Liouville type theorems for mappings with bounded (co)-distortion
type: Item
instance of: scholarly article
title: Liouville type theorems for mappings with bounded (co)-distortion
zbMATH Open document ID: 1019.30022
author: Marc Troyanov
author: S. K. Vodop'yanov
publication date: 2003-03-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116026
full work available at URL: Liouville.pdf
review text: Let \(M\) and \(N\) be oriented \(n\)-dimensional Riemannian manifolds. A mapping \(f:M\to N\) has bounded \(s\)-distortion \((1\leq s<\infty)\) if \(f\in W^{1,1}_{\text{loc}} (M,N)\) and \(|df_x|^s \leq KJ_f(x)\) a.e. \(x \in M\). This paper deals with a number of questions for such kinds of mappings. The main result is the following Liouville type theorem: Theorem: Let \(M\) and \(N\) be oriented \(n\)-dimensional Riemannian manifolds and let \(f\in W^{1, 1}_{\text{loc}}(M,N)\) be a mapping with bounded \(s\)-distortion with \(s>(n-1)\). Assume that \(M\) is \(p\)-parabolic, where \(1/p+(n-1)/s=1\). Then either \(f\) is constant a.e. or \(N\) is also \(p\)-parabolic.
reviewed by: Bodo Dittmar
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 26B10
zbMATH DE Number: 1883509
zbMATH Keywords: mapping with bounded distortion
zbMATH Keywords: capacity
zbMATH Keywords: parabolicity
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: Liouville type theorem
MaRDI profile type: Publication
OpenAlex ID: W3382305
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DOI: 10.5802/AIF.1933
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1883509