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

TL;DR

This paper introduces a new non-local integral representation for symmetric gradient semi-norms in functions with bounded deformation, extending previous formulas for vector fields and functions of bounded variation.

Contribution

It develops a non-local integral difference quotient representation for symmetric gradient semi-norms in $BD()$ and $LD()$, avoiding distributional derivatives.

Findings

Provides a new integral representation for symmetric gradient semi-norms.

Extends Bourgain-Brezis-Mironescu formulas to functions with bounded deformation.

Simplifies analysis by avoiding distributional derivatives.

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 D\'avila in $BV(\Omega)$.