Which special functions of bounded deformation have bounded variation?

TL;DR

This paper explores the relationship between functions of bounded deformation ($BD$) and functions of bounded variation ($BV$), establishing conditions for inclusion and providing counterexamples to highlight differences.

Contribution

It demonstrates that certain $BD$ functions are in $GSBV$ under regularity assumptions and constructs examples of $BD$ functions not in $BV$, clarifying their relationship.

Findings

$BD$ functions piecewise affine on Caccioppoli partitions are in $GSBV$

$SBD^p$ functions are approximately continuous off the jump set

Counterexamples show $BD$ functions not in $BV$ with specific strain parts

Abstract

Functions of bounded deformation ($BD$) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation ($BV$), but are less well understood. We discuss here the relation to $BV$ under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that $BD$ functions which are piecewise affine on a Caccioppoli partition are in $GSBV$, and we prove that $SBD^p$ functions are approximately continuous ${\mathcal{H}}^{n-1}$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in $BV$ and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.