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On the relaxation of integral functionals depending on the symmetrized gradient

Published online by Cambridge University Press:  08 April 2020

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])

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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