dacorogna b direct methods in the calculus of variations sinhvienzone com

Applied Mathematical Sciences
Volume 78
Editors
S.S. Antman J.E. Marsden L. Sirovich

Advisors

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J.K. Hale P. Holmes J. Keener
J. Keller B.J. Matkowsky A. Mielke
C.S. Peskin K.R. Sreenivasan

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Applied Mathematical Sciences

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33. Grenander: Regular Structures: Lectures
in Pattern Theory, Vol. III.
34. Kevorkian/Cole: Perturbation Methods
in Applied Mathematics.
35. Carr: Applications of Centre Manifold Theory
36. Bengtsson/Ghil/Källén: Dynamic Meteorology:
Data Assimilation Methods.
37. Saperstone: Semidynamical Systems in Infinite
Dimensional Spaces.
38. Lichtenberg/Lieberman: Regular and Chaotic
Dynamics, 2nd ed.
39. Piccini/Stampacchia/Vidossich: Ordinary
Differential Equations in Rn.
40. Naylor/Sell: Linear Operator Theory in
Engineering and Science.
41. Sparrow: The Lorenz Equations: Bifurcations,
Chaos, and Strange Attractors.
42. Guckenheimer/Holmes: Nonlinear
Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields.
43. Ockendon/Taylor: Inviscid Fluid Flows.
44. Pazy: Semigroups of Linear Operators and
Applications to Partial Differential Equations.
45. Glashoff/Gustafson: Linear Operations and

Approximation: An Introduction to the
Theoretical Analysis and Numerical Treatment of
Semi-Infinite Programs.
46. Wilcox: Scattering Theory for Diffraction
Gratings.
47. Hale: Dynamics in Infinite Dimensions/Magalhāes/
Oliva, 2nd ed.
48. Murray: Asymptotic Analysis.
49. Ladyzhenskaya: The Boundary-Value Problems of
Mathematical Physics.
50. Wilcox: Sound Propagation in Stratified Fluids.
51. Golubitsky/Schaeffer: Bifurcation and Groups
in Bifurcation Theory, Vol. I.
52. Chipot: Variational Inequalities and Flow
in Porous Media.
53. Majda: Compressible Fluid Flow and
Systems of Conservation Laws in
Several Space Variables.
54. Wasow: Linear Turning Point Theory.
55. Yosida: Operational Calculus: A Theory
of Hyperfunctions.
56. Chang/Howes: Nonlinear Singular Perturbation
Phenomena: Theory and Applications.
57. Reinhardt: Analysis of Approximation Methods for
Differential and Integral Equations.
58. Dwoyer/Hussaini/Voigt (eds): Theoretical
Approaches to Turbulence.
59. Sanders/Verhulst: Averaging Methods in
Nonlinear Dynamical Systems.
60. Ghil/Childress: Topics in Geophysical Dynamics:

Atmospheric Dynamics, Dynamo Theory and
Climate Dynamics.

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1. John: Partial Differential Equations, 4th ed.
2. Sirovich: Techniques of Asymptotic Analysis.
3. Hale: Theory of Functional Differential
Equations, 2nd ed.
4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics.
6. Freiberger/Grenander: A Short Course in
Computational Probability and Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory.
8. Giacaglia: Perturbation Methods in
Non-linear Systems.
9. Friedrichs: Spectral Theory of Operators in Hilbert
Space.
10. Stroud: Numerical Quadrature and Solution
of Ordinary Differential Equations.
11. Wolovich: Linear Multivariable Systems.
12. Berkovitz: Optimal Control Theory.
13. Bluman/Cole: Similarity Methods for
Differential Equations.

14. Yoshizawa: Stability Theory and the
Existence of Periodic Solution and
Almost Periodic Solutions.
15. Braun: Differential Equations and Their
Applications, 3rd ed.
16. Lefschetz: Applications of Algebraic Topology.
17. Collatz/Wetterling: Optimization Problems
4th ed.
18. Grenander: Pattern Synthesis: Lectures
in Pattern Theory, Vol. I.
19. Marsden/McCracken: Hopf Bifurcation and Its
Applications.
20. Driver: Ordinary and Delay Differential
Equations.
21. Courant/Friedrichs: Supersonic Flow and
Shock Waves.
22. Rouche/Habets/Laloy: Stability Theory by
Liapunov’s Direct Method.
23. Lamperti: Stochastic Processes: A Survey
of the Mathematical Theory.
24. Grenander: Pattern Analysis: Lectures
in Pattern Theory, Vol. II.
25. Davies: Integral Transforms and Their
Applications, 2nd ed.
26. Kushner/Clark: Stochastic Approximation Methods for
Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines: Revised
Edition.
28. Keilson: Markov Chain Models–Rarity and
Exponentiality.

