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Γ -convergence of concentration problems

Micol Amar, Adriana Garroni (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we use Γ -convergence techniques to study the following variational problem S ε F ( Ω ) : = sup ε - 2 * Ω F ( u ) d x : Ω | u | 2 d x ε 2 , u = 0 on Ω , where 0 F ( t ) | t | 2 * , with 2 * = 2 n n - 2 , and Ω is a bounded domain of n , n 3 . We obtain a Γ -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 ε F ( Ω ) . Finally, a second order expansion in Γ -convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

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