On the Homogeneity of Global Minimizers for the Mumford-Shah Functional when $K$ is a Smooth Cone

Abstract

We show that if (u,K) is a global minimizer forthe Mumford-Shah functional in RN, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in R3 as for instance that an angular sector cannot be the singular set of a global minimizer, that if K is a half-plane then u is the corresponding cracktip function of two variables, or that if K is a cone that meets S2 with an union of _C_∞ curvilinear convex polygones, then it is a P, Y or T.