Regularity up to the Crack-Tip for the Mumford-Shah problem

John Andersson; Hayk Mikayelyan

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arXiv · 1512.05094

Regularity up to the Crack-Tip for the Mumford-Shah problem

John Andersson·Hayk Mikayelyan

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Abstract

We prove that if $(u,\Gamma)$ is a minimizer of the functional $$ J(u,\Gamma)=\int_{B_1(0)\setminus \Gamma}|\nabla u|^2dx +\H^1(\Gamma) $$ and $\Gamma$ connects $\partial B_1(0)$ to a point in the interior, then $\Gamma$ satisfies a point-wise $C^{2,\alpha}$-estimate at the crack-tip. This means that the Mumford-Shah functional satisfies an additional, and previously unknown, Euler-Lagrange condition. ******* The previous version of the paper contained some mistakes, which has been fixed. More explanations/details has been added in Section 6.

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John Andersson, Hayk Mikayelyan. 2015-12-16. Regularity up to the Crack-Tip for the Mumford-Shah problem. https://arxiv.org/abs/1512.05094

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