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

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia

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).

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

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 in C^1(Omega).
Paper Structure (6 sections, 4 theorems, 51 equations)

This paper contains 6 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The abstract transversality result
  3. The case of Problem (\ref{['eq:Pb-coeff1']})
  4. The case of Problem (\ref{['eq:Pb-coeff2']})
  5. Proof of Theorem \ref{['thm:coeff']}
  6. Proof of Theorem \ref{['thm:dom']}

Key Result

Theorem 1.1

Let $\lambda_{0}$ be an eigenvalue for Problem (eq:Pb-coeff1) (respectively Problem (eq:Pb-coeff2)) with multiplicity $\nu>1$, and let $\varphi_{1},\dots,\varphi_{\nu}$ be an $L^{2}$-orthonormal basis for the eigenspace relative to $\lambda_{0}$. Assume that $a\in C^{1}(\Omega)$ and $\min_{\bar{\Ome Then the set $\mathscr{I}$ of the $b$'s close to $0$ in $C^1(\Omega)$ such that the perturbed probl

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 5.1
  • Remark 6.1