Existence of minimizers for the $2$d stationary Griffith fracture model

TL;DR

This paper proves the existence of strong minimizers for the 2D stationary Griffith fracture model, showing that weak minimizers in the SBD^2 space are actually strong minimizers, extending scalar techniques to vector-valued cases.

Contribution

It demonstrates that weak minimizers in the SBD^2 space are actually strong minimizers for the 2D Griffith fracture model, adapting scalar methods to vector-valued functions.

Findings

Existence of strong minimizers in the 2D Griffith model.

Weak minimizers in SBD^2 are also strong minimizers.

Extension of scalar Mumford-Shah techniques to vector-valued fracture models.

Abstract

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $SBD^2$ and for which existence is well-known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford-Shah problem.