Regularity of the singular set for Mumford-Shah minimizers in ${ℝ}^3$ near a minimal cone

Lemenant, Antoine

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 561-609.

We prove that if $(u,K)$ is a minimizer of the Mumford-Shah functional in an open set $\Omega$ of ${ℝ}^3$, and if $x\in K$ and $r>0$ are such that $K$ is close enough to a minimal cone of type $\mathbb{P}$ (a plane), $𝕐$ (three half planes meeting at $x$ with 120${}^{\circ }$ angles) or $𝕋$ (cone over the 6 edges of a regular tetrahedron centered at $x$) in terms of Hausdorff distance in $B(x,r)$, then $K$ is $C^{1,\alpha }$ equivalent to the minimal cone in $B(x,cr)$ where $c<1$ is a universal constant.

Classification : 49Q20, 49Q05

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