$\Gamma$-convergence of constrained Dirichlet functionals

Leonardi, Gian Paolo

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 339-351

Résumé

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.