Page 6 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Contents

Preface 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 11
1.1 Introduction ... ........ ........ ....... .... 11
1.2 Continuous and Hölder continuous functions . ....... .... 12
1.2.1 Exercises ........ ........ ....... .... 16
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1.3 L spaces .... ........ ........ ....... .... 16
1.3.1 Exercises ........ ........ ....... .... 23
1.4 Sobolev spaces . ........ ........ ....... .... 25
1.4.1 Exercises ........ ........ ....... .... 38
1.5 Convex analysis . ........ ........ ....... .... 40
1.5.1 Exercises ........ ........ ....... .... 43

2 Classical methods 45
2.1 Introduction ... ........ ........ ....... .... 45
2.2 Euler-Lagrange equation .... ........ ....... .... 47
2.2.1 Exercises ........ ........ ....... .... 57
2.3 Second form of the Euler-Lagrange equation . ....... .... 59
2.3.1 Exercises ........ ........ ....... .... 61
2.4 Hamiltonian formulation .... ........ ....... .... 61
2.4.1 Exercises ........ ........ ....... .... 68
2.5 Hamilton-Jacobi equation ... ........ ....... .... 69
2.5.1 Exercises ........ ........ ....... .... 72

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