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

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"


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Mariano Giaquinta

Giuseppe Modica
Jir i Soucek

Cartesian Currents
in the Calculus
of Variations I
Cartesian Currents

Springer


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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

Cecilia and Laura,
Giulia, Francesca and Sandra,
Eva and Sonia.


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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


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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.


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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


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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


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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


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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


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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


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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

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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)


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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


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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)


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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


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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)


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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)


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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


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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

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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,


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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.

.