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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 functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution) among these minimizers. We provide two different proofs of this 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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