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Existence theorem for a minimum problem with free discontinuity set. (English) Zbl 0682.49002
Summary: We study the variational problem \[ \min \{\int_{\Omega \setminus K}| \nabla u|^ 2dx+\mu \int_{\Omega \setminus K}| u- g|^ qdx+\lambda H_{n-1}(K\cap \Omega);\quad K\subset {\mathbb{R}}^ n\quad closed\quad set,\quad u\in C^ 1(\Omega \setminus K)\}, \] where \(\Omega\) is an open set in \({\mathbb{R}}^ n\), \(n\geq 2\), \(g\in L^ q(\Omega)\cap L^{\infty}(\Omega)\), \(1\leq q<+\infty\), \(0<\lambda\), \(\mu <+\infty\) and \(H_{n-1}\) is the (n-1)-dimensional Hausdorff measure.

MSC:
49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation
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