Weak density of smooth maps for the Dirichlet energy between manifolds
type: Item
instance of: scholarly article
title: Weak density of smooth maps for the Dirichlet energy between manifolds
zbMATH Open document ID: 1028.58008
published in: Geometric and Functional Analysis. GAFA
publication date: 2003-08-27T00:00:00Z
review text: Let \(M\) and \(N\) be compact smooth manifolds and assume that \(M\) is simply connected. In the main result of the article, the authors show that smooth maps are dense in the Sobolev space \(W^{1,2}(M,N)\) for the sequential weak topology. Moreover, if \(\varphi\) denotes an element of \(C^\infty (\partial M,N)\) smoothly extendable to \(M\), then every \(u\in W^{1,2} (M,N)\) satisfying \(u=\varphi\) on \(\partial M\) can be weakly approximated by a sequence of smooth maps \(u_m\) such that \(u_m=\varphi\) on \(\partial M\). Besides a delicate study of chains associated with the singular sets of mappings, the proof involves recent results of \textit{F. Hang} and \textit{F. Lin} [Math. Res. Lett. 8, No. 3, 321--330 (2001; Zbl 1049.46018)]. In the same spirit, the paper also contains several statements on the density of smooth maps in \(W^{1,p}({\mathbf B}^n,N)\).
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 58A25
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 1971830
zbMATH Keywords: Sobolev space
zbMATH Keywords: density of smooth maps
author: Mohammad Reza Pakzad
author: Tristan Rivière
reviewed by: Vincent Thilliez
MaRDI profile type: Publication
full work available at URL: s000390300006
OpenAlex ID: W2047649026
Wikidata QID: Q125938413
DOI: 10.1007/S000390300006
description: scientific article; zbMATH DE number 1971830