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

TL;DR

This paper introduces a new matrix-valued measure for functions in GBD, extending the symmetric gradient concept, and characterizes GSBD functions via measure decomposition, enhancing understanding of deformation spaces.

Contribution

It defines a novel measure associated with GBD functions, decomposes it into singular parts, and characterizes GSBD functions through measure properties, advancing the mathematical framework of deformation analysis.

Findings

The measure $u$ admits a decomposition into absolutely continuous, Cantor, and jump parts.

GSBD functions are characterized by the absence of the Cantor part in the measure decomposition.

The measure generalizes the symmetric gradient for functions of 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 $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega)$ functions. We then characterise the space $GSBD(\Omega)$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega)$ such that $\mu^c_u=0$.