Volume 17, Number 1, January-March 2011
|Page(s)||222 - 242|
|Published online||24 March 2010|
Monotonicity properties of minimizers and relaxation for autonomous variational problems
Dipartimento di Matematica,
Università di Bologna,
Piazza di Porta S.Donato 5, 40126 Bologna, Italy. email@example.com
2 Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy. firstname.lastname@example.org
We consider the following classical autonomous variational problem
where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.
Mathematics Subject Classification: 49K05 / 49J05
Key words: Nonconvex variational problems / autonomous variational problems / existence of minimizers / DuBois-Reymond necessary condition / relaxation
© EDP Sciences, SMAI, 2010
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