Characterization of generalized Young measures generated by symmetric gradients

TL;DR

This paper characterizes generalized Young measures generated by symmetric gradients of functions of bounded deformation, linking them to symmetric-quasiconvex functions with linear growth, and explores their structural properties.

Contribution

It provides a new characterization theorem for Young measures in BD, extending classical results to the symmetric gradient setting with a novel proof approach.

Findings

Young measures in BD are dual to symmetric-quasiconvex functions with linear growth

The proof combines blow-up techniques with the singular structure theorem in BD

Atomic parts in BD-Young measures can be isolated in generating sequences

Abstract

This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer-Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The "local" proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti's rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.