On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

Lotfi Hermi, Naoki Saito

Research output: Contribution to journalArticle

Abstract

In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form . |x-y|ρ, . 0<ρ≤1, . x,y∈[-a,a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when . ρ=1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Original languageEnglish (US)
JournalApplied and Computational Harmonic Analysis
DOIs
StateAccepted/In press - Mar 7 2016

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Nonlocal Boundary Value Problems
Integral Operator
Rayleigh
Boundary value problems
Mathematical operators
Boundary conditions
Eigenvalue
kernel
Sums of Powers
Recursion Formula
Recursive Method
Nonlocal Boundary Conditions
Distance Matrix
Recursive Formula
Commute
Higher Dimensions
Simplicity
Existence and Uniqueness
Harmonic

Keywords

  • Eigenvalues
  • Integral operator
  • Laplacian
  • Non-local boundary value problem
  • Rayleigh function

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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