called Caratheodory's measures Such a method

essen-in fact works startessen-ing 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

0"

00

a* (E) := infa(Gk) I Gk E , U Gk D E}

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

Definition 5 A measure A on X is said to be c-regular, where 9 is a collection of 2X, if for every E C X we have

sub-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 holdfor all sequences {Ek} not necessarily in Ma Moreover the following measura-

bility 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 thatA(F) = A(E) Observe that if A is regular and E ¢ Ma, then there is F E Mawith 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-almostevery x in X, if there is a null set A C X, p(A) = 0, such that P(x) holds foreach xEX\A.

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 classicalapproach 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-measurablefunctions

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

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 whichtakes only a finite number of values

Let f : X -+ [-oo, +oc] be any function and let Sµ denote the set of allµ-integrable step functions We define

ffd µ := inf{I(g; µ) I g E SN, f< g µ -a.e}

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

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) L' (X; µ) is a linear space; moreover If I E L' (X ; µ) if f E Ll (X; µ)

(ii) The operator

fk(x) -> f (x) uniformly in B .

(ii) Beppo Levi theorem If 0 < fl < f2 < and f (x) := limk-,, fk(x),then

(iii) Fatou's lemma If fk > 0, k = 1, 2, , then

r

flim f fk dµ < lim n f fk dµ .

(iv) Lebesgue's dominated convergence theorem If fk(x) -+ f (x) for

p-a e x E X and if there exists a p-summable function g such that

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

we deduce that fk - f in measure p provided f I fk - f I dp -+ 0, i.e.,

Ll-convergence implies Ll-convergence in measure

Also, because of Egoroff's theorem, convergence a.e in a set of finite measureimplies convergence in measure

Proposition 4 Let {fk} be a sequence of p-measurable functions, which

con-verges to f in measure p Then there exists a subsequence { fk, } such that

r 1 if 2 < x< 2±1 _

A (x) ' SI 0 otherwise Then

but

lim fk(x)kooexists for no x E [0, 1] If we instead set

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

meas{xER I fk(x)- f(x)I >e}<2-"-+0 ask >oo

Definition 11 Let p be a measure on X

(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 calledcountably ,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 andv-measurable sets Bk C Y such that M C UkAk x Bk The set function µ x v iscalled 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, X E X

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 v-

measurable function, then f (x, ) : y E Y -+ f (x, y) is v-integrable for p-a.e

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

1 (ff(xv)()) dp(x)

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 algebra containing all open sets in X is called the Borel algebra and is denoted by B(X) A function f : X -> Y, with values in atopological space Y is called a Borel function if, for every open set U C Y,

o-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

n-dimensional 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 givesa 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 metricspace 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,,

Next theorem states the basic approximation property in terms of Borelmeasures

Theorem 2 Let p be a Borel measure on the metric space X and let B C X

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 pose that X C Uk 1Uk where Uk are open sets with p(Uk) < +oo Then

Sup-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 separablespace 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 ameasurable set by a continuous map f in general is not measurable For instanceCantor-Vitali function in Sec 1.1.3 below maps the Cantor set, which is a nullset, 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

The following is then easily proved taking into account Lusin's theorem low: Let µ be a Borel-regular measure in X and let f : X -, ]R'n be a Borelmap

be-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 Recallthat 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 beobtained 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 functionscpk 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

(iii) Let X = Ian, let A C Rn be p-measurable with µ(A) < +oo, and let e > 0

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 closedcubeswith disjoint interiors and with side one, one easily sees that Lusin's theoremholds for any measurable set A of R', not necessarily of finite measure, if p isthe 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-dimensionalmeasures

in JR'1 for any s, 0 < s < n and actually in any metric space They will becontinuously used in the sequel

Let s be a nonnegative real number We denote by w9 the volume of the unitball in IRS f o r s = 1, 2, 3, , we set wo := 1 and we let w9 any convenient fixedconstant for non-integers s Since the measure of the unit n- ball is given by

rn/2(n/2)I'(n/2)where F(t) is Euler's gamma function

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

Observe that 71 > 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 todefine H' The naive definition

in fact would have very unpleasant features.

In (1) we may also require that the sets Ck be closed or open without changingthe 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

thus 7{s (A) = +oo if 7lr (A) > 0 and ?ir (A) = 0 if 'Hs(A) < oo This inparticular shows that 71s (A) can be positive and finite for at most one (possiblynone) 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-regularfor 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

Using the so-called isodiametric inequality

(diarn A '

which says that among sets A C ][fin with a given diameter p, the ball with

diam-eter p has the largest Lebesgue measure, one also proves the following importanttheorem

Theorem 1 The Hausdorff measure H-ln in R' coincides with the Lebesguemeasure 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 aset and even harder to find its Hausdorff measure Of course the most difficultpart 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 oflength 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+ifor 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

{

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 ofthe Cantor set C inductively by

Fig 1.3 Self-similarity of Cantor sets in two dimensions.

sense that certain homothetic expansions of C are locally identical to the originalset; more precisely we have

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 similarity of C can be also expressed by

self-cc

c = n rk([O,1]) ,

k=0

i.e., 7- (C) = C

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 theright and left end points of any interval I We have

if a E I,,,,,i, we set a' = R(I00,i); a' > a

if aEJk,j, we set a'=L(Jk,j); a'>a-8kSimilarly

k

k "`

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