Γ -convergence of concentration problems

Micol Amar; Adriana Garroni

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 1, page 151-179
  • ISSN: 0391-173X

Abstract

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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 ℝ 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.

How to cite

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Amar, Micol, and Garroni, Adriana. "$\Gamma $-convergence of concentration problems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.1 (2003): 151-179. <http://eudml.org/doc/84494>.

@article{Amar2003,
abstract = {In this paper, we use $\Gamma $-convergence techniques to study the following variational problem\[ S^F\_\{\varepsilon \}(\Omega ) := \sup \left\lbrace \{\varepsilon \}^\{-2^*\}\int \_\Omega F(u) ~dx \ :\ \int \_\Omega \vert \nabla u\vert ^2~dx \le \{\varepsilon \}^2\ , \ u=0\ \{\rm on\}\ \partial \Omega \right\rbrace \, , \]where $0\le F(t)\le \vert t\vert ^\{2^*\}$, with $2^*=\{2n \over n-2\}$, and $\Omega $ is a bounded domain of $\{\{\mathbb \{R\}\}^n\}$, $n\ge 3$. 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 $S^F_\{\varepsilon \}(\Omega )$. Finally, a second order expansion in $\Gamma $-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.},
author = {Amar, Micol, Garroni, Adriana},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {151-179},
publisher = {Scuola normale superiore},
title = {$\Gamma $-convergence of concentration problems},
url = {http://eudml.org/doc/84494},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Amar, Micol
AU - Garroni, Adriana
TI - $\Gamma $-convergence of concentration problems
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 1
SP - 151
EP - 179
AB - In this paper, we use $\Gamma $-convergence techniques to study the following variational problem\[ S^F_{\varepsilon }(\Omega ) := \sup \left\lbrace {\varepsilon }^{-2^*}\int _\Omega F(u) ~dx \ :\ \int _\Omega \vert \nabla u\vert ^2~dx \le {\varepsilon }^2\ , \ u=0\ {\rm on}\ \partial \Omega \right\rbrace \, , \]where $0\le F(t)\le \vert t\vert ^{2^*}$, with $2^*={2n \over n-2}$, and $\Omega $ is a bounded domain of ${{\mathbb {R}}^n}$, $n\ge 3$. 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 $S^F_{\varepsilon }(\Omega )$. Finally, a second order expansion in $\Gamma $-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.
LA - eng
UR - http://eudml.org/doc/84494
ER -

References

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