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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., Comment: 61 pages, 6 figures

We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, \Gamma)=\int_{B_1\backslash\Gamma} |\nabla u|^2 dx + \lambda^2 \frac{\pi}{2} \mathcal{H}^1(\Gamma). $$ It is known that if the minimizer has a crack-tip in the ball $B_1$ (assume at the origin), then $u\approx \lambda \Im \sqrt{z}$ at this point. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation., Comment: This was the first attempt of the authors to analyze the behavior of the function and the discontinuity set near the crack-tip. The paper contains a sign mistake in the equation (17) on page 12, as well as some wrong arguments in the proofs. The authors would like to thank Alessio Figalli and Camillo DeLellis for pointing out these mistakes on March 11th, 2016. In a later work (see arXiv:1512.05094) the authors still had the same sign mistake in the first version, which was corrected in subsequent versions