29. de Veubeke: A Course in Elasticity.
30. Sniatycki: Geometric Quantization and
Quantum Mechanics.
31. Reid: Sturmian Theory for Ordinary
Differential Equations.
32. Meis/Markowitz: Numerical Solution
of Partial Differential Equations.

(continued after index)

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Bernard Dacorogna

om

Direct Methods in the
Calculus of Variations

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Second Edition

ABC
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Bernard Dacorogna
´
´
Departement
de Mathematiques
´
´ ´
Ecole
Polytechnique Federale
de Lausanne
CH-1015 Lausanne, Switzerland
Editors:
L. Sirovich
Division of Applied
Mathematics
Brown University
Providence, RI 02912
USA

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J.E. Marsden
Control and Dynamical
Systems, 107-81
California Institute of
Technology
Pasadena, CA 91125
USA


S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015

USA


ISBN: 978-0-387-35779-9

e-ISBN: 978-0-387-55249-1

Library of Congress Control Number: 2007938908

Mathematics Subject Classification (2000): 74S05

© 2008 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
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Printed on acid-free paper
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Contents

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Preface

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1 Introduction
1.1 The direct methods of the calculus of variations . . . . . .
1.2 Convex analysis and the scalar case . . . . . . . . . . . . .
1.2.1 Convex analysis . . . . . . . . . . . . . . . . . . . .
1.2.2 Lower semicontinuity and existence results . . . .
1.2.3 The one dimensional case . . . . . . . . . . . . . .
1.3 Quasiconvex analysis and the vectorial case . . . . . . . .
1.3.1 Quasiconvex functions . . . . . . . . . . . . . . . .
1.3.2 Quasiconvex envelopes . . . . . . . . . . . . . . . .
1.3.3 Quasiconvex sets . . . . . . . . . . . . . . . . . . .
1.3.4 Lower semicontinuity and existence theorems . . .
1.4 Relaxation and non-convex problems . . . . . . . . . . . .
1.4.1 Relaxation theorems . . . . . . . . . . . . . . . . .
1.4.2 Some existence theorems for differential inclusions

1.4.3 Some existence results for non-quasiconvex
integrands . . . . . . . . . . . . . . . . . . . . . . .
1.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 H¨older and Sobolev spaces . . . . . . . . . . . . . .
1.5.2 Singular values . . . . . . . . . . . . . . . . . . . .
1.5.3 Some underdetermined partial differential
equations . . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Extension of Lipschitz maps . . . . . . . . . . . . .

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Convex analysis and the scalar case

2 Convex sets and convex functions
2.1 Introduction . . . . . . . . . . . . . . .
2.2 Convex sets . . . . . . . . . . . . . . .

2.2.1 Basic definitions and properties
2.2.2 Separation theorems . . . . . .

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2.3

2.2.3 Convex hull and Carath´eodory theorem
2.2.4 Extreme points and Minkowski theorem
Convex functions . . . . . . . . . . . . . . . . .
2.3.1 Basic definitions and properties . . . . .
2.3.2 Continuity of convex functions . . . . .

2.3.3 Convex envelope . . . . . . . . . . . . .
2.3.4 Lower semicontinuous envelope . . . . .
2.3.5 Legendre transform and duality . . . . .
2.3.6 Subgradients and differentiability
of convex functions . . . . . . . . . . . .
2.3.7 Gauges and their polars . . . . . . . . .
2.3.8 Choquet function . . . . . . . . . . . . .

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3 Lower semicontinuity and existence theorems
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 Weak lower semicontinuity . . . . . . . . . . . . . .
3.2.1 Preliminaries . . . . . . . . . . . . . . . . .
3.2.2 Some approximation lemmas . . . . . . . .
3.2.3 Necessary condition: the case without lower
order terms . . . . . . . . . . . . . . . . . .
3.2.4 Necessary condition: the general case . . .
3.2.5 Sufficient condition: a particular case . . .
3.2.6 Sufficient condition: the general case . . . .
3.3 Weak continuity and invariant integrals . . . . . .
3.3.1 Weak continuity . . . . . . . . . . . . . . .

