Mappings of finite distortion: Hausdorff measure of zero sets
type: Item
instance of: scholarly article
title: Mappings of finite distortion: Hausdorff measure of zero sets
zbMATH Open document ID: 1017.30030
author: Stanislav Hencl
author: Jan Malý
published in: Mathematische Annalen
publication date: 2003-03-27T00:00:00Z
review text: Let \(f\) be a continuous mapping of the Sobolev class \(W^{1,n}_{\text{loc}}(\Omega,\mathbb{R}^n)\) in a domain \(\Omega\) of \(\mathbb{R}^n\). The mapping \(f\) is said to be quasi-light if for every \(y\) the components of the set \(f^{-1}(y)\) are compact and \(f\) is said to be of finite distortion \(K\) if there is a measurable function \(K: \Omega\to [0,\infty]\) such that \(|Df(x)|^n\leq K(x) J(x,t)\) a.e. If \(f\) satisfies these conditions and if \(K\in L^{n-1}_{\text{loc}}(\Omega)\), then it is shown that \(f\) is discrete and open. This is an extension of the Reshetnyak result and improves a result of \textit{J. Heinonen} and \textit{P. Koskela} [Arch. Ration. Mech. Anal. 125, No. 1, 81-97 (1993; Zbl 0792.30016)] to the borderline exponent \(n-1\) but here the condition on quasi-lightness has been added. The authors also study the Hausdorff dimension of the set \(f^{-1}(y)\) in the above class of mappings under various local integrability conditions for \(K\). The methods rest on the area formula and topological degree formulas for integrals.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 26B10
zbMATH DE Number: 1889141
zbMATH Keywords: mappings of finite distribution
zbMATH Keywords: quasi-light mappings
reviewed by: O. Martio
MaRDI profile type: Publication
full work available at URL: s00208-002-0347-z
OpenAlex ID: W2075549197
Wikidata QID: Q116538759
DOI: 10.1007/S00208-002-0347-Z
description: scientific article; zbMATH DE number 1889141