Existence of strong minimizers for the Griffith static fracture model in dimension two

TL;DR

This paper proves the existence of strong minimizers in the two-dimensional Griffith fracture model, featuring closed jump sets and smooth deformation fields, by extending decay estimates and approximation techniques to vectorial functions.

Contribution

It generalizes decay estimates to the vectorial case and develops approximation methods for $SBD^p$ functions, enabling the proof of strong minimizers in 2D fracture models.

Findings

Existence of strong minimizers with closed jump sets

Decomposition of the problem into approximation and regularity results

Extension of decay estimates to vectorial functions

Abstract

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation. This is based on replacing the coarea formula by a method to approximate $SBD^p$ functions with small jump set by Sobolev functions and is restricted to two dimensions. The other two ingredients are contained in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSBD^p$ functions by $SBV^p$ ones.