3.3.2 Invariant integrals . . . . . . . . . . . . . .
3.4 Existence theorems and Euler-Lagrange equations
3.4.1 Existence theorems . . . . . . . . . . . . . .
3.4.2 Euler-Lagrange equations . . . . . . . . . .
3.4.3 Some regularity results . . . . . . . . . . . .
4 The
4.1
4.2
4.3

4.4

4.5
4.6
4.7

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one dimensional case
Introduction . . . . . . . . . . . . . . . . . . . . . . .
An existence theorem . . . . . . . . . . . . . . . . .
The Euler-Lagrange equation . . . . . . . . . . . . .
4.3.1 The classical and the weak forms . . . . . . .
4.3.2 Second form of the Euler-Lagrange equation .
Some inequalities . . . . . . . . . . . . . . . . . . . .
4.4.1 Poincar´e-Wirtinger inequality . . . . . . . . .
4.4.2 Wirtinger inequality . . . . . . . . . . . . . .
Hamiltonian formulation . . . . . . . . . . . . . . . .
Regularity . . . . . . . . . . . . . . . . . . . . . . . .
Lavrentiev phenomenon . . . . . . . . . . . . . . . .

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vii

CONTENTS

II

Quasiconvex analysis and the vectorial case

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6 Polyconvex, quasiconvex and rank one convex envelopes
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The polyconvex envelope . . . . . . . . . . . . . . . . . . . .
6.2.1 Duality for polyconvex functions . . . . . . . . . . .
6.2.2 Another representation formula . . . . . . . . . . . .
6.3 The quasiconvex envelope . . . . . . . . . . . . . . . . . . .
6.4 The rank one convex envelope . . . . . . . . . . . . . . . . .
6.5 Some more properties of the envelopes . . . . . . . . . . . .
6.5.1 Envelopes and sums of functions . . . . . . . . . . .
6.5.2 Envelopes and invariances . . . . . . . . . . . . . . .
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 Duality for SO (n) × SO (n) and O (N ) × O (n)
invariant functions . . . . . . . . . . . . . . . . . . .
6.6.2 The case of singular values . . . . . . . . . . . . . .
6.6.3 Functions depending on a quasiaffine function . . . .
6.6.4 The area type case . . . . . . . . . . . . . . . . . . .
6.6.5 The Kohn-Strang example . . . . . . . . . . . . . . .
6.6.6 The Saint Venant-Kirchhoff energy function . . . . .
6.6.7 The case of a norm . . . . . . . . . . . . . . . . . . .

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5 Polyconvex, quasiconvex and rank one convex functions
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Definitions and main properties . . . . . . . . . . . . . . . . . . .
5.2.1 Definitions and notations . . . . . . . . . . . . . . . . . .
5.2.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Further properties of polyconvex functions . . . . . . . . .
5.2.4 Further properties of quasiconvex functions . . . . . . . .
5.2.5 Further properties of rank one convex functions . . . . . .
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Quasiaffine functions . . . . . . . . . . . . . . . . . . . . .
5.3.2 Quadratic case . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Convexity of SO (n) × SO (n) and O (N ) × O (n)
invariant functions . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Polyconvexity and rank one convexity of SO (n) × SO (n)
and O (N ) × O (n) invariant functions . . . . . . . . . . .
5.3.5 Functions depending on a quasiaffine function . . . . . . .
5.3.6 The area type case . . . . . . . . . . . . . . . . . . . . . .
5.3.7 The example of Sverak . . . . . . . . . . . . . . . . . . . .
5.3.8 The example of Alibert-Dacorogna-Marcellini . . . . . . .
5.3.9 Quasiconvex functions with subquadratic growth. . . . . .
5.3.10 The case of homogeneous functions of degree one . . . . .

5.3.11 Some more examples . . . . . . . . . . . . . . . . . . . . .
5.4 Appendix: some basic properties of determinants . . . . . . . . .

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7 Polyconvex, quasiconvex and rank one convex sets
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Polyconvex, quasiconvex and rank one convex sets . . . . . .
7.2.1 Definitions and main properties . . . . . . . . . . . . .
7.2.2 Separation theorems for polyconvex sets . . . . . . . .
7.2.3 Appendix: functions with finitely many gradients . . .
7.3 The different types of convex hulls . . . . . . . . . . . . . . .
7.3.1 The different convex hulls . . . . . . . . . . . . . . . .
7.3.2 The different convex finite hulls . . . . . . . . . . . . .
7.3.3 Extreme points and Minkowski type theorem
for polyconvex, quasiconvex and rank one convex sets
7.3.4 Gauges for polyconvex sets . . . . . . . . . . . . . . .
7.3.5 Choquet functions for polyconvex and rank one

convex sets . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 The case of singular values . . . . . . . . . . . . . . .
7.4.2 The case of potential wells . . . . . . . . . . . . . . . .
7.4.3 The case of a quasiaffine function . . . . . . . . . . . .
7.4.4 A problem of optimal design . . . . . . . . . . . . . .

