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

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

2008-09-24

arxiv.org/abs/0809.4174v1

Mathematics

Analysis of PDEs

28 pages

Scientific paper

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 $T$.

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