Orienting method for obstacle problems
type: Item
instance of: scholarly article
title: Orienting method for obstacle problems
zbMATH Open document ID: 1029.47044
DOI: 10.4171/ZAA/1075
author: R. Smith
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-01-25T00:00:00Z
review text: Let \(u\) be a minimizer of \(\int_\Omega F(x,v,\nabla v) dx\), where \(\Omega\subset\mathbb R^n\) is an open bounded set and \(v\in W^{1,p}(\Omega)\) satisfies \(v\geq r\) in \(\Omega\), \(v=g\) on \(\partial\Omega\). If \(\widetilde\Omega\subset\Omega\), \(\widetilde g=r\) on \(\partial\widetilde\Omega\cap\Omega\), \(r\leq\widetilde g\leq g\) on \(\partial\widetilde\Omega\cap\partial\Omega\), and the minimizer \(\widetilde u\) of \(\int_{\widetilde\Omega}F(x,v,\nabla v) dx\) in \(\{v\in W^{1,p}(\widetilde\Omega) : v=\widetilde g\text{ on }\partial\widetilde\Omega\}\) is unique, then the authors show the lower estimate \(u\geq\widetilde u\) in \(\widetilde\Omega\). In particular, the set \(\{x\in\widetilde\Omega : \widetilde u(x)>r(x)\}\) is a subset of the non-coincidence set \(\{x\in\Omega : u(x)>r(x)\}\). Similar upper estimates are also derived and the results are applied to the problem with \(n=1\), \(\Omega=(-2,3)\), \(F=4v+|v'|^2\), \(r(x)=|x|\), \(g(-2)=4\), \(g(3)=5\).
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 1779546
zbMATH Keywords: obstacle problem
zbMATH Keywords: variational inequality
zbMATH Keywords: coincidence set
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1075
OpenAlex ID: W2170817118
description: scientific article; zbMATH DE number 1779546