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

Guido De Philippis, Filip Rindler

Research output: Contribution to journalArticlepeer-review

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 L1-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.

Original languageEnglish (US)
Pages (from-to)386-422
Number of pages37
JournalMathematics In Engineering
Volume2
Issue number3
DOIs
StatePublished - 2020

Keywords

  • BD-functions
  • Fine properties
  • PDE-constrained measures
  • Plasticity
  • Relaxation

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematical Physics
  • Analysis

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