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8 Lower semi continuity and existence theorems in the
vectorial case
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Weak lower semicontinuity . . . . . . . . . . . . . . . . .
8.2.1 Necessary condition . . . . . . . . . . . . . . . .
8.2.2 Lower semicontinuity for quasiconvex functions
without lower order terms . . . . . . . . . . . . .
8.2.3 Lower semicontinuity for general quasiconvex
functions for p = ∞ . . . . . . . . . . . . . . . .
8.2.4 Lower semicontinuity for general quasiconvex
functions for 1 ≤ p < ∞ . . . . . . . . . . . . . .
8.2.5 Lower semicontinuity for polyconvex functions .
8.3 Weak Continuity . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Necessary condition . . . . . . . . . . . . . . . .
8.3.2 Sufficient condition . . . . . . . . . . . . . . . . .
8.4 Existence theorems . . . . . . . . . . . . . . . . . . . . .
8.4.1 Existence theorem for quasiconvex functions . . .

8.4.2 Existence theorem for polyconvex functions . . .
8.5 Appendix: some properties of Jacobians . . . . . . . . .

III

Relaxation and non-convex problems

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9.2 Relaxation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 416

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9.2.1
9.2.2

The case without lower order terms . . . . . . . . . . . . 416
The general case . . . . . . . . . . . . . . . . . . . . . . . 424

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465
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467
472
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487
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10 Implicit partial differential equations
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Existence theorems . . . . . . . . . . . . . . . . . . . . .
10.2.1 An abstract theorem . . . . . . . . . . . . . . . .
10.2.2 A sufficient condition for the relaxation property
10.2.3 Appendix: Baire one functions . . . . . . . . . .
10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 The scalar case . . . . . . . . . . . . . . . . . . .
10.3.2 The case of singular values . . . . . . . . . . . .
10.3.3 The case of potential wells . . . . . . . . . . . . .
10.3.4 The case of a quasiaffine function . . . . . . . . .
10.3.5 A problem of optimal design . . . . . . . . . . .

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11 Existence of minima for non-quasiconvex integrands
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Sufficient conditions . . . . . . . . . . . . . . . . . . .
11.3 Necessary conditions . . . . . . . . . . . . . . . . . . .
11.4 The scalar case . . . . . . . . . . . . . . . . . . . . . .
11.4.1 The case of single integrals . . . . . . . . . . .
11.4.2 The case of multiple integrals . . . . . . . . . .
11.5 The vectorial case . . . . . . . . . . . . . . . . . . . .
11.5.1 The case of singular values . . . . . . . . . . .
11.5.2 The case of quasiaffine functions . . . . . . . .
11.5.3 The Saint Venant-Kirchhoff energy . . . . . . .
11.5.4 A problem of optimal design . . . . . . . . . .
11.5.5 The area type case . . . . . . . . . . . . . . . .
11.5.6 The case of potential wells . . . . . . . . . . . .

IV

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Miscellaneous

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12 Function spaces
12.1 Introduction . . . . . . . . . . . . . . . . . . . .
12.2 Main notation . . . . . . . . . . . . . . . . . . .
12.3 Some properties of H¨
older spaces . . . . . . . .
12.4 Some properties of Sobolev spaces . . . . . . .
12.4.1 Definitions and notations . . . . . . . .
12.4.2 Imbeddings and compact imbeddings . .
12.4.3 Approximation by smooth and piecewise

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affine functions

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503
503
503
506
509
510
510
512


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CONTENTS

13 Singular values
515
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
13.2 Definition and basic properties . . . . . . . . . . . . . . . . . . . 515
13.3 Signed singular values and von Neumann type inequalities . . . . 519
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529

529
529
529
531
533
535
535
539
541
543

15 Extension of Lipschitz functions on Banach
15.1 Introduction . . . . . . . . . . . . . . . . . .
15.2 Preliminaries and notation . . . . . . . . . .
15.3 Norms induced by an inner product . . . .
15.4 Extension from a general subset of E to E .
15.5 Extension from a convex subset of E to E .

