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Zbl 0960.49024
Rigot, Séverine
Big pieces of $C^{1,\alpha}$-graphs for minimizers of the Mumford-Shah functional. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No.2, 329-349 (2000). ISSN 0391-173X

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 $\bbfR^n$ $(n\ge 2)$, $g$ 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 $r\le 1$, then there is a ball $B$ centered on $K$, contained in $B(x,r)$, with radius $\ge C^{-1}r$, 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 2000:: *49N60 Regularity of solutions in the calculus of variations
49J10 Free problems in several independent variables (existence)
49Q20 Variational problems in geometric measure-theoretic setting

Keywords: $C^{1,\alpha}$-regularity; Mumford-Shah functional; $C^{1,\alpha}$-constant; Hausdorff dimension; $C^{1,\alpha}$-hypersurface

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