$\Gamma$-convergence of concentration problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 151-179.

In this paper, we use $\Gamma$-convergence techniques to study the following variational problem

 ${S}_{\epsilon }^{F}\left(\Omega \right):=sup\left\{{\epsilon }^{-{2}^{*}}{\int }_{\Omega }F\left(u\right)\phantom{\rule{3.33333pt}{0ex}}dx\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}{\int }_{\Omega }{|\nabla u|}^{2}\phantom{\rule{3.33333pt}{0ex}}dx\le {\epsilon }^{2}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}u=0\phantom{\rule{4pt}{0ex}}\mathrm{on}\phantom{\rule{4pt}{0ex}}\partial \Omega \right\}\phantom{\rule{0.166667em}{0ex}},$
where $0\le F\left(t\right)\le {|t|}^{{2}^{*}}$, with ${2}^{*}=\frac{2n}{n-2}$, and $\Omega$ is a bounded domain of ${ℝ}^{n}$, $n\ge 3$. We obtain a $\Gamma$-convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem ${S}_{\epsilon }^{F}\left(\Omega \right)$. Finally, a second order expansion in $\Gamma$-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

Classification : 35J60,  35C20,  49J45
@article{ASNSP_2003_5_2_1_151_0,
author = {Amar, Micol and Garroni, Adriana},
title = {$\Gamma$-convergence of concentration problems},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {151--179},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {1},
year = {2003},
zbl = {1121.35048},
mrnumber = {1990977},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_151_0/}
}
Amar, Micol; Garroni, Adriana. $\Gamma$-convergence of concentration problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 151-179. http://www.numdam.org/item/ASNSP_2003_5_2_1_151_0/

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