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549
549
549
551
558
565

Bibliography

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611

Index

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569

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Notation

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14 Some underdetermined partial differential equations
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
14.2 The equations div u = f and curl u = f . . . . . . . . .
14.2.1 A preliminary lemma . . . . . . . . . . . . . .
14.2.2 The case div u = f . . . . . . . . . . . . . . . .
14.2.3 The case curl u = f . . . . . . . . . . . . . . . .
14.3 The equation det ∇u = f . . . . . . . . . . . . . . . .
14.3.1 The main theorem and some corollaries . . . .
14.3.2 A deformation argument . . . . . . . . . . . . .
14.3.3 A proof under a smallness assumption . . . . .
14.3.4 Two proofs of the main theorem . . . . . . . .

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Preface

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The present monograph is a revised and augmented edition to Direct Methods
in the Calculus of Variations [179] which is now out of print. The core and the
structure of the present book are essentially the one of [179], although it has now
almost doubled its size. While writing the present volume, it clearly appeared to
me that a new subject has emerged and that it deserves to be called “quasiconvex
analysis”. This name, of course, refers to “convex analysis”, although the new
subject is still in its infancy when compared with the classical one.
The calculus of variations is an immense and very active field. It is therefore,
when writing a book, necessary to make a severe selection. This was already
the case for [179] and is even more so for this new edition. Rather than superficially covering a lot of materials, I preferred to privilege only some aspects
of the field. Here are some main features of the book. I strongly emphasized
the resemblances between convex and quasiconvex analysis as well as the “algebraic” aspect of the field, notably through the determinants and singular values.
Besides the classical results on lower semicontinuity and relaxation, an important feature of the monograph is the emphasis on the existence of minimizers
for non convex problems.
In doing so I missed several important aspects of the calculus of variations such as regularity theory, study of stationary points, existence and
relaxation in BV spaces, minimal surfaces, Young measures and the mathematical study of microstructures, Γ convergence and homogenization. However there are already several excellent books on these subjects, some of
them very classical, such as: Almgren [18], Ambrosio-Fusco-Pallara [25],
Braides-Defranceschi [101], Buttazzo [112], Buttazzo-Giaquinta-Hildebrandt

[117], Dal Maso [217], Dierkes-Hildebrandt-K¨
uster-Wohlrab [248], Dolzmann
[249], Ekeland [263], Ekeland-Temam [264], Evans [271], Fonseca-Leoni [284],
Giaquinta [307], Giaquinta-Hildebrandt [309], Giaquinta-Modica-Soucek [312],
Gilbarg-Trudinger [313], Giusti [315], [316], Ladyzhenskaya-Uraltseva [388],
Mawhin-Willem [440], Morrey [455], M¨
uller [462], Nitsche [476], Pedregal [492],
Roubicek [517] or Struwe [546], [547]. I have also added in the bibliography
several articles which present important developments that I did not discuss in
the present monograph, but are still closely related.
For a reader not very familiar with the calculus of variations, it might be
advisable to start with an introductory book such as [180], which could be considered as a companion to the present one. Nevertheless, the present monograph,

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xii

Preface

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which is essentially a reference book on the subject of quasiconvex analysis, can
be used, as was [179], for an advanced course on the calculus of variations.
I would next like to reiterate my thanks to the people who helped me while
writing the earlier version [179], namely J.M. Ball, L. Boccardo, P. Ciarlet,
I. Ekeland, J.C. Evard, B. Kawohl, P. Marcellini, J. Moser, C.A. Stuart,
E. Zehnder and B. Zwahlen.
However, since then I have benefited of many other important discussions.
Surely the most influential ones were with P. Marcellini, with whom I have a long
standing collaboration. We have written together several articles and a book
[202], which helped me in writing Part III of the present monograph. I want also
to recall fruitful discussions with E. Acerbi, J.J. Alibert, N. Ansini, G. Aubert,
S. Bandyopadhyay, A.C. Barroso, H. Br´ezis, G. Buttazzo, P. Cardaliaguet,
A. Cellina, G. Croce, G. Dal Maso, F. De Blasi, E. De Giorgi, O. Dosly,
J. Douchet, A. Ferriero, I. Fonseca, N. Fusco, W. Gangbo, N. Georgy,
F. Gianetti, J.P. Haeberly, H. Hartwig, S. Hildebrandt, T. Iwaniec, O. Kneuss,
H. Koshigoe, P.L. Lions, J. Maly, P. Mar´echal, A. Martinaglia, E. Mascolo,
J. Matias, P. Metzener, G. Mingione, G. Modica, S. M¨
uller, F. Murat,
G. Pianigiani, G. Pisante, L. Poggiolini, A.M. Ribeiro, N. Rochat, C. Sbordone,
K.D. Semmler, V. Sverak, M. Sychev, R. Tahraoui, C. Tanteri, L. Tartar,
M. Troyanov and K. Zhang.
My thanks also go to Mme. G. Rime, who typed the manuscript of [179],
and to Mme. M.F. De Carmine, who typed an earlier version of the present
monograph. Finally, M. H¨
agler and C. Hebeisen prepared for me all the figures

