Calculus of Variations and Geometric Measure Theory

A. Arroyo-Rabasa - P. Bonicatto

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

created by bonicatto on 15 Jun 2021



Inserted: 15 jun 2021

Year: 2021

ArXiv: 2106.06954 PDF


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.