preprint

Inserted: 15 jun 2021

Year: 2021

Abstract:

We establish a difference quotient integral representation for symmetric gradient semi-norms in $W^{1,p}(\Omega)$, $LD(\Omega)$ and $BD(\Omega)$. The representation, which is inspired by the formulas for the $W^{1,p}(\Omega)$ semi-norm introduced by Bourgain, Brezis and Mironescu and for the total variation semi-norm of $BV(\Omega)$ by Davila, provides a criterion for the $L^p$ and total-variation boundedness of symmetric gradients that does not require the understanding of distributional derivatives.