included in the book.
During the past several years, I have benefited from grants from the Fonds
National Suisse and the Troisi`eme Cycle Romand. Of course, particular thanks
go to the Section de Math´ematiques of the Ecole Polytechnique F´ed´erale de
Lausanne.

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Chapter 1

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The direct methods of the calculus
of variations

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Introduction

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The main problem that we will be investigating throughout the present monograph is the following. Consider the functional

I (u) :=

f (x, u (x) , ∇u (x)) dx

en

Ω

where

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- Ω ⊂ Rn , n ≥ 1, is a bounded open set and a point in Ω is denoted by
x = (x1 , ..., xn ) ;


- u : Ω → RN , N ≥ 1, u = u1 , · · · , uN , and hence
∇u =



∂uj
∂xi

1≤j≤N
1≤i≤n

∈ RN ×n ;

- f : Ω × RN × RN ×n → R, f = f (x, u, ξ) , is a given function.

We say that the problem under consideration is scalar if either N = 1 or
n = 1; otherwise we speak of the vectorial case.
Associated to the functional I is the minimization problem
(P )

m := inf {I (u) : u ∈ X} ,

meaning that we wish to find u ∈ X such that
m = I (u) ≤ I (u) for every u ∈ X.
Here X is the space
of admissible functions (in most parts, it is the Sobolev

space u0 + W01,p Ω; RN , where u0 is a given function).

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2

Introduction

We now give several examples.
(1) The classical calculus of variations dealt essentially with the case n =
N = 1, where the most celebrated examples are the Fermat principle in geometrical optics, where

f (x, u, ξ) := g (x, u) 1 + ξ 2 ,
the Newton problem, where

ξ3
,
1 + ξ2

or the brachistochrone problem, where

om

f (x, u, ξ) = f (u, ξ) := 2πu

.C


1 + ξ2
.
f (x, u, ξ) = f (u, ξ) := √
2gu

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(2) When turning our attention to the case n > N = 1 (in our terminology,
it is still part of the scalar case), the Dirichlet integral surely plays a central
role; we have there
1 2
f (x, u, ξ) = f (ξ) := |ξ| .
2
A natural generalization is when 1 < p < ∞ and

en

f (x, u, ξ) = f (ξ) :=

1 p
|ξ| .
p

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The minimal surface in non-parametric form enters also in this framework; we
have in this case

2
f (x, u, ξ) = f (ξ) := 1 + |ξ| .
In geometrical terms, the integral represents the area of the surface given by
(x, u (x)) ∈ Rn+1 when x ∈ Ω ⊂ Rn .
(3) In the vectorial case n, N ≥ 2, the first example is the case of minimal
surfaces in parametric form, a geometrical framework more general than the
preceding one. In this case, we have N = n + 1 and therefore the matrix
ξ ∈ R(n+1)×n . We denote by adjn ξ ∈ Rn+1 the vector formed by all the n × n
minors of the matrix ξ. Finally, we let
f (x, u, ξ) = f (ξ) := |adjn ξ| ,

where |.| stands for the Euclidean norm. In geometrical terms, the integral
represents the area of the surface given by u (x) ∈ Rn+1 when x ∈ Ω ⊂ Rn ;
moreover, adjn ∇u represents the normal to the surface.
Other important examples in the vectorial case are motivated by non-linear

elasticity. A particularly simple one is when N = n and
f (x, u, ξ) = f (ξ) := g (det ξ) ,

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3

Convex analysis and the scalar case
where g : R → R is a given function.

We do not discuss the history of the calculus of variations and we refer for this
matter to the books of Dierkes-Hildebrandt-K¨
uster-Wohlrab [248], GiaquintaHildebrandt [309], Goldstine [319] and Monna [449].

