One-dimensional Variational Problems

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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