Page 8 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

CONTENTS vii

7 Solutions to the Exercises 169
7.1 Chapter 1: Preliminaries .... ........ ....... .... 169
7.1.1 Continuous and Hölder continuous functions ... .... 169
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7.1.2 L spaces ........ ........ ....... .... 170
7.1.3 Sobolev spaces ..... ........ ....... .... 175
7.1.4 Convex analysis ..... ........ ....... .... 179
7.2 Chapter 2: Classical methods . ........ ....... .... 184
7.2.1 Euler-Lagrange equation ........ ....... .... 184
7.2.2 Second form of the Euler-Lagrange equation .... .... 190
7.2.3 Hamiltonian formulation ........ ....... .... 191
7.2.4 Hamilton-Jacobi equation ....... ....... .... 193
7.2.5 Fields theories ..... ........ ....... .... 195
7.3 Chapter 3: Direct methods .. ........ ....... .... 196
7.3.1 The model case: Dirichlet integral .. ....... .... 196
7.3.2 A general existence theorem ...... ....... .... 196
7.3.3 Euler-Lagrange equations ....... ....... .... 198
7.3.4 The vectorial case ... ........ ....... .... 199
7.3.5 Relaxation theory ... ........ ....... .... 204
7.4 Chapter 4: Regularity ..... ........ ....... .... 205
7.4.1 The one dimensional case ....... ....... .... 205
7.4.2 The model case: Dirichlet integral .. ....... .... 207
7.5 Chapter 5: Minimal surfaces .. ........ ....... .... 210
7.5.1 Generalities about surfaces ...... ....... .... 210
7.5.2 The Douglas-Courant-Tonelli method ....... .... 213
7.5.3 Nonparametric minimal surfaces ... ....... .... 213
7.6 Chapter 6: Isoperimetric inequality ...... ....... .... 214
7.6.1 The case of dimension 2 ........ ....... .... 214
7.6.2 The case of dimension n ........ ....... .... 217
Bibliography 219

Index 227