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Partial regularity of free discontinuity sets. I. (English) Zbl 0896.49023
The authors consider a class of perturbations of the functional \[ F(u,K)=\int_{\Omega\setminus K}|\nabla u|^2dx+{\mathcal H}^{n-1}(K) \] where \(\Omega\in\mathbb{R}^n\) is an open set, \(K\) is a variable closed subset of \(\Omega\), \(u\) is a smooth (\(C^1\)) function outside \(K\) and \({\mathcal H}^{n-1}\) stands for the \((n-1)\)-dimensional Hausdorff measure. This class encompasses the so-called Mumford-Shah functional \[ G(u,K)=\int_{\Omega\setminus K}[|\nabla u|^2+(u-g)^2]\;dx+{\mathcal H}^{n-1}(K), \] (with \(g\in L^\infty(\Omega)\)), which in the two dimensional case was proposed as a variational model of image segmentation. The authors prove that if \((u,K)\) is a quasi minimizing pair for the functional \(F\) and \(|\nabla u|\) belongs to the Morrey space \(L^{2,\lambda}(\Omega)\) for some \(\lambda>n-1\), then the set \(K\) is a \(C^{1,\alpha}\) hypersurface outside a relatively closed, \({\mathcal H}^{n-1}\)-negligible singular set. The problem of regularity of the “minimal” set \(K\) (of quasi-minimizers) will be treated in full generality (without assumption \(|\nabla u|\in L^{2,\lambda}(\Omega)\)) in the second part of the paper (see the review of the Part II below). Some indications on possible applications of the techniques developed in this paper to more general classes of functionals are also given.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49J10 Existence theories for free problems in two or more independent variables
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