An epsilon-regularity result for Griffith almost-minimizers in the plane

We present regularity results for the crack set of a minimizer for the Griffith fracture energy, arising in the variational modeling of brittle materials. In the planar setting, we prove an epsilon-regularity theorem showing that the crack is locally a $C^{1,1\mathrm{/}2}$ curve outside of a singular set of zero Hausdorff measure. The main novelty is that, in contrast to previous results, no topological constraints on the crack are required. The results also apply to almost-minimizers.