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

2024/02/06 by Marco Ghimenti, Ghimenti, Marco, Anna Maria Micheletti +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2402.04164

openalex publication_date 2024/02/06 · openalex created_date 2024/02/08 · openalex updated_date 2026/07/28

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 C¹(Omega).

Related

• Generic properties of eigenvalues of the fractional Laplacian
• Minimizing Eigenvalues of the Fractional Laplacian
• On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains--with some numerics--
• Principal Eigenvalue of Mixed Problem for the Fractional Laplacian:\n Moving the Boundary Conditions
• On the Eigenvalues of the p&q- Fractional Laplacian