$\Gamma$-convergence of constrained Dirichlet functionals

Leonardi, Gian Paolo

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 339-351 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

Given an open, bounded and connected set $\Omega\subset \mathbb{R}^{n}$ with Lipschitz boundary and volume $|\Omega|$, we prove that the sequence $\mathcal{F}_{k}$ of Dirichlet functionals defined on $H^{1}(\Omega; \mathbb{R}^{d})$, with volume constraints $v^{k}$ on $m\geq2$ fixed level-sets, and such that $\sum_{i=1}^{m}v_{i}^{k}< |\Omega|$ for all $k$, $\Gamma$-converges, as $v^{k}\rightarrow v$ with $\sum_{i=1}^{m}v_{i}^{k}=|\Omega|$, to the squared total variation on $BV(V; \mathbb{R}^{d})$, with $v$ as volume constraint on the same level-sets.

Résumé

Dato $\Omega\subset \mathbb{R}^{n}$ aperto, limitato e connesso, con frontiera Lipschitziana e volume $|\Omega|$, si prova che la successione $\mathcal{F}_{k}$ di funzionali di Dirichlet definiti in $H^{1}(\Omega; \mathbb{R}^{d})$, con vincoli di volume $v^{k}$ su $m\geq2$ insiemi di livello prescritti, tali che $\sum_{i=1}^{m}v_{i}^{k}< |\Omega|$ per ogni $k$, $\Gamma$-converge, quando $v^{k}\rightarrow v$ con $\sum_{i=1}^{m}v_{i}^{k}=|\Omega|$, al quadrato della variazione totale in $BV(V; \mathbb{R}^{d})$, con vincoli di volume $v$ sui medesimi insiemi di livello.

Détail

Publié le : 2003-06-01

MR 1988209 | Zbl 1177.49026

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