On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions
type: Item
instance of: scholarly article
title: On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions
zbMATH Open document ID: 0852.35104
DOI: 10.1007/BF01197537
author: Alexander Kozhevnikov
author: Sasun Yakubov
published in: Integral Equations and Operator Theory
publication date: 1996-12-09T00:00:00Z
review text: Let \(\Omega\subset \mathbb{R}^N\) be a bounded connected open set, \(A\) a linear elliptic differential operator of order \(\mu\), \(\ell\) a fixed natural number between 1 and \(\mu/2\), \(B_j u\), \(j= 1,\dots, \ell\) and \(C_k u\), \(k= 1,\dots, \mu/2\) linear boundary operators. The authors study eigenvalue problems of the form: \[ Au= \lambda u\text{ in } \Omega,\;B_j u= \lambda C_j u,\;j= 1,\dots,\ell,\;C_k u= 0,\;k= \ell+ 1,\dots, \mu/2,\text{ on } \partial \Omega.\tag{1} \] \(\lambda_0\) is called an eigenvalue of problem (1), if (1) with \(\lambda_0\) has at least one non-trivial solution \(u_0\in H^\mu(\Omega)\). Let \(\lambda_0\) and \(u_0\) be an eigenvalue resp. eigenfunction. Consider the problems: \[ (A- \lambda_0 I) u_p= u_{p- 1}\text{ in } \Omega,\;(B_j- \lambda_0 C_j) u_p= C_j u_{p- 1},\;j= 1,\dots, \ell,\tag{2} \] \[ C_k u_p= 0,\;k= \ell+ 1,\dots, \mu/2\text{ on } \partial \Omega,\;p= 1,2,\dots\;. \] A solution of problem (2) is called an associated function of the \(p\)th rank to the eigenfunction \(u_0\). Eigenfunctions and associated functions of problem (1) are combined under the general name root functions. The authors establish completeness and summability properties of the root functions.
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 811613
zbMATH Keywords: root functions
zbMATH Keywords: summability properties
MaRDI profile type: Publication
cites work: Properties of solutions of ordinary differential equations in banach space
cites work: ELLIPTIC PROBLEMS WITH A PARAMETER AND PARABOLIC PROBLEMS OF GENERAL TYPE
cites work: Boundary problems for pseudo-differential operators
cites work: Q5792996
cites work: Spectral theory for operators generated by elliptic boundary problems with eigenvalue parameter in boundary conditions, I
cites work: Q5571194
cites work: Caractérisation de quelques espaces d'interpolation
cites work: Functional calculus of pseudo-differential boundary problems
cites work: Remaider estimates for eigenvalues and complex powers of the doulis-nirenberg elliptic system
cites work: Strongly definitizable linear pencils in Hilbert space
cites work: Q4740221
cites work: Norms and Domains of the Complex Powers A B z
cites work: Q5542418
cites work: Q4169288
cites work: Q4325522
full work available at URL: https://doi.org/10.1007/bf01197537
OpenAlex ID: W1999670990
description: scientific article; zbMATH DE number 811613