Partial regularity for the crack set minimizing the two-dimensional Griffith energy

In this paper we prove a ${\mathcal{C}}^{1,\alpha }$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a ${\mathcal{C}}^{1,\alpha }$ curve outside a set of zero Hausdorff measure.