$\Gamma$-convergence of concentration problems

Amar, Micol; Garroni, Adriana

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

$$S_{\epsilon }^F\left(\Omega \right):=sup\left\{{\epsilon }^{-2^{*}}{\int }_{\Omega }F\left(u\right)dx:{\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\le {\epsilon }^2,u=0\mathrm{on}\partial \Omega \right\},$$

where $0\le F\left(t\right)\le {\left|t\right|}^{2^{*}}$, 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

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