Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane

Abstract

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than $1$. This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional.