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

Abstract

We associate to every function u ∈ GBD(Ω) a measure µu 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 µu admits a decomposition as the sum of three mutually singular matrix-valued measures µau, µcu, and µju, 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 ∈ GBD(Ω) such that µcu = 0.