Unilateral eigenvalue problems for nonlinearly elastic plates: An approach via pseudo-monotone operators
type: Item
instance of: scholarly article
title: Unilateral eigenvalue problems for nonlinearly elastic plates: An approach via pseudo-monotone operators
zbMATH Open document ID: 0952.35080
DOI: 10.1007/BF02484188
author: Liliana Gratie
published in: Chinese Annals of Mathematics. Series B
publication date: 2001-01-11T00:00:00Z
review text: The paper concerns abstract variational inequalities of the type \[ u\in U,\;\lambda\in\mathbb{R}_+: \langle Au- \lambda L(u)+ C(u)- f, v-u\rangle\geq 0\text{ for all }v\in K.\tag{1} \] Here \(K\) is a closed convex cone in a real Hilbert space \(X\) with scalar product \(\langle\cdot,\cdot\rangle\), \(U\subset K\) is bounded and open, \(A: X\to X\) is linear, bounded and coercive, \(L: K\to X\) (resp. \(C:K\to X\)) is continuous and positively homogeneous of order one (resp. three), and \(\langle C(u),u\rangle> 0\) for all \(u\in K\setminus\{0\}\). Using topological \(S\)-degree, it is proved that, if \(\langle f,u_0\rangle> 0\) for certain \(u_0\in K\), then for all \(\lambda\in \mathbb{R}_+\) there exists a solution to (1). The result is applied to von Karman's equations for a thin elastic plate, clamped on a part of its boundary and simply supported on the remaining part, with one-sided obstacle zero everywhere in the domain.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35L85
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 74K20
Mathematics Subject Classification ID: 74M15
zbMATH DE Number: 1462257
zbMATH Keywords: abstract variational inequalities
zbMATH Keywords: topological \(S\)-degree
zbMATH Keywords: von Karman's equations
MaRDI profile type: Publication
cites work: On the topological degree for mappings of monotone type
cites work: Équations et inéquations non linéaires dans les espaces vectoriels en dualité
cites work: Degree of mapping for nonlinear mappings of monotone type
cites work: Q3992497
cites work: Nonlinear eigenvalue problems governed by a variational inequality of von Karman's type: A degree theoretic approach
cites work: Q4371364
cites work: Unilateral eigenvalue problems for nonlinearly elastic plates and pseudo-monotone operators
cites work: Q4447139
cites work: Modelling and control in solid mechanics
cites work: Global bifurcation in variational inequalities: applications to obstacle and unilateral problems
cites work: Q4049449
description: scientific article; zbMATH DE number 1462257