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

Abstract

We prove that if $(u,\Gamma )$ is a minimizer of the functional $$ J(u,Γ)=\int_{B_1(0)\setminus Γ}|\nabla u|^2dx +\H^1(Γ) $$ 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.

61 pages, 6 figures

References in corpus (1)

• A model for the quasi-static growth of brittle fractures: existence and approximation results

Cited by in corpus (1)

• Stationarity of the crack-front for the Mumford-Shah problem in 3D