$\Gamma $-convergence of concentration problems

Amar, Micol; Garroni, Adriana

Γ

-convergence of concentration problems

Amar, Micol; Garroni, Adriana

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 151-179

Abstract

In this paper, we use $Γ$ -convergence techniques to study the following variational problem $S_{ε}^{F} (Ω) : = sup \{ε^{- 2^{*}} \int_{Ω} F (u) d x : \int_{Ω} {| \nabla u |}^{2} d x \leq ε^{2}, u = 0 on \partial Ω\},$ where $0 \leq F (t) \leq {| t |}^{2^{*}}$ , with $2^{*} = \frac{2 n}{n - 2}$ , and $Ω$ is a bounded domain of $ℝ^{n}$ , $n \geq 3$ . 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.

MR 1990977 | Zbl 1121.35048 | 1 citation in Numdam

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},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     pages = {151-179},
     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, Serie 5, Volume 2 (2003) no. 1, pp. 151-179. http://www.numdam.org/item/ASNSP_2003_5_2_1_151_0/

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