Calculus of Variations and Geometric Measure Theory

C. Labourie - A. Lemenant

Epsilon-Regularity for Griffith almost-minimizers in any dimension under a separating condition

created by lemenant on 24 Nov 2022



Inserted: 24 nov 2022
Last Updated: 24 nov 2022

Year: 2022


In this paper we prove that if $(u,K)$ is an almost-minimizer of the Griffith functional and $K$ is epsilon-close to a plane in some ball B in $\mathbb{R}^N$ while separating the ball $B$ in two big parts, then $K$ is $C^{1,\alpha}$ in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of Babadjian, Iurlano, Lemenant (2022), with a different and more sophisticate approach inspired by Lemenant (2011), using also Labourie (2021) in order to adapt a part of the argument to Griffith minimizers.