Direct Methods in the Calculus of Variations

Contents

Preface

Introduction

1.1 The direct methods of the calculus of variations

1.2 Convex analysis and the scalar case

1.3 Quasiconvex analysis and the vectorial case

1.4 Relaxation and non-convex problems

1.5 Miscellaneous

Convex analysis and the scalar case

Convex sets and convex functions

2.1 Introduction

2.2 Convex sets

2.3 Convex functions

Lower semicontinuity and existence theorems

3.1 Introduction

3.2 Weak lower semicontinuity

3.3 Weak continuity and invariant integrals

3.4 Existence theorems and Euler-Lagrange equations

The one dimensional case

4.1 Introduction

4.2 An existence theorem

4.3 The Euler-Lagrange equation

4.4 Some inequalities

4.5 Hamiltonian formulation

4.6 Regularity

4.7 Lavrentiev phenomenon

Quasiconvex analysis and the vectorial case

Polyconvex, quasiconvex and rank one convex functions

5.1 Introduction

5.2 Definitions and main properties

5.3 Examples

5.4 Appendix: some basic properties of determinants

Polyconvex, quasiconvex and rank one convex envelopes

6.1 Introduction

6.2 The polyconvex envelope

6.3 The quasiconvex envelope

6.4 The rank one convex envelope

6.5 Some more properties of the envelopes

6.6 Examples

Polyconvex, quasiconvex and rank one convex sets

7.1 Introduction

7.2 Polyconvex, quasiconvex and rank one convex sets

7.3 The different types of convex hulls

7.4 Examples

Lower semi continuity and existence theorems in the vectorial case

8.1 Introduction

8.2 Weak lower semicontinuity

8.3 Weak Continuity

8.4 Existence theorems

8.5 Appendix: some properties of Jacobians

Relaxation and non-convex problems

Relaxation theorems

9.1 Introduction

9.2 Relaxation Theorems

Implicit partial differential equations

10.1 Introduction

10.2 Existence theorems

10.3 Examples

Existence of minima for non- quasiconvex integrands

11.1 Introduction

11.2 Sufficient conditions

11.3 Necessary conditions

11.4 The scalar case

11.5 The vectorial case

Miscellaneous

Function spaces

12.1 Introduction

12.2 Main notation

12.3 Some properties of Hölder spaces

12.4 Some properties of Sobolev spaces

Singular values

13.1 Introduction

13.2 Definition and basic properties

13.3 Signed singular values and von Neumann type inequalities

Some underdetermined partial differential equations

14.1 Introduction

14.2 The equations div u = f and curl u = f

14.3 The equation det. u = f

Extension of Lipschitz functions on Banach spaces

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

Bibliography

Notation

General notation

Convex analysis

Determinants and singular values

Quasiconvex analysis

Function spaces

Index