*Accepted Paper*

**Inserted:** 8 mar 2014

**Last Updated:** 10 feb 2015

**Journal:** Journal de math. pures et appliquÃ©es

**Year:** 2014

**Abstract:**

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.

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