On the eigenvalue problem for hemivariational inequalities: Existence and multiplicity of solutions
type: Item
instance of: scholarly article
title: On the eigenvalue problem for hemivariational inequalities: Existence and multiplicity of solutions
zbMATH Open document ID: 0864.49008
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-06-22T00:00:00Z
review text: The paper deals with the existence of solutions and the multiplicity of solutions for hemivariational inequalities of the form \[ a(u,v)- \langle f,v\rangle_V+\int_\Omega j^0 (x,u;v)dx\geq \lambda(u,v)_V\quad \forall v\in V, \] where \(u\in S_r:= \{v\in V:|v|_V=r\}\) and \(\lambda\in R\) are the basic unknowns. The novelties of the presented approach in comparison with the classical results of the critical point theory are the following: 1. The eigensolution \(u\) is assumed to lie in the prescribed sphere \(S_r\); 2. The mountain-pass theorem of Ambrosetti and Rabinowitz is not applicable due to the existence of a linear part \(\langle f,\cdot\rangle_V\); 3. \(V\) is a vector-valued function space which does not allow \(j\) to be of the form of a primitive. The main results of the paper establish both the existence and multiplicity of solutions. In particular, the conditions ensuring a countable set of solutions for hemivariational inequalities under consideration have been derived. One can learn from the paper how to efficiently extend and modify the methods of the classical critical point theory to the nonsmooth case represented by hemivariational inequalities involving locally Lipschitz functionals. Interesting examples of mechanical problems concerning the behavior of a plane elastic body subject to nonmonotone possibly multivalued skin friction conditions and the behavior of two adhesively connected plates illustrate the theoretical results. The paper is very interesting not only for those who deal with hemivariational inequalities but also for readers working in the field of nonsmooth analysis.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49R50
zbMATH DE Number: 902428
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: hemivariational inequalities
zbMATH Keywords: critical point theory
author: Dumitru Motreanu
author: Panagiotis D. Panagiotopoulos
reviewed by: Z. Naniewicz
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0008
OpenAlex ID: W2154354687
DOI: 10.1006/JMAA.1996.0008
description: scientific article; zbMATH DE number 902428