Introduction to The Calculus of Variations

Bernard Dacorogna9781860944994, 1-86094-499-X

The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology.
This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist — mathematicians, physicists, engineers, students or researchers — in discovering the subjects most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions.
The book, containing more than seventy exercises with detailed solutions, is well designed for a course both at the undergraduate and graduate levels.

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Table of contents :
Introduction to the calculus of variations……Page 5
Contents……Page 6
Preface to the English Edition……Page 10
0.1 Brief historical comments……Page 14
0.2 Model problem and some examples……Page 16
0.3 Presentation of the content of the mono-graph……Page 20
1.1 Introduction……Page 24
1.2 Continuous and Hölder continuous functions……Page 25
1.3 Lp spaces……Page 29
1.4 Sobolev spaces……Page 38
1.5 Convex analysis……Page 53
2.1 Introduction……Page 58
2.2 Euler-Lagrange equation……Page 60
2.3 Second form of the Euler-Lagrange equation……Page 72
2.4 Hamiltonian formulation……Page 74
2.5 Hamilton-Jacobi equation……Page 82
2.6 Fields theories……Page 85
3.1 Introduction……Page 92
3.2 The model case: Dirichlet integral……Page 94
3.3 A general existence theorem……Page 97
3.4 Euler-Lagrange equations……Page 105
3.5 The vectorial case……Page 111
3.6 Relaxation theory……Page 120
4.1 Introduction……Page 124
4.2 The one dimensional case……Page 125
4.3 The model case: Dirichlet integral……Page 130
4.4 Some general results……Page 137
5.1 Introduction……Page 140
5.2 Generalities about surfaces……Page 143