Approximation of flat \(W ^{2,2}\) isometric immersions by smooth ones
type: Item
instance of: scholarly article
title: Approximation of flat \(W ^{2,2}\) isometric immersions by smooth ones
zbMATH Open document ID: 1242.46040
author: Peter Hornung
published in: Archive for Rational Mechanics and Analysis
publication date: 2011-10-04T00:00:00Z
review text: Let \(S\subset {\mathbb R}^2\) be a bounded Lipschitz domain in the euclidean plane. On this domain consider the Sobolev space \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\) of Sobolev maps \(u:S\to {\mathbb R}^3\) into euclidean 3-space which are isometric immersions almost everywhere on \(S\). In the main theorem it is proved that the strong \(W^{2,2}\) closure of \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\cap C^{\infty}(\bar S; {\mathbb R}^3)\) agrees with \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\), i.e., the flat \(W^{2,2}\) isometric immersions of a planar Lipschitz domain into euclidean 3-space can be approximated by smooth ones. The result has applications in nonlinear elasticity where such immersions form the natural class of admissible deformations in Kirchhoff's plate theory.
reviewed by: Vagn Lundsgaard Hansen
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 58D10
zbMATH DE Number: 5952962
zbMATH Keywords: Lipschitz domains
zbMATH Keywords: flat isometric immersions
zbMATH Keywords: nonlinear elasticity theory
zbMATH Keywords: Sobolev space
MaRDI profile type: Publication
full work available at URL: s00205-010-0374-y
OpenAlex ID: W2042954301
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DOI: 10.1007/S00205-010-0374-Y
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description: scientific article; zbMATH DE number 5952962