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The first question that arises in conjunction with problem (P ) is, of course,
the existence of minimizers. This strongly depends on the choice of admissible
functions,
by X. A natural choice would be a subspace of

which we denoted


C 1 Ω; RN , or even C 2 Ω; RN , if we want to be able to write the differential
equation naturally associated to the minimization problem and known as the
Euler-Lagrange equation. This turns out to be a strategy too hard to implement
in most problems, particularly those dealing with partial derivatives (i.e. n > 1).
The essence of the direct methods of the calculus of variations is to split the
problem into two parts. First to enlarge the space of admissible functions, for
example by considering spaces such as the Sobolev spaces W 1,p so as to get a
general existence theorem and then to prove some regularity results that should
satisfy any minimizer of (P ). In the present book, we are essentially concerned
only with the first problem. In most cases, the space of admissible functions is


X = u0 + W01,p Ω; RN ,

en

where u0 is a given function and the notation u ∈ X is a shortcut meaning that
u = u0 on ∂Ω and u ∈ W 1,p Ω; RN .
The existence of minimizers in the above space relies on the fundamental
property of (sequential) weak lower semicontinuity, meaning that
uν ⇀ u in W 1,p ⇒ lim inf I (uν ) ≥ I (u) ,

(1.1)

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ν→∞

where ⇀ stands for weak convergence. This property is thoroughly investigated,
notably in Chapters 3 and 8.
It turns out that the property (1.1) is intimately related to the convexity of
the function ξ → f (x, u, ξ) in the scalar case where N = 1 or n = 1 and to the
quasiconvexity (in the sense of Morrey) of the same function in the vectorial
case.
This leads us to the study of convex analysis in Chapter 2 and quasiconvex
analysis in Chapters 5, 6 and 7.

We now discuss in more details the content of the monograph and outline
some of the main results in every chapter. We state them, most of the time,
under slightly stronger hypotheses than needed, but we refer to the precise
theorems at each step.

1.2

Convex analysis and the scalar case

We start with the scalar case where n = 1 or N = 1. The first one corresponds
to the case of one single independent variable and is much easier to deal with,
in particular from the point of view of regularity. It is discussed in the general

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4

Introduction

framework of the scalar case in Chapter 3 but also has a special treatment in
Chapter 4. The second case, n > N = 1, involves partial derivatives and is
considerably harder; it is discussed in Chapter 3. However, since both cases use
in a significant way many results of convex analysis, we start with the study of
this classical subject.

1.2.1

Convex analysis

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In Chapter 2, we present the most important results of convex analysis. Even
though many excellent books exist on the subject, we have decided, for the
convenience of the reader, to state and to prove all the results that we need.
Another motivation in the presentation of this chapter has been to stress both
the similarities and the differences with quasiconvex analysis, which is discussed
in Part II.
Traditionally, convex analysis starts with the notion of a convex set and then
continues with that of convex functions. This is also the path we have followed,
in contrast with the quasiconvex case.
We start by recalling the notion of a convex set. A set E ⊂ RN is said to be
convex if for every x, y ∈ E and every t ∈ [0, 1]

Zo

tx + (1 − t) y ∈ E.

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We then give several elementary properties concerning the interior, closure and
boundary of convex sets. We next turn to two of the most useful results for convex sets, namely the separation theorems (see Corollary 2.11) and Carath´eodory
theorem (see Theorem 2.13). A typical separation theorem is, for example, the
following.
Theorem 1.1 Let E ⊂ RN be convex and x ∈ ∂E. Then there exists a ∈ RN ,
a = 0, so that

x; a
≤
x; a
for every x ∈ E,
where
.; .
denotes the scalar product in RN .

We also recall that the convex hull of a set E ⊂ RN is the smallest convex
set containing E and is denoted by co E. Carath´eodory theorem then states the
following.
Theorem 1.2 Let E ⊂ RN . Then



N +1

N +1
co E = x ∈ RN : x = i=1 λi xi , xi ∈ E, λi ≥ 0 with
i=1 λi = 1 .

We then conclude this brief account on convex sets by recalling the notion
of extreme points of a convex set and Minkowski theorem, ensuring that if E is
compact and Eext denotes the set of extreme points of co E, then
co E = co Eext .

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5

Convex analysis and the scalar case

We next discuss the concept of a convex function. We recall that a function
f : RN → R ∪ {+∞} is said to be convex if
f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y)

om

for every x, y ∈ RN and every t ∈ [0, 1] . An important property of convex functions that take only finite values (i.e. f : RN → R) is that they are everywhere
continuous (see Theorem 2.31).
The notions of convex set and function are related through the indicator
function of a set E defined by

0
if x ∈ E
χE (x) =
+∞ if x ∈
/ E.

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Indeed the function χE is convex if and only if the set E is convex.
As we defined the notion of a convex hull for a set, a natural concept is
the convex envelope of a given function f, which is, by definition, the largest
convex function below f and is denoted by Cf. We can therefore write, for every
x ∈ RN ,
Cf (x) := sup {g (x) : g ≤ f and g convex} .

Zo

Of central importance in convex analysis is the concept of a conjugate function
(or Legendre transform). The conjugate of a function f is a function f ∗ : RN →
R ∪ {+∞} defined by

en

f ∗ (x∗ ) := sup {
x; x∗
− f (x)} ,

x∈RN

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which is a convex function, independently of the convexity of f. Iterating the
process, we define the biconjugate of f as f ∗∗ : RN → R ∪ {±∞} , it is given by
f ∗∗ (x) = sup {
x; x∗
− f ∗ (x∗ )} .
x∗ ∈RN

It turns out that if f takes only finite values then (see Theorem 2.43)
Cf = f ∗∗ .

Finally, we also investigate the differentiability of convex functions, discussing, in particular, the notion of a subgradient.

1.2.2

Lower semicontinuity and existence results

The main result of Chapter 3 is the following (more general ones are found in
Theorem 3.15 and Corollary 3.24).
Theorem 1.3 Let n, N ∈ N, p ≥ 1, Ω ⊂ Rn be a bounded open set with a
Lipschitz boundary, f : Ω × RN × RN ×n → R be a non-negative continuous
function and

I (u) =
f (x, u (x) , ∇u (x)) dx.

Ω

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6

Introduction

Part 1. If the function ξ → f (x, u, ξ) is convex, then I is (sequentially)
weakly lower semicontinuous in W 1,p (meaning that (1.1) is satisfied).
Part 2. Conversely, if either N = 1 or n = 1 and I is (sequentially) weakly
lower semicontinuous in W 1,p , then the function ξ → f (x, u, ξ) is convex.
We should emphasize that in the vectorial case, n, N ≥ 2, Part 1 of the
theorem is valid but the conclusion of Part 2 does not hold.
This theorem, in the scalar case, has as a first direct consequence that the
functional is (sequentially) weakly continuous in W 1,p , meaning that

om

uν ⇀ u in W 1,p ⇒ lim I (uν ) = I (u)
ν→∞

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if and only if ξ → f (x, u, ξ) is affine. This result again strongly contrasts with
the vectorial case.

The main implication of the lower semicontinuity theorem is on the existence
of minimizers for the problem



(P ) inf I (u) : u ∈ u0 + W01,p Ω; RN
= m.

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Indeed we have, as a special case of our general theorem (see Theorem 3.30),
the following result.

Theorem 1.4 Let Ω be a bounded open set of Rn with a Lipschitz boundary.
Let f : Ω×RN ×RN ×n → R be continuous and satisfying the coercivity condition
p

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f (x, u, ξ) ≥ α1 |ξ| − α2 , ∀ (x, u, ξ) ∈ Ω × RN × RN ×n ,

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for some α1 > 0, α2 ∈ R and p > 1. Assume that ξ → f (x, u, ξ) is convex and
that I (u0 ) < ∞. Then (P ) has at least one minimizer.

This theorem is also valid in the vectorial case, but can then be improved a
great deal.

As is well known, associated with any variational problem is the differential
equation known as the Euler-Lagrange equation. Under appropriate regularity
hypotheses on the function f and on a minimizer u of (P ) , we find that u should
satisfy, for every x ∈ Ω,
(E)

n

∂
∂f
∂f
[ i (x, u, ∇u) ] =
(x, u, ∇u) , i = 1, · · · , N.
i
∂x
∂ξ
∂u
α
α
α=1

The differential equation is a second order ordinary differential equation if n =
N = 1, a system of such equations if N > n = 1, a single second order partial
differential equation if n > N = 1 and a system of such equations when n, N ≥ 2.
In any case, the convexity of the function ξ → f (x, u, ξ) ensures the ellipticity of
the Euler-Lagrange equations. The prototype example is the Dirichlet integral
where n > N = 1,
1
f (x, u, ξ) = f (ξ) := |ξ|2 ,
2

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Convex analysis and the scalar case
and the associated equation is nothing other than the Laplace equation
∆u = 0.

1.2.3

The one dimensional case

b

(P )

inf

I (u) =

a

om

In Chapter 4, we specialize to the case where N = n = 1, although most of the
results are also valid if N > n = 1. We are therefore considering the problem