Weakly elliptic systems of variational inequalities: A \(2 \times 2\) model problem with obstacles in both components
type: Item
instance of: scholarly article
title: Weakly elliptic systems of variational inequalities: A \(2 \times 2\) model problem with obstacles in both components
zbMATH Open document ID: 0855.35048
DOI: 10.1007/BF01759353
author: David R. Adams
author: Helena J. Nussenzveig Lopes
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1997-02-03T00:00:00Z
review text: The authors solve the problem with the obstacle \(\psi= (\psi_1, \psi_2): \Omega\to \mathbb{R}^2\) such that \(\psi_i< 0\) near \(\partial \Omega\) for \(i= 1,2\): Find \(U\in K\) such that \(\langle A\Delta U- BU, V- U\rangle\geq 0\) for all \(V\in K\), \(K= \{W\in H^1_0(\Omega)\times H^1_0(\Omega)\mid W_i\geq \psi_i\), \(i= 1,2\}\), \(A\) and \(B\) are \(2\times 2\) matrices. Various assumptions and types of positivity of the matrix \(-A\) are considered. Using the penalization method, existence theorems for systems which are not strongly elliptic are obtained. An example of non-uniqueness in the case of \(\Omega= B(0, 1)\) the unit ball in \(\mathbb{R}^n\), \(A= (\begin{smallmatrix} 0\\ -1\end{smallmatrix} \begin{smallmatrix} -1\\ 0\end{smallmatrix})\) and \(B= 0\) is given.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 878429
zbMATH Keywords: obstacle problem
zbMATH Keywords: penalization method
zbMATH Keywords: non-uniqueness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878429