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Monotonicity properties of minimizers and relaxation for autonomous variational problems

Published online by Cambridge University Press:  24 March 2010

Giovanni Cupini
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy. giovanni.cupini@unibo.it
Cristina Marcelli
Affiliation:
Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy. marcelli@dipmat.univpm.it
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Abstract

We consider the following classical autonomous variational problem

\[ \textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \d x\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta \right\},\]

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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