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

Guido De Philippis; Filip Rindler; Guido De Philippis; Filip Rindler

doi:10.3934/mine.2020018

Mathematics in Engineering

2020, Volume 2, Issue 3: 386-422. doi: 10.3934/mine.2020018

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

Guido De Philippis ¹,
Filip Rindler ^{2
,
,}

1.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA
2.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK, and The Alan Turing Institute, British Library, 96 Euston Road, London NW1 2DB, UK

Received: 04 November 2019 Accepted: 06 February 2020 Published: 27 February 2020

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.
- BD-functions,
- fine properties,
- PDE-constrained measures,
- plasticity,
- relaxation
Citation: Guido De Philippis, Filip Rindler. Fine properties of functions of bounded deformation-an approach via linear PDEs[J]. Mathematics in Engineering, 2020, 2(3): 386-422. doi: 10.3934/mine.2020018

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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.

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