Compactness and lower semicontinuity in $GSBD$

TL;DR

This paper establishes compactness and semicontinuity results in the space $GSBD$ for sequences with bounded Griffith energy, enabling the existence of solutions in crack growth models and analyzing related variational energies.

Contribution

It generalizes classical compactness and semicontinuity results from $(G)SBV$ and $SBD$ to $GSBD$, facilitating analysis of crack growth problems.

Findings

Proves compactness in $GSBD$ for Griffith energy sequences.

Establishes semicontinuity results in $GSBD$.

Enables existence results for crack growth models.

Abstract

In this paper, we prove a compactness and semicontinuity result in $GSBD$ for sequences with bounded Griffith energy. This generalises classical results in $(G)SBV$ by Ambrosio and $SBD$ by Bellettini-Coscia-Dal Maso. As a result, the static problem in Francfort-Marigo's variational approach to crack growth admits (weak) solutions. Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies, for which we have recently shown the $\Gamma$-convergence to Griffith energy.