A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation

Abstract

We associate to every function $u\in GBD(Ω)$ a measure $μ_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $μ_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $μ^a_u$, $μ^c_u$, and $μ^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(Ω)$ functions. We then characterise the space $GSBD(Ω)$, originally defined only by slicing, as the space of functions $u\in GBD(Ω)$ such that $μ^c_u=0$.