Level surfaces of parametric eigenvalue problems
type: Item
instance of: scholarly article
title: Level surfaces of parametric eigenvalue problems
zbMATH Open document ID: 0558.34019
DOI: 10.1016/0022-247X(84)90009-X
author: Mao Xia Ning
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: This paper concerns an eigenvalue problem containing the vector parameters \(\alpha\) and \(\beta\) : \(-(p(x,\alpha)u')'+q(x,\beta)u=\lambda u\), \(ru(0)-p(0,\alpha)u'(0)=0\), \(su(1)+p(1,\alpha)u'(1)=0\) where r and s are given numbers. We consider the eigenvalue as a function of the parameters \(\alpha\) and \(\beta\) and denote it by \(\lambda\) (\(\alpha\),\(\beta)\). Then its gradient can be evaluated by numerical procedures. If the gradient does not vanish, then the level surface for the function \(\lambda (\alpha,\beta)=\lambda_ c=const\). can be obtained numerically. In some cases the level surface splits the whole (\(\alpha\),\(\beta)\) space into two subspaces. When (\(\alpha\),\(\beta)\) are in different subspaces, the values of \(\lambda\) (\(\alpha\),\(\beta)\) are greater or less than \(\lambda_ c\), respectively. When \(\alpha\) and \(\beta\) are linear, then the smallest eigenvalue \(\lambda\) (\(\alpha\),\(\beta)\) is concave and its values in different subspaces split by the level surface can be controlled easily. These results are illustrated by 3 examples.
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 3889930
zbMATH Keywords: concave function
zbMATH Keywords: level surface
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: examples
MaRDI profile type: Publication
cites work: Q5574850
cites work: State Constraints in Convex Control Problems of Bolza
cites work: Q5584775
description: scientific article; zbMATH DE number 3889930