Seminari M.A.P.

Abstract

We associate to every function of generalised bounded deformation $u\in {\rm GBD}(\Omega)$ 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 $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of ${\rm BD}(\Omega)$ functions. We then characterise the space ${\rm GSBD}(\Omega)$, originally defined only by slicing, as the space of functions $u\in {\rm GBD}(\Omega)$ such that $\mu^c_u=0$. This is joint work with Gianni Dal Maso (SISSA).