Invertibility of Sobolev mappings under minimal hypotheses
type: Item
instance of: scholarly article
title: Invertibility of Sobolev mappings under minimal hypotheses
zbMATH Open document ID: 1190.30019
author: Leonid V. Kovalev
author: Jani Onninen
author: Kai Rajala
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2010-04-08T00:00:00Z
review text: The paper deals with a version of the inverse function theorem for continuous weakly differentiable mappings \(f : \Omega \rightarrow {\mathbb R}^n\) with \(\Omega\) being a bounded domain in \({\mathbb R}^n\). The main result of the paper is the following Theorem. Suppose \(f\in W_{\text{loc}}^{1, n}(\Omega, {\mathbb R}^n\) is a nonconstant mapping such that the mean inner distortion \({\mathcal K}_{\Omega} [f] < \infty\). If there exists \(\delta > - 1\) such that \(D f\in {\mathcal M}(\delta)\) for almost every \(x\in \Omega\), then \(f\) is a local homeomorphism. The proof of this theorem is based on two results of independent interest. The first step consists in the proof that the mapping is discrete and open; that is, preimages of points are discrete sets, and images of open sets are open. Another ingredient of the proof is an estimate for the multiplicity \(N(y, f, A)\) \(:=\) \( \sharp \left(f^{-1}(y)\cap A\right)\) of a local homeomorphism \(f\) in terms of the integral of the inner distortion \({\mathcal K}_{I}(\cdot, f)\) in dimension \(n\geq 3\).
reviewed by: Sergei V. Rogosin
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 5690770
zbMATH Keywords: local homeomorphism
zbMATH Keywords: differential inclusion
zbMATH Keywords: finite distortion
MaRDI profile type: Publication
full work available at URL: j.anihpc.2009.09.010
OpenAlex ID: W2002691568
Wikidata QID: Q110086149
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DOI: 10.1016/J.ANIHPC.2009.09.010
description: scientific article; zbMATH DE number 5690770