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Fine properties of functions of bounded deformation - an approach via linear PDES
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De Philippis, Guido and Rindler, Filip (2020) Fine properties of functions of bounded deformation - an approach via linear PDES. Mathematics in Engineering, 2 (3). pp. 386-422. doi:10.3934/mine.2020018 ISSN 2640-3501.
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Official URL: https://doi.org/10.3934/mine.2020018
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.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Boundary value problems, Functions of bounded variation, Calculus of variations | ||||||
Journal or Publication Title: | Mathematics in Engineering | ||||||
Publisher: | AIMS Press | ||||||
ISSN: | 2640-3501 | ||||||
Official Date: | 27 February 2020 | ||||||
Dates: |
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Volume: | 2 | ||||||
Number: | 3 | ||||||
Page Range: | pp. 386-422 | ||||||
DOI: | 10.3934/mine.2020018 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 5 March 2020 | ||||||
Date of first compliant Open Access: | 11 March 2020 | ||||||
RIOXX Funder/Project Grant: |
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