One-Dimensional Variational Problems: An Introduction

✍ Scribed by Giuseppe Buttazzo, Mariano Giaquinta, Stefan Hildebrandt

• Publisher: Oxford Univ Pr
• Year: 1999
• Tongue: English
• Leaves: 272
• Series: Oxford Lecture Series in Mathematics and Its Applications
• Category: Library

✦ Synopsis

While easier to solve and accessible to a broader range of students, one-dimensional variational problems and their associated differential equations exhibit many of the same complex behavior of higher-dimensional problems. This book, the first moden introduction, emphasizes direct methods and provides an exceptionally clear view of the underlying theory.

✦ Table of Contents

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
CONTENTS......Page 8
Introduction ......Page 10
1.1 The Euler equation and other necessary conditions for optimality ......Page 19
1.2 Calibrators and sufficient conditions for minima ......Page 34
1.3 Some classical problems ......Page 48
2.1 Sobolev spaces in dimension 1 ......Page 63
2.2 Absolutely continuous functions ......Page 89
2.3 Functions of bounded variation ......Page 99
3.1 A lower semicontinuity theorem ......Page 113
3.2 Existence results in Sobolev spaces ......Page 123
3.3 Lower semicontinuity in the space of measures ......Page 133
3.4 Existence results in the space BV ......Page 137
4.1 The regular case ......Page 143
4.2 Tonelli's partial regularity theorem ......Page 148
4.3 The Lavrentiev phenomenon and the singular set ......Page 155
5.1 Boundary value problems ......Page 165
5.2 The Sturm-Liouville eigenvalue problem ......Page 172
5.3 The vibrating string ......Page 193
5.4 Variational problems with obstacles ......Page 196
5.5 Periodic solutions of variational problems ......Page 202
5.6 Periodic solutions of Hamiltonian systems ......Page 208
5.7 Non-coercive variational problems ......Page 212
5.8 An existence result in optimal control theory ......Page 220
5.9 Parametric variational problems ......Page 225
6.1 Additional remarks on the calculus of variations ......Page 234
6.2 Semicontinuity and compactness ......Page 235
6.3 Absolutely continuous functions ......Page 237
6.4 Sobolev spaces ......Page 239
6.5 Non-convex functionals on measures and bounded variation functions ......Page 240
6.6 Direct methods ......Page 241
6.7 Lavrentiev phenomenon ......Page 244
6.8 The vibrating string problem ......Page 251
6.10 Periodic solutions ......Page 253
References ......Page 255
Index ......Page 270
Back Cover......Page 272