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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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