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Regularity results for anisotropic image segmentation models. (English) Zbl 0899.49018
The paper is concerned with existence of minimizing pairs $$(u,K)$$ for the generalized Mumford-Shah functional $\int_{\Omega\setminus K}F(\nabla u)+\alpha(u-g)^2 dx +\beta{\mathcal H}^{n-1}(K\cap\Omega).$ The idea of the proof is to estabilish a density lower bound ${\mathcal H}^{n-1}(S_u\cap B_\rho(x))\geq\theta\rho^{n-1} \qquad\forall x\in \overline{S}_u,\quad B_\rho(x)\subset \Omega,$ for minimizers $$u$$ of the weak formulation of the problem in the space of special functions with bounded variation. The density lower bound implies that $${\mathcal H}^{n-1}(\Omega\cap\overline{S}_u\setminus S_u)=0$$, and hence that $$(u,\overline{S}_u)$$ is a minimizing pair. The proof of the density lower bound is related to the rate of decay in $$\rho$$ of the energy $\int_{B_\rho(x)}F(\nabla v(y)) dy$ for local minimizers $$v$$ of $$\int F(\nabla v)$$ in a Sobolev space. A general decay property is proved under a strict convexity assumption on $$F$$ (no smoothness is required). By a perturbation argument, also some existence results in the vectorial problem are presented.
Reviewer: L.Ambrosio (Pavia)

##### MSC:
 49N60 Regularity of solutions in optimal control 49J10 Existence theories for free problems in two or more independent variables
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