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Quantitative spectral stability for the Neumann Laplacian in domains with small holes

Veronica Felli, Lorenzo Liverani, Roberto Ognibene

TL;DR

The paper addresses quantitative spectral stability of the Neumann Laplacian under interior perturbations by small holes in a Lipschitz domain. It introduces a Sobolev $f$-torsional rigidity that governs first-order eigenvalue shifts for simple limit eigenvalues and proves an explicit asymptotic expansion for perturbed Neumann eigenvalues, complemented by convergence results for eigenfunctions. A blow-up analysis yields sharp rates for holes shrinking to a point, depending on whether the hole center lies on the singular set of the limit eigenfunction, with dimension-specific refinements and explicit spherical-hole formulas. The results illuminate how hole location relative to nodal structure influences the sign and rate of spectral variation, and provide detailed 2D/3D expansions, including an explicit interface $\Gamma$ in the spherical-hole case. These findings advance precise spectral stability understanding in perforated domains and have implications for shape optimization and numerical analysis of cavities.

Abstract

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are "small" in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation.

Quantitative spectral stability for the Neumann Laplacian in domains with small holes

TL;DR

The paper addresses quantitative spectral stability of the Neumann Laplacian under interior perturbations by small holes in a Lipschitz domain. It introduces a Sobolev -torsional rigidity that governs first-order eigenvalue shifts for simple limit eigenvalues and proves an explicit asymptotic expansion for perturbed Neumann eigenvalues, complemented by convergence results for eigenfunctions. A blow-up analysis yields sharp rates for holes shrinking to a point, depending on whether the hole center lies on the singular set of the limit eigenfunction, with dimