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

TL;DR

This paper establishes that Griffith almost-minimizers that are close to a plane and separate a ball into two large parts are smoothly regular in a smaller region, generalizing previous 2D results to higher dimensions.

Contribution

It extends epsilon-regularity results for Griffith almost-minimizers to any dimension under a separating condition, using a new, sophisticated approach.

Findings

K is C^{1,α} in a smaller ball under the given conditions

Generalizes 2D epsilon-regularity results to higher dimensions

Introduces a new approach inspired by previous works

Abstract

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 $\subset$ 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 [4], with a different and more sophisticate approach inspired by [23, 24], using also [20] in order to adapt a part of the argument to Griffith minimizers.