On the total variation of the Jacobian.
type: Item
instance of: scholarly article
title: On the total variation of the Jacobian.
zbMATH Open document ID: 1041.49016
DOI: 10.1016/S0022-1236(03)00111-3
author: Irene Fonseca
author: N. Fusco
author: Paolo Marcellini
published in: Journal of Functional Analysis
publication date: 2004-03-29T00:00:00Z
review text: Let \(\Omega\) be an open subset of \({\mathbb R}^2\), and \(u=(u^1,u^2)\in L^\infty_{\text{ loc}}(\Omega;{\mathbb R}^2)\cap W^{1,p}(\Omega;{\mathbb R}^2)\) for some \(p>1\). In the paper, a comparison is carried out among the classical Jacobian determinant \(\det Du\) defined a.e. in \(\Omega\), the distributional Jacobian determinant \(\text{ Det} Du\) defined by \[ \langle \text{ Det} Du,\varphi\rangle= -\int_\Omega \{u^1D_2u^2D_1\varphi-u^1D_1u^2D_2\varphi\} \,dx,\quad\varphi\in C_0^\infty(\Omega), \] and the total variation \(TV(u,\Omega)\) of the Jacobian determinant given by \[ TV(u,\Omega)=\inf\left\{\liminf_{h\to+\infty} \int_\Omega| \det Du_h| dx : \{u_h\}\subseteq W^{1,2}(\Omega; {\mathbb R}^2),u_h\to u\text{ weakly in }W^{1,p}(\Omega;{\mathbb R}^2)\right\}. \] An explicit characterization of \(TV(u,\Omega)\) is given under additional assumptions on \(u\), and it is proved that \(TV(u,\Omega)\) can be expressed by means of the topological degree, provided \(u\) is also locally Lipschitz outside a finite number of points of \(\Omega\). Some examples are also discussed proving nonidentity among the three notions of Jacobian determinant.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74G70
zbMATH DE Number: 2062721
zbMATH Keywords: Jacobian determinant
zbMATH Keywords: total variation
zbMATH Keywords: topological degree
MaRDI profile type: Publication
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full work available at URL: s0022-1236(03)00111-3
OpenAlex ID: W2154869945
description: scientific article; zbMATH DE number 2062721