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

Abstract

We associate to every function $u\in GBD(\Omega )$ a measure ${\mu }_u$ 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 ${\mu }_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures ${\mu }_u^a$, ${\mu }_u^c$, and ${\mu }_u^j$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega )$ functions. We then characterise the space $GSBD(\Omega )$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega )$ such that ${\mu }_u^c=0$.

50 pages, 3 figures