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

We prove that if $(u,\mathrm{\Gamma })$ is a minimizer of the functional J(u,\Gamma)=\int_{B_1(0)\setminus \Gamma}|\nabla u|^2dx +\H^1(\Gamma) and $\mathrm{\Gamma }$ connects $\mathrm{\partial }{B}_1(0)$ to a point in the interior, then $\mathrm{\Gamma }$ 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.