Some spectral properties and isoperimetric inequalities for a nonlocal Laplacian problem

TL;DR

This paper investigates spectral properties and isoperimetric inequalities for a nonlocal Laplacian problem in symmetric domains, extending classical results to non-rectangular geometries and establishing foundational spectral bounds.

Contribution

It introduces a nonlocal Laplacian problem in multidimensional symmetric domains, proves self-adjointness, constructs eigenfunctions, and derives spectral inequalities analogous to classical Rayleigh bounds.

Findings

Proved self-adjointness of the nonlocal Laplacian problem

Constructed eigenfunctions for the problem

Derived spectral inequalities for the first eigenvalue

Abstract

In this paper we consider a non-local problem for a Laplace operator in a multidimensional bounded symmetric domain. The investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular domain. We prove self-adjointness of the problem and show a method of constructing eigenfunctions. We obtain an analogue of the Rayleigh type inequality and some spectral inequalities for the first eigenvalue of the nonlocal problem.