  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.
