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On the relaxation of integral functionals depending on the symmetrized gradient

Part of: Existence theories

Published online by Cambridge University Press:  08 April 2020

Kamil Kosiba  and
Filip Rindler
Kamil Kosiba
Affiliation:
Mathematics Institute, University of Warwick, CoventryCV4 7AL, UK ([email protected])
Filip Rindler
Affiliation:
Mathematics Institute, University of Warwick, CoventryCV4 7AL, UK and The Alan Turing Institute, British Library, 96 Euston Road, LondonNW1 2DB, UK ([email protected])
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Abstract

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form

$${\cal F}[u]: = \int_\Omega f \left( {\displaystyle{1 \over 2}\left( {\nabla u(x) + \nabla u{(x)}^T} \right)} \right) \,{\rm d}x,\quad u:\Omega \subset {\mathbb R}^d\to {\mathbb R}^d,$$
over the space BD(Ω) of functions of bounded deformation or over the Temam–Strang space
$${\rm U}(\Omega ): = \left\{ {u\in {\rm BD}(\Omega ):\;\,{\rm div}\,u\in {\rm L}^2(\Omega )} \right\},$$
depending on the growth and shape of the integrand f. Such functionals are interesting, for example, in the study of Hencky plasticity and related models.

Keywords

Relaxationlower semicontinuityintegral functionalsfunctions of bounded deformationHencky plasticity

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 473 - 508
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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