Fine properties of functions of bounded deformation-an approach via linear PDEs
- Resource Type
- Authors
- Filip Rindler; Guido De Philippis
- Source
- Mathematics in Engineering, Vol 2, Iss 3, Pp 386-422 (2020)
- Subject
- Pure mathematics
Applied Mathematics
lcsh:T57-57.97
010102 general mathematics
bd-functions
Bounded deformation
fine properties
01 natural sciences
010101 applied mathematics
Mathematics - Analysis of PDEs
pde-constrained measures
relaxation
Bounded function
plasticity
Radon measure
Compatibility (mechanics)
lcsh:Applied mathematics. Quantitative methods
FOS: Mathematics
0101 mathematics
QA
Mathematical Physics
Analysis
Mathematics
Analysis of PDEs (math.AP)
- Language
- English
- ISSN
- 2640-3501
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.
33 pages