Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions
type: Item
instance of: scholarly article
title: Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions
zbMATH Open document ID: 0727.35097
DOI: 10.1155/S0161171290000904
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46594
review text: For a bounded region \(\Omega \subset {\mathbb{R}}^ n\) with smooth boundary, the following eigenvalue problem is considered: \(-(1/r)\Delta_ nu=\lambda u\text{ in } \Omega,\quad \partial u/\partial \gamma +h(x)u=\lambda u\text{ on } \partial \Omega.\) Here, \(\Delta_ n\) denotes the Laplacian, r a positive weight function satisfying some Hölder condition, \(\lambda\) the eigenvalue parameter, and h a non-negative function on \(\partial \Omega\) which may be discontinuous. Introducing an appropriate Hilbert space, the problem is reduced to an operator equation \(Au=\lambda u\), where the linear operator A is symmetric, strictly positive and unbounded from above. It is shown: The spectrum consists of real eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq..\)., \(\lambda_ n\to \infty\) as \(n\to \infty\), and the corresponding eigenfunctions form a complete system in the Hilbert space.
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 4200778
zbMATH Keywords: eigenparameter
zbMATH Keywords: boundary conditions
zbMATH Keywords: operator equation
author: Elsayed M. E. Zayed
author: Said F. M. Ibrahim
reviewed by: Waldemar Velte
MaRDI profile type: Publication
OpenAlex ID: W2105415689
description: scientific article; zbMATH DE number 4200778