On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone

TL;DR

This paper proves that global minimizers of the Mumford-Shah functional with smooth cone singular sets are homogeneous of degree 1/2, leading to classification results for their singular sets in three dimensions.

Contribution

It establishes the homogeneity of minimizers with smooth cone singularities and classifies possible singular sets in three dimensions.

Findings

Homogeneity of minimizers with smooth cone singularities.

Certain geometric configurations cannot be singular sets of minimizers.

Classification of singular sets as P, Y, or T shapes in 3D.

Abstract

We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, 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 $R^3$ 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 $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$.