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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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