In this paper, we prove that the approximants naturally associated to a supremal functional -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.
Mots clés : supremal functionals, lower semicontinuity, generalized Jensen inequality, absolute minimizer (AML, local minimizer), $L^p$ approximation
@article{COCV_2004__10_1_14_0,
author = {Champion, Thierry and Pascale, Luigi De and Prinari, Francesca},
title = {$\Gamma $-convergence and absolute minimizers for supremal functionals},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {14--27},
publisher = {EDP-Sciences},
volume = {10},
number = {1},
year = {2004},
doi = {10.1051/cocv:2003036},
mrnumber = {2084253},
zbl = {1068.49007},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2003036/}
}
TY - JOUR AU - Champion, Thierry AU - Pascale, Luigi De AU - Prinari, Francesca TI - $\Gamma $-convergence and absolute minimizers for supremal functionals JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 14 EP - 27 VL - 10 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2003036/ DO - 10.1051/cocv:2003036 LA - en ID - COCV_2004__10_1_14_0 ER -
%0 Journal Article %A Champion, Thierry %A Pascale, Luigi De %A Prinari, Francesca %T $\Gamma $-convergence and absolute minimizers for supremal functionals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 14-27 %V 10 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2003036/ %R 10.1051/cocv:2003036 %G en %F COCV_2004__10_1_14_0
Champion, Thierry; Pascale, Luigi De; Prinari, Francesca. $\Gamma $-convergence and absolute minimizers for supremal functionals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 14-27. doi : 10.1051/cocv:2003036. http://www.numdam.org/articles/10.1051/cocv:2003036/
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