Remarks on the spectrum of a nonlocal Dirichlet problem

TL;DR

This paper investigates the spectrum of nonlocal Dirichlet problems with non-singular kernels, demonstrating eigenvalue continuity under domain changes and eigenvalue differentiability under smooth conditions.

Contribution

It introduces the continuity of eigenvalues with respect to domain perturbations and establishes eigenvalue differentiability under smoothness assumptions.

Findings

Eigenvalues are continuous with respect to domain perturbations.

Simple eigenvalues are differentiable with respect to domain changes.

First derivatives of eigenvalues are explicitly computed.

Abstract

In this paper we analyse the spectrum of nonlocal Dirichlet problems with non-singular kernels in bounded open sets. The novelty is the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Also, under additional smooth condition on the kernel and domain, we prove differentiability of simple eigenvalues computing their first derivative.