Γ-convergence and absolute minimizers for supremal functionals

¹ 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; champion@unic.tln.fr.
² Dipartimento di Matematica Applicata, Universitá di Pisa, Via Bonanno Pisano 25/B, 56126 Pisa, Italy.
³ Dipartimento di Matematica, Universitá di Pisa, Via Buonarroti 2,56127 Pisa, Italy.

Received: 26 September 2002
Revised: 8 April 2003

Abstract

In this paper, we prove that the L^p 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.