A Bourgain-Brezis-Mironescu representation for functions with bounded deformation

Abstract

We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in $BD(\Omega )$ and $LD(\Omega )$, which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in $W^{1,p}(\Omega ;{\mathbb{R}}^d)$, which are inspired by the gradient semi-norm formulas introduced by Bourgain, Brezis and Mironescu in $W^{1,p}(\Omega )$ and by Dávila in $BV(\Omega )$.

21 pages. After uploading the first version, we were made aware that some of the results had already been proven by Mengesha. We have therefore restructured the paper accordingly