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 for the Mumford–Shah functional in ${\mathbb{R}}^N$, and if $K$ is a smooth enough cone, then (modulo constants) $u$ is a homogenous function of degree $\frac{1}{2}$. We deduce some applications in ${\mathbb{R}}^3$ 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 $S^2$ with an union of $C^{\infty }$ curvilinear convex polygones, then it is a $\mathbb{P}$, $\mathbb{Y}$ or $\mathbb{T}$.