A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

TL;DR

This paper investigates conditions under which the multiplicity of eigenvalues of the fractional Laplacian remains unchanged under perturbations of the domain or coefficients, extending recent results on eigenvalue simplicity.

Contribution

It provides a specific condition for preserving eigenvalue multiplicity under perturbations and characterizes the set of such perturbations as a smooth manifold.

Findings

Identifies conditions preserving eigenvalue multiplicity under perturbations.

Shows the set of perturbations maintaining multiplicity is a smooth codimension-2 manifold.

Extends understanding of eigenvalue stability for fractional Laplacians.

Abstract

We consider the eigenvalues problem for the the fractional Laplacian in a bounded domain Omega with Dirichlet boundary condition. A recent result by Fall, Ghimenti, Micheletti and Pistoia (CVPDE (2023)) states that under generic small perturbations of the coefficient of the equation or of the domain Omega all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity 2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension $2$ in C^1(Omega).