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On a volume constrained variational problem in \(SBV{^2(\Omega)}\). I. (English) Zbl 1047.49016
Authors’ abstract: “We consider the problem of minimizing the energy \(E(u):=\int _{\Omega}| \nabla u(x)| ^{2}\,dx+\int_{S_{u}\cap\Omega}(1+| [u](x)| )\,dH^{n-1}(x)\) among all functions \(u\in SBV^{2}(\Omega)\) for which two level sets \(\{u=l_{i}\}\) have prescribed Lebesgue measure \(\alpha_{i}.\) Subject to this volume constraint the existence of minimizers for \(E(\cdot)\) is proved and the asymptotic behaviour of the solutions is investigated.”
MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35R35 Free boundary problems for PDEs
35A15 Variational methods applied to PDEs
49K10 Optimality conditions for free problems in two or more independent variables
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