Rough isometry and \(p\)-harmonic boundaries of complete Riemannian manifolds
type: Item
instance of: scholarly article
title: Rough isometry and \(p\)-harmonic boundaries of complete Riemannian manifolds
zbMATH Open document ID: 1082.31005
author: Yong Hah Lee
published in: Potential Analysis
publication date: 2005-05-03T00:00:00Z
review text: The \(p\)-harmonic boundary of a complete Riemannian manifold \(M\) is introduced. This is a certain subset of \(\widehat{M}\setminus M\), where \(\widehat{M}\) is the so-called Royden \(p\)-compactification of \(M\). The author proves that every bounded, energy-finite \(p\)-harmonic function on \(M\) whose distributional gradient belongs to \(L^p(M)\) is uniquely determined by its limit values on the \(p\)-harmonic boundary. It is also proven that a rough isometry between complete Riemannian manifolds always induces a homeomorphism between their \(p\)-harmonic boundaries and a bijection between their spaces of bounded, energy-finite \(p\)-harmonic functions with distributional gradient in \(L^p\). In particular, in the case of the Laplacian, a rough isometry induces a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral.
reviewed by: Eleutherius Symeonidis
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 31C45
zbMATH DE Number: 2164739
zbMATH Keywords: \(p\)-harmonic function
zbMATH Keywords: Royden \(p\)-compactification
MaRDI profile type: Publication
full work available at URL: s11118-004-3261-z
OpenAlex ID: W2067457267
Wikidata QID: Q115381453
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DOI: 10.1007/S11118-004-3261-Z
description: scientific article; zbMATH DE number 2164739