On mappings in the Orlicz--Sobolev classes on Riemannian manifolds
type: Item
instance of: scholarly article
title: On mappings in the Orlicz--Sobolev classes on Riemannian manifolds
zbMATH Open document ID: 1268.58008
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-27T00:00:00Z
review text: The authors' purpose in this paper is to prove the extendability of lower \(Q\)-homeomorphisms of domains on smooth Riemannian manifolds. They begin by establishing a criterion for a homeomorphism to satisfy their lower \(Q\)-condition: Theorem (1, Thm. 3.1). Let \(D\) and \(D_*\) be domains on smooth Riemannian manifolds \((\mathbb M,g)\) and \((\mathbb{M}^n_*, g^*)\), \(n\geq 3\), respectively, and let \(\varphi: [0,\infty)\to [0,\infty)\) be an increasing function under condition (2.1). Then every homeomorphism \(f : D\to D_*\) with finite distortion of the class \(W^{1,\varphi}_{\text{loc}}(D)\) is a lower \(Q\)-homeomorphism in \(D\) with \(Q(x)= K^f(X)\). Here \(W^{1,\varphi}_{\text{loc}}(D)\) refers to the Orlicz-Sobolev class associated with the growth function \(\varphi\) and domain \(D\). The authors proceed in Section 6 to show that under the assumption of finite mean oscillation, homeomorphisms in a given Orlicz-Sobolev class extend. In the final section, finite mean oscillation is replaced by finite distortion, with similar conclusions. The flavor of these extendability theorems is represented by Theorem (1, Thm. 7.1). Let \(D\) be locally connected on the boundary, let \(\overline D\) be compact, and let \(\partial D_*\) be weakly flat. If \(f: D\to D_*\) is a homeomorphism with finite distortion of the Orlicz-Sobolev class \(W^{1,\varphi}_{\text{loc}}\) under condition (2.1), and \(K(x,f)\in L^{n-1}(D)\), then \(f^{-1}\) has a continuous extension to \(\overline D\).
reviewed by: Crichton Ogle
Mathematics Subject Classification ID: 58C07
zbMATH DE Number: 6110536
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: finite mean oscillation
zbMATH Keywords: finite distortion
zbMATH Keywords: lower \(Q\)-homeomorphisms
zbMATH Keywords: Orlicz-Sobolev classes
author: E. S. Afanas'eva
author: Vladimir I. Ryazanov
MaRDI profile type: Publication
Wikidata QID: Q115382403
full work available at URL: s10958-012-0672-z
OpenAlex ID: W2058099583
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DOI: 10.1007/S10958-012-0672-Z
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description: scientific article; zbMATH DE number 6110536