Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes
type: Item
instance of: scholarly article
title: Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes
zbMATH Open document ID: 1208.30022
author: Flavia Giannetti
author: Antonia Passarelli Di Napoli
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-07-07T00:00:00Z
review text: A homeomorphism \(f:\Omega\to\Omega'\) between open sets in \(\mathbb{R}^n\) is said to be a \(W^{1,p}\)-bi-Sobolev mapping if \(f\in W^{1,p}(\Omega,\Omega')\) and \(f^{-1}\in W^{1,p}(\Omega',\Omega)\). Let \(K_f(x)=|Df(x)|^n/J_f(x)\) if \(J_f(x)\neq 0\), and let \(K_f(x)=1\) otherwise. Here \(J_f\) is the Jacobian of \(f\). The authors investigate the connection between the properties of \(Df^{-1}\) and \(K_f\) for such mappings in \(\mathbb{R}^n\). We cite two of the main theorems. Theorem~1.1. Let \(f:\Omega\to\Omega'=f(\Omega)\) be a \(W^{1,2}\)-bi-Sobolev mapping such that \(|Df^{-1}|\in L^2\log^\alpha L(\Omega')\) for some \(\alpha\geq0\). Then \(K_f\in L\log^\alpha L(\Omega)\), and there exists a nonnegative constant \(c\) depending only on \(\alpha\) such that \[ \int_\Omega K_f(x)\log^\alpha\big(e+K_f(x)\big)dx\leq c\int_{\Omega'}|Df^{-1}(y)|^2\log^\alpha\big(1+|Df^{-1}(y)|\big)dy+\int_\Omega |Df(x)|^2dx. \] Theorem 1.2. Let \(f:\Omega\to\Omega'=f(\Omega)\) be a homeomorphism in \(W^{1,1}\) with finite distortion such that \(K_f\in L\log^\alpha L(\Omega)\) for some \(\alpha\geq0\). Then \(|Df^{-1}|\in L^2\log^2\log_{\text{loc}}L(\Omega')\).
reviewed by: Samyon R. Nasyrov
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 5731691
zbMATH Keywords: bi-Sobolev mappings
zbMATH Keywords: Orlicz classes
zbMATH Keywords: area formula
zbMATH Keywords: maps of finite distortion
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.03.034
OpenAlex ID: W1975501632
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DOI: 10.1016/J.JMAA.2010.03.034
description: scientific article; zbMATH DE number 5731691