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On a Volume Constrained Variational Problem in SBV²(Ω): Part I

Published online by Cambridge University Press:  15 September 2002

Ana Cristina Barroso  and
José Matias
Ana Cristina Barroso
Affiliation:
CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal and Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal; abarroso@lmc.fc.ul.pt.
José Matias
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal; jmatias@math.ist.utl.pt.
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Abstract

We consider the problem of minimizing the energy $$ E(u):= \int_{\Omega}|\nabla u(x)|^2 \, {\rm d}x + \int_{S_u \cap \Omega}\left (1 + |[u](x)|\right) \, {\rm d}H^{N - 1}(x)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u = l_i\}$ have prescribed Lebesgue measure $\alpha_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.

Keywords

Special functions of bounded variationlevel setslower semicontinuityΓ-limit.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 223 - 237
Copyright
© EDP Sciences, SMAI, 2002

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