Table of Contents
Prefaces to the English Edition ix
Preface to the French Edition xi
0 Introduction 1
0.1 Brief historical comments 1
0.2 Model problem and some examples 3
0.3 Presentation of the content of the monograph 7
1 Preliminaries 13
1.1 Introduction 13
1.2 Continuous and Höet;lder continuous functions 14
1.2.1 Exercises 18
1.3 L spaces 19
1.3.1 Exercises 26
1.4 Sobolev spaces 29
1.4.1 Exercises 42
1.5 Convex analysis 45
1.5.1 Exercises 48
2 Classical methods 51
2.1 Introduction 51
2.2 Euler-Lagrange equation 53
2.2.1 Exercises 64
2.3 Second form of the Euler-Lagrange equation 66
2.3.1 Exercises 68
2.4 Hamiltonian formulation 69
2.4.1 Exercises 76
2.5 Hamilton-Jacobi equation 77
2.5.1 Exercises 81
2.6 Fields theories 81
2.6.1 Exercises 86
3 Direct methods: existence 87
3.1 Introduction 87
3.2 The model case: Dirichlet integral 89
3.2.1 Exercise 92
3.3 A general existence theorem 92
3.3.1 Exercises 99
3.4 Euler-Lagrange equation 101
3.4.1 Exercises 107
3.5 The vectorial case 107
3.5.1 Exercises 115
3.6 Relaxation theory 118
3.6.1 Exercises 121
4 Direct methods: regularity 125
4.1 Introduction 125
4.2 The one dimensional case 126
4.2.1 Exercises 131
4.3 The difference quotient method: interior regularity 133
4.3.1 Exercises 139
4.4 The difference quotient method: boundary regularity 140
4.4.1 Exercises 143
4.5 Higher regularity for the Dirichlet integral 144
4.5.1 Exercises 146
4.6 Weyl lemma 147
4.6.1 Exercise 150
4.7 Some general results 150
4.7.1 Exercises 152
5 Minimal surfaces 155
5.1 Introduction 155
5.2 Generalities about surfaces 158
5.2.1 Exercises166
5.3 The Douglas-Courant-Tonelli method 167
5.3.1 Exercise 173
5.4 Regularity, uniqueness and non-uniqueness 173
5.5 Nonparametric minimal surfaces 175
5.5.1 Exercise 180
6 Isoperimetric inequality 181
6.1 Introduction 181
6.2 The case of dimension 2 182
6.2.1 Exercises 188
6.3 The case of dimension n 189
6.3.1 Exercises 196
7 Solutions to the Exercises 199
7.1 Chapter 1. Preliminaries 199
7.1.1 Continuous and Höet;lder continuous functions 203
7.1.2 L spaces 210
7.1.3 Sobolev spaces 217
7.1.4 Convex analysis 217
7.2 Chapter 2 Classical methods 224
7.2.1 Euler-Lagrange equation 224
7.2.2 Second form of the Euler-Lagrange equation 230
7.2.3 Hamiltonian formulation 231
7.2.4 Hamiltion-Jacobi equation 232
7.2.5 Fields theories 234
7.3 Chapter 3 Direct methods: existence 236
7.3.1 The model case: Dirichlet integral 236
7.3.2 A general existence theorem 236
7.3.3 Euler-Lagrange equation 239
7.3.4 The vectorial case 240
7.3.5 Relaxation theory 247
7.4 Chapter 4 Direct methods: regularity 251
7.4.1 The one dimensional case 251
7.4.2 The difference quotient method: interior regularity 254
7.4.3 The difference quotient method: boundary regularity 256
7.4.4 Higher regularity for the Dirichlet Integral 257
7.4.5 Weyl lemma 259
7.4.6 Some general results 260
7.5 Chapter 5 Minimal surfaces 263
7.5.1 Generalities about surfaces 263
7.5.2 The Douglas-Courant-Tonelli method 266
7.5.3 Nonparametric minimal surfaces 267
7.6 Chapter 6 Isoperimetric inequality 268
7.6.1 The case of dimension 2 268
7.6.2 The case of dimension n 271
Bibliography 275
Index 283
