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Γ -convergence and absolute minimizers for supremal functionals

Published online by Cambridge University Press:  15 February 2004

Thierry Champion ,
Luigi De Pascale  and
Francesca Prinari
Thierry Champion
Affiliation:
Laboratoire d'Analyse Non Linéaire Appliquée, U.F.R. des Sciences et Techniques, Université de Toulon et du Var, Avenue de l'Université, BP. 132, 83957 La Garde Cedex, France; [email protected].
Luigi De Pascale
Affiliation:
Dipartimento di Matematica Applicata, Universitá di Pisa, Via Bonanno Pisano 25/B, 56126 Pisa, Italy.
Francesca Prinari
Affiliation:
Dipartimento di Matematica, Universitá di Pisa, Via Buonarroti 2,56127 Pisa, Italy.
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Abstract

In this paper, we prove that the Lp approximants naturally associated to a supremal functionalΓ-converge to it. This yields a lower semicontinuity result for supremalfunctionals whose supremand satisfy weak coercivity assumptions aswell as a generalized Jensen inequality. The existence of minimizersfor variational problems involving such functionals (together with aDirichlet condition) then easily follows. In the scalarcase we show the existence of at least one absolute minimizer (i.e. localsolution) among these minimizers. We provide two different proofs ofthis fact relying on different assumptions and techniques.

Keywords

Supremal functionalslower semicontinuitygeneralized Jensen inequalityabsolute minimizer (AMLlocal minimizer)Lp approximation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 14 - 27
Copyright
© EDP Sciences, SMAI, 2004

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