The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford-Shah functional in the plain

Abstract

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.