Γ -convergence of constrained Dirichlet functionals

Gian Paolo Leonardi

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 339-351
  • ISSN: 0392-4041

Abstract

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Given an open, bounded and connected set Ω ⊂ R n with Lipschitz boundary and volume Ω , we prove that the sequence F k of Dirichlet functionals defined on H 1 ⁢ Ω ; R d , with volume constraints v k on m ≥ 2 fixed level-sets, and such that ∑ i = 1 m v i k < Ω for all k , Γ -converges, as v k → v with ∑ i = 1 m v i k = Ω , to the squared total variation on B ⁢ V ⁢ V ; R d , with v as volume constraint on the same level-sets.

How to cite

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Leonardi, Gian Paolo. "$\Gamma$-convergence of constrained Dirichlet functionals." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 339-351. <http://eudml.org/doc/195247>.

@article{Leonardi2003,
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.},
author = {Leonardi, Gian Paolo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {339-351},
publisher = {Unione Matematica Italiana},
title = {$\Gamma$-convergence of constrained Dirichlet functionals},
url = {http://eudml.org/doc/195247},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Leonardi, Gian Paolo
TI - $\Gamma$-convergence of constrained Dirichlet functionals
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 339
EP - 351
AB - 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.
LA - eng
UR - http://eudml.org/doc/195247
ER -

References

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