Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with \(L^{p}\) data and applications to nonlinear elasticity
type: Item
instance of: scholarly article
title: Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with \(L^{p}\) data and applications to nonlinear elasticity
zbMATH Open document ID: 1406.35493
author: Julian Fischer
author: Olivier Kneuss
published in: Journal of Differential Equations
publication date: 2018-11-06T00:00:00Z
review text: The authors establish existence results in Sobolev spaces \(W^{1,q}(\Omega;\mathbb{R}^2)\) for the Jacobian inequality \[ \det\,\nabla\,\phi\geq f\;\text{in}\;\Omega;\quad \phi=\text{id}\;\text{on}\,\partial \Omega. \] For \(f\geq 0,\;\int_\Omega f\,dx < |\Omega|\) have the solutions the following properties: 1. For every \(f\in L^p(\Omega),\;p>1\) and every \(1<q<(p+1)/2)\) there exists a homeomorphism \(\phi :\bar\Omega \to \bar\Omega\) solving the Jacobian inequality such that \(\phi,\;\phi^{-1}\,\in W^{1,q}(\Omega;\Omega).\) 2. For \(f\in L^\infty(\Omega)\) there exists a bi-Lipschitz solution \(\phi :\bar\Omega \to \bar\Omega\). The bi-Lipschitz regularity is in general sharp. The results are applied to a model functional in nonlinear elasticity, whose integrand features a fast blow up as the Jacobian determinant of the deformation becomes small.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J47
zbMATH DE Number: 6973894
zbMATH Keywords: prescribed Jacobian inequality
zbMATH Keywords: nonlinear elasticity
zbMATH Keywords: bi-Sobolev solution
zbMATH Keywords: bi-Lipschitz regularity
MaRDI profile type: Publication
full work available at URL: j.jde.2018.07.045
OpenAlex ID: W2886957492
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Wikidata QID: Q129481740
DOI: 10.1016/J.JDE.2018.07.045
description: scientific article; zbMATH DE number 6973894