Fine properties of functions of bounded deformation -- an approach via linear PDEs

TL;DR

This survey explores recent advances in understanding the detailed structure of functions of bounded deformation (BD) through PDE techniques, highlighting new ellipticity-based methods and open problems in the field.

Contribution

It introduces a PDE-based approach to analyze BD functions, providing new insights and complementing classical methods in the study of their fine properties.

Findings

PDE methods offer a new perspective on BD functions.

Ellipticity arguments underpin recent progress.

Open problems highlight future research directions.

Abstract

In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are $\mathrm{L}^1$-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems.