Calculus of Variations and Geometric Measure Theory

A. Lemenant

A rigidity result for global Mumford-Shah minimizers in dimension three

created by lemenant on 08 Mar 2014
modified on 10 Feb 2015


Accepted Paper

Inserted: 8 mar 2014
Last Updated: 10 feb 2015

Journal: Journal de math. pures et appliquées
Year: 2014


We study global Mumford-Shah minimizers in $\mathbb{R}^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u,K)$ in $\mathbb{R}^3$, if $K$ is contained in a half-plane and touching its edge, then it is the half-plane itself. This partially answers a question of Guy David.