An approximation result for special functions with bounded deformation
type: Item
instance of: scholarly article
title: An approximation result for special functions with bounded deformation
zbMATH Open document ID: 1084.49038
author: Antonin Chambolle
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2004-11-23T00:00:00Z
full work available at URL: https://basepub.dauphine.fr/handle/123456789/6549
review text: A \textit{special displacement with bounded deformation} is a function \(u:\Omega \subset \mathbb R^N \to \mathbb R^N\) whose symmetrized gradient is a bounded measure which coincides, outside an \((N-1)\)-dimensional rectifiable \textit{jump set} \(J_u\), with a summable function \(e(u)\). In the paper the author proves that in dimension \(N=2\) if \(u\) and \(e(u)\) are square integrable and \({\mathcal H}^{N-1}(J_u)\) is finite then \(u\) can be approximated with a sequence \(\{u_n\}\) of piecewise continuous functions whose jump sets \(J_{u_n}\) are relatively closed, with \(u_n\) and \(e(u_n)\) converging strongly in \(L^2\), respectively, to \(u\) and \(e(u)\) and the lengths \({\mathcal H}^{N-1}(J_{u_n})\) converging to \({\mathcal H}^{N-1}(J_u)\). As an application it is given an Ambrosio-Tortorelli type approximation of a functional which appears in the theory of brittle fracture in linearized elasticity. In the addendum the results are extended to dimension \(N\geq 3\) by a careful application of a discretization technique.
reviewed by: Luigi De Pascale
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74G05
Mathematics Subject Classification ID: 74R10
zbMATH DE Number: 2118455
zbMATH Keywords: functions with bounded deformation
zbMATH Keywords: free discontinuity problems
zbMATH Keywords: brittle fracture
zbMATH Keywords: elliptic approximations
MaRDI profile type: Publication
OpenAlex ID: W1970124681
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DOI: 10.1016/J.MATPUR.2004.02.004
DOI: 10.1016/J.MATPUR.2004.11.001
description: scientific article; zbMATH DE number 2118455