Calculus of Variations and Geometric Measure Theory

G. De Philippis - F. Rindler

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

created by dephilipp on 04 Nov 2019
modified on 05 Feb 2020


Survey paper

Inserted: 4 nov 2019
Last Updated: 5 feb 2020

Year: 2019


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 of this PDE approach. We also present some open problems.