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γ-limit.

ESAIM: Control, Optimisation and Calculus of Variations (1)

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On a Volume Constrained Variational Problem in SBV²(Ω): Part I
Ana Cristina Barroso, José Matias
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 7 / 2002

Published online by Cambridge University Press:

15 September 2002, pp. 223-237

Print publication:

2002
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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.