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.
Author(s): Giuseppe Buttazzo, Mariano Giaquinta, Stefan Hildebrandt
Series: Oxford Lecture Series in Mathematics and Its Applications
Publisher: Oxford University Press, USA
Year: 1999
Language: English
Pages: 272
Front Cover......Page p0000.djvu
Title......Page p0003.djvu
Copyright......Page p0004.djvu
Preface......Page p0005.djvu
CONTENTS......Page p0007.djvu
Introduction ......Page p0009.djvu
1.1 The Euler equation and other necessary conditions for optimality ......Page p0018.djvu
1.2 Calibrators and sufficient conditions for minima ......Page p0033.djvu
1.3 Some classical problems ......Page p0047.djvu
2.1 Sobolev spaces in dimension 1 ......Page p0062.djvu
2.2 Absolutely continuous functions ......Page p0088.djvu
2.3 Functions of bounded variation ......Page p0098.djvu
3.1 A lower semicontinuity theorem ......Page p0112.djvu
3.2 Existence results in Sobolev spaces ......Page p0122.djvu
3.3 Lower semicontinuity in the space of measures ......Page p0132.djvu
3.4 Existence results in the space BV ......Page p0136.djvu
4.1 The regular case ......Page p0142.djvu
4.2 Tonelli's partial regularity theorem ......Page p0147.djvu
4.3 The Lavrentiev phenomenon and the singular set ......Page p0154.djvu
5.1 Boundary value problems ......Page p0164.djvu
5.2 The Sturm-Liouville eigenvalue problem ......Page p0171.djvu
5.3 The vibrating string ......Page p0192.djvu
5.4 Variational problems with obstacles ......Page p0195.djvu
5.5 Periodic solutions of variational problems ......Page p0201.djvu
5.6 Periodic solutions of Hamiltonian systems ......Page p0207.djvu
5.7 Non-coercive variational problems ......Page p0211.djvu
5.8 An existence result in optimal control theory ......Page p0219.djvu
5.9 Parametric variational problems ......Page p0224.djvu
6.1 Additional remarks on the calculus of variations ......Page p0233.djvu
6.2 Semicontinuity and compactness ......Page p0234.djvu
6.3 Absolutely continuous functions ......Page p0236.djvu
6.4 Sobolev spaces ......Page p0238.djvu
6.5 Non-convex functionals on measures and bounded variation functions ......Page p0239.djvu
6.6 Direct methods ......Page p0240.djvu
6.7 Lavrentiev phenomenon ......Page p0243.djvu
6.8 The vibrating string problem ......Page p0250.djvu
6.10 Periodic solutions ......Page p0252.djvu
References ......Page p0254.djvu
Index ......Page p0269.djvu
Back Cover......Page p0271.djvu
