Compactness for $GSBV^p$ via concentration-compactness

TL;DR

This paper introduces a new, transparent proof of compactness for $GSBV^p$ functions using concentration-compactness, specifically applied to variational models for fracture, marking a novel connection in the field.

Contribution

It provides the first explicit use of concentration-compactness in fracture mechanics problems, simplifying and clarifying the proof of compactness for $GSBV^p$ functions.

Findings

New proof of compactness for $GSBV^p$ functions without bounds

Explicit connection between concentration-compactness and fracture models

Simplified and transparent proof strategy

Abstract

Motivated by variational models for fracture, we provide a new proof of compactness for $GSBV^p$ functions without a priori bounds on the function itself. Our proof is based on the classical idea of concentration-compactness, making it transparent in strategy and simple in implementation. Further, so far as we are aware, this is the first time the connection to concentration-compactness has been made explicit for problems in fracture mechanics.