Direct Methods in the Calculus of Variations (eBook)

Contents 6
Preface 12
Introduction 14
1.1 The direct methods of the calculus of variations 14
1.2 Convex analysis and the scalar case 16
1.3 Quasiconvex analysis and the vectorial case 22
1.4 Relaxation and non-convex problems 30
1.5 Miscellaneous 36
Convex analysis and the scalar case 41
Convex sets and convex functions 42
2.1 Introduction 42
2.2 Convex sets 43
2.3 Convex functions 55
Lower semicontinuity and existence theorems 83
3.1 Introduction 83
3.2 Weak lower semicontinuity 84
3.3 Weak continuity and invariant integrals 111
3.4 Existence theorems and Euler-Lagrange equations 115
The one dimensional case 128
4.1 Introduction 128
4.2 An existence theorem 129
4.3 The Euler-Lagrange equation 134
4.4 Some inequalities 141
4.5 Hamiltonian formulation 146
4.6 Regularity 152
4.7 Lavrentiev phenomenon 157
Quasiconvex analysis and the vectorial case 161
Polyconvex, quasiconvex and rank one convex functions 162
5.1 Introduction 162
5.2 Definitions and main properties 163
5.3 Examples 185
5.4 Appendix: some basic properties of determinants 256
Polyconvex, quasiconvex and rank one convex envelopes 271
6.1 Introduction 271
6.2 The polyconvex envelope 272
6.3 The quasiconvex envelope 277
6.4 The rank one convex envelope 283
6.5 Some more properties of the envelopes 286
6.6 Examples 291
Polyconvex, quasiconvex and rank one convex sets 319
7.1 Introduction 319
7.2 Polyconvex, quasiconvex and rank one convex sets 321
7.3 The different types of convex hulls 329
7.4 Examples 353
Lower semi continuity and existence theorems in the vectorial case 373
8.1 Introduction 373
8.2 Weak lower semicontinuity 374
8.3 Weak Continuity 399
8.4 Existence theorems 409
8.5 Appendix: some properties of Jacobians 413
Relaxation and non-convex problems 418
Relaxation theorems 419
9.1 Introduction 419
9.2 Relaxation Theorems 420
Implicit partial differential equations 442
10.1 Introduction 442
10.2 Existence theorems 443
10.3 Examples 454
Existence of minima for non- quasiconvex integrands 467
11.1 Introduction 467
11.2 Sufficient conditions 469
11.3 Necessary conditions 474
11.4 The scalar case 485
11.5 The vectorial case 489
Miscellaneous 502
Function spaces 503
12.1 Introduction 503
12.2 Main notation 503
12.3 Some properties of Hölder spaces 506
12.4 Some properties of Sobolev spaces 509
Singular values 514
13.1 Introduction 514
13.2 Definition and basic properties 514
13.3 Signed singular values and von Neumann type inequalities 518
Some underdetermined partial differential equations 527
14.1 Introduction 527
14.2 The equations div u = f and curl u = f 527
14.3 The equation det. u = f 533
Extension of Lipschitz functions on Banach spaces 546
15.1 Introduction 546
15.2 Preliminaries and notation 546
15.3 Norms induced by an inner product 548
15.4 Extension from a general subset of E to E 555
15.5 Extension from a convex subset of E to E 562
Bibliography 565
Notation 606
General notation 606
Convex analysis 606
Determinants and singular values 607
Quasiconvex analysis 609
Function spaces 609
Index 610