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

Antoine Lemenant (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We study global Mumford-Shah minimizers in $\R^N$, 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 $\R^3$, if $K$ is contained in a half-plane and touching its edge, then it is the half-plane itself. This partially answers to a question of Guy David.

Domaines: Mathématiques générales [math.GM]

Antoine Lemenant. A rigidity result for global Mumford-Shah minimizers in dimension three. Journal de Mathématiques Pures et Appliquées, 2015. ⟨hal-00957699⟩