On the relaxation of integral functionals depending on the symmetrized gradient. Issue 2 (April 2021)

• Record Type: Journal Article
• Authors: Kosiba, Kamil; Rindler, Filip
• Is Part Of: Proceedings. Volume 151:Issue 2(2021)
• Page Start: 473
• Page End: 508
• Publication Date: 2021-04
• Subjects: Relaxation, -- lower semicontinuity, -- integral functionals, -- functions of bounded deformation, -- Hencky plasticity; 49J45; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.22
• Languages: English
• ISSNs: 0308-2105

Abstract

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.