One-dimensional Variational Problems

✍ Scribed by Giuseppe Buttazzo, Mariano Giaquinta, Stefan Hildebrandt

• Publisher: Clarendon
• Year: 1998
• Tongue: English
• Leaves: 269
• Category: Library

✦ Table of Contents

Cover
Title page
Introduction
1 Classical problems and indirect methods
1.1 The Euler equation and other necessary conditions for optimality
1.2 Calibrators and sufficient conditions for minima
1.3 Some classical problems
2 Absolutely continuous fonctions and Sobolev spaces
2.1 Sobolev spaces in dimension 1
2.2 Absolutely continuons functions
2.3 Functions of bounded variation
3 Semicontinuity and existence results
3.1 A lower semicontinuity theorem
3.2 Existence results in Sobolev spaces
3.3 Lower semicontinuity in the space of measures
3.4 Existence results in the space BV
4 Regularity or minimizers
4.1 The regular case
4.2 Tonelli's partial regularity theorem
4.3 The Lavrentiev phenomenon and the singular set
5 Some applications
5.1 Boundary value problems
5.2 The Sturm-Liouville eigenvalue problem
5.3 The vibrating string
5.4 Variational problems with obstacles
5.5 Periodic solutions of variational problems
5.6 Periodic solutions of Hamiltonian systems
5.7 Non-coercive variational problems
5.8 An existence result in optimal control theory
5.9 Parametric variational problems
6 Scholia
6.1 Additional remarks on the calculus of variations
6.2 Semicontinuity and compactness
6.3 Absolutely continuous functions
6.4 Sobolev spaces
6.5 Non-convex functionals on measures and bounded variation functions
6.6 Direct methods
6.7 Lavrentiev phenomenon
6.8 The vibrating string problem
6.9 Variational inequalities and the obstacle problem. Non-coercive problems
6.10 Periodic solutions
References
Index