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\(C^ 1\)-arcs for minimizers of the Mumford-Shah functional. (English) Zbl 0870.49020
The paper is concerned with regularity of minimizing pairs \((u,K)\) of the Mumford-Shah functional \[ J(u,K)=\int_{\Omega\setminus K}|\nabla u|^2+ \int_{\Omega\setminus K}(u-g)^2+ {\mathcal H}^1(K) \] (with \(\Omega\subset{\mathbb{R}}^2\) open and bounded, \(g\in L^\infty(\Omega)\)) where \(u\in C^1(\Omega\setminus K)\) and \(K\subset\Omega\) is relatively closed. D. Mumford and J. Shah [Commun. Pure Appl. Math. 42, No. 5, 577-685 (1989; Zbl 0691.49036)] proposed, in connection with a variational approach to image segmentation problems, the minimization of this kind of functionals. They conjectured that, for any minimizing pair \((u,K)\), \(K\) consists of a locally finite number of \(C^{1,1}\) arcs. This conjecture is still open. The main result of this paper is that any ball centered in \(K\) contains a ball \(B\) still centered on \(K\) with comparable radius such that \(K\cap B\) is a \(C^{1,1/6}\) curve. As a byproduct, the set of points around which \(K\) is not a \(C^{1,1/6}\) curve is \({\mathcal H}^1\)-negligible. Related regularity results have been proved by Bonnet (under an a priori topological assumption on \(K\)) and Ambrosio, Fusco and Pallara (even for the higher dimensional analogue of \(J\)).
Reviewer: L.Ambrosio (Pavia)

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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