Applications of a one-dimensional Sobolev inequality to eigenvalue problems
type: Item
instance of: scholarly article
title: Applications of a one-dimensional Sobolev inequality to eigenvalue problems
zbMATH Open document ID: 0842.34083
author: Štefan Schwabik
author: Richard C. Brown
author: Don B. Hinton
published in: Differential and Integral Equations
publication date: 1996-07-28T00:00:00Z
review text: The paper deals with upper and lower bounds for the eigenvalues of regular Sturm-Liouville boundary eigenvalue problems \(- y'' + qy = \lambda y\) where \(q \in L_s (a,b)\), \(1 \leq s \leq \infty\), with boundary conditions \(y(a) = y(b) = 0\) or \(y(a) = y'(b) = 0\). The case of a Sturm Liouville problem of this type with a positive weight function \(k \in L_s (a,b)\) is also considered. To this end a one-dimensional Sobolev type inequality is used, supplemented with a Prüfer transformation argument. In particular, a shorter proof for a known result on the maximization or minimization of an eigenvalue \(\lambda_n\) subject to a norm constraint on \(q\) is given. The inequalities obtained for the eigenvalues are applied to estimate the number of eigenvalues in \((\infty, \lambda]\), the gaps between eigenvalues and to derive Lyapunov and de La Vallée Poussin type inequalities.
reviewed by: Christiane Tretter
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D20
zbMATH DE Number: 856933
zbMATH Keywords: upper and lower bounds for the eigenvalues
zbMATH Keywords: Sturm-Liouville boundary eigenvalue problems
zbMATH Keywords: one-dimensional Sobolev type inequality
zbMATH Keywords: Prüfer transformation
zbMATH Keywords: number of eigenvalues
zbMATH Keywords: gaps
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 856933