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Big pieces of \(C^{1,\alpha}\)-graphs for minimizers of the Mumford-Shah functional. (English) Zbl 0960.49024
Summary: We consider the generalization of the Mumford-Shah functional defined by \[ J(u, K)= \int_{\Omega\setminus K}|u-g|^2+ \int_{\Omega\setminus K}|\nabla u|^2+ H^{n- 1}(K), \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) \((n\geq 2)\), \(g\) a bounded measurable function on \(\Omega\), \(K\) a relatively closed subset of \(\Omega\), \(H^{n-1}(K)\) denotes the \((n-1)\)-dimensional Hausdorff measure on \(K\) and \(u\in W^{1,2}(\Omega\setminus K)\). We prove here that there exist \(\alpha\in (0,1)\) and \(C>1\) such that if \((u,K)\) is an irreducible minimizer for \(J\) and \(B(x,r)\) a ball centered on \(K\), contained in \(\Omega\), with radius \(r\leq 1\), then there is a ball \(B\) centered on \(K\), contained in \(B(x,r)\), with radius \(\geq C^{-1}r\), such that \(K\cap B\) is a \(C^{1,\alpha}\)-hypersurface. Moreover, the constants \(\alpha\), \(C\) and the \(C^{1,\alpha}\)-constant for \(K\cap B\) depend only on \(n\) and \(\|g\|_\infty\). In particular, the Hausdorff dimension of the set of points in \(K\) around which \(K\) is a \(C^{1,\alpha}\)-hypersurface is strictly less than \(n-1\).

MSC:
49N60 Regularity of solutions in optimal control
49J10 Existence theories for free problems in two or more independent variables
49Q20 Variational problems in a geometric measure-theoretic setting
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