Calculus of Variations and Geometric Measure Theory
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M. Amar - A. Garroni

$\Gamma$-convergence of concentration problems with critical growth

created on 07 Jan 2002
modified on 23 Jul 2003

[BibTeX]

Published Paper

Inserted: 7 jan 2002
Last Updated: 23 jul 2003

Journal: Ann. Scuola Norm. Sup. Pisa Cl. Sci.
Volume: 2 (Serie V)
Number: 1
Pages: 151-179
Year: 2003

Abstract:

In this paper, we use $\Gamma$-convergence techniques to study the following variational problem $$ SF\epsilon(\Omega) := \sup \left\{ \epsilon{-2}\int\Omega F(u) dx \ :\ \int\Omega \vert\nabla u\vert2 dx \leq \epsilon2\ , \ u=0\ {\rm on}\ \partial\Omega\right\}\ , $$ where $0\leq F(t)\leq \vert t\vert^{2^*}$, with $2^*={2n \over n-2}$, and $\Omega$ is a bounded domain of $R^n$, $n\geq 3$. 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 $S^F_\epsilon(\Omega)$. Finally, a second order expansion in $\Gamma$-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.


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