Epsilon-Regularity for Griffith Almost-Minimizers in Any Dimension Under a Separating Condition

In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is \(\varepsilon \)-close to a plane in some ball \(B\subset {\mathbb {R}}^N\) 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 et al. (J Eur Math Soc 24(7):2443–2492, 2022), with a different and more sophisticate approach inspired by Lemenant (Ann Sc Norm Super Pisa Cl Sci 9(2):351–384, 2010; Ann Sc Norm Super Pisa Cl Sci 10(3):561–609, 2011), using also Labourie (J Geom Anal 31(10):10024–10135, 2021) in order to adapt a part of the argument to Griffith minimizers.