On a volume constrained variational problem in SBV${}^{2}\left(\Omega \right)$ : part I
ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 223-237.

We consider the problem of minimizing the energy

 $\phantom{\rule{-42.67912pt}{0ex}}E\left(u\right):={\int }_{\Omega }{|\nabla u\left(x\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+{\int }_{{S}_{u}\cap \Omega }\left(1+|\left[u\right]\left(x\right)|\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{H}^{N-1}\left(x\right)$
among all functions $u\in SB{V}^{2}\left(\Omega \right)$ for which two level sets $\left\{u={l}_{i}\right\}$ have prescribed Lebesgue measure ${\alpha }_{i}$. Subject to this volume constraint the existence of minimizers for $E\left(·\right)$ is proved and the asymptotic behaviour of the solutions is investigated.

DOI: 10.1051/cocv:2002009
Classification: 49J45,  35R35,  76T05
Keywords: special functions of bounded variation, level sets, lower semicontinuity, $\Gamma$-limit
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author = {Barroso, Ana Cristina and Matias, Jos\'e},
title = {On a volume constrained variational problem in {SBV}${^2(\Omega )}$ : part {I}},
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Barroso, Ana Cristina; Matias, José. On a volume constrained variational problem in SBV${^2(\Omega )}$ : part I. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 223-237. doi : 10.1051/cocv:2002009. http://archive.numdam.org/articles/10.1051/cocv:2002009/

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