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

TL;DR

This paper demonstrates that functions with small jump sets can be approximated by smooth functions, enabling the extension of decay estimates and establishing the closedness of jump sets for minimizers of Griffith's energy.

Contribution

It generalizes decay estimates to the linearized setting and proves the existence of strong minimizers for Griffith's energy with small jump sets.

Findings

Functions with small jump sets are close to smooth functions in energy.

Decay estimates are extended to the linearized context.

Jump sets of local minimizers are shown to be closed.

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 of the jump set for local minimizers of the Griffith energy.