Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy

Abstract

We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness (up to negligible sets for the (n−1)-dimensional Hausdorff measure) of the jump set for local minimizers of the Griffith energy.