Global estimates for the Jacobians of orientation preserving mappings
type: Item
instance of: scholarly article
title: Global estimates for the Jacobians of orientation preserving mappings
zbMATH Open document ID: 1114.46023
author: Shusen Ding
author: Tadeusz Iwaniec
publication date: 2006-02-16T00:00:00Z
review text: In accordance with the notation of the authors, let \(\Omega\) be an open subset of \({\mathbb R}^n\), \(n \geq 2\), and let \(f = (f^1, \dots, f^n) : \Omega \to {\mathbb R}^n\) be a mapping of Sobolev class \(W^{1,p}_{\text{loc}} (\Omega, {\mathbb R}^n ), 1 \leq p < \infty ,\) whose distributional differential \(Df = [\frac{\partial f^i}{\partial x_j}] : \Omega \to \text{GL}(n) \) is a locally integrable function on \(\Omega\) with values in the space \(\text{GL}(n)\) of all \(n \times n\)-matrices. In addition, we say that \(f\) is orientation preserving (respectively, reversing) if its Jacobian determinant \(J(x, f) = \det Df(x)\) is nonnegative (respectively, nonpositive) almost everywhere. Then the authors presented a result of \textit{S. Müller} [J. Reine Angew. Math. 412, 20--34 (1990; Zbl 0713.49004)] that can be restated as the following Theorem. Let \(\Omega \subset {\mathbb R}^n\) be a bounded domain and \(f\) be an orientation-preserving map of Sobolev class \( W^{1, n}_{\text{loc}}(\Omega, {\mathbb R}^n)\). Then \[ \int_E J(x, f) \log \left(e + \frac{J(x, f)}{\int_{\Omega}J(y,f) \,dy} \right) dx \leq C \int_{\Omega} {| Df(x) | }^n\, dx, \tag{1} \] for each compact \(E \subset \Omega\), where \(C\) is a constant and \(| Df(x)| \) denotes the norm \(\max\{ | Df(x)h|\): \(| h| =1\} \). Shortly after, \textit{T. Iwaniec} and \textit{C. Sbordone} proved in [Arch. Ration. Mech. Anal. 119, 129--143 (1992; Zbl 0766.46016)] the following inequality. Let \(f\) be an orientation-preserving map of Orlicz-Sobolev class \(D^n \log^{-1} D(\Omega , {\mathbb R}^n).\) Then the Jacobian of \(f\) is locally integrable and for each compact \(E \subset \Omega\), the following estimate holds \[ \int_E J(x, f) \,dx \leq C \left( \int_{\Omega} \frac{{| Df(x)| }^n}{\log (e + | Df(x)| / | Df| _{\Omega})} \,dx \right), \tag{2} \] where \({| Df| }_{\Omega} \) denotes the integral mean of \(| Df| \) over \(\Omega\) and \(C\) is a constant. In this paper, the authors answer the follo…
reviewed by: F. Colin
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 5006960
zbMATH Keywords: Jacobians
zbMATH Keywords: John domains
zbMATH Keywords: orientation preserving mappings
MaRDI profile type: Publication
full work available at URL: j.camwa.2005.04.014
OpenAlex ID: W2027595455
DOI: 10.1016/J.CAMWA.2005.04.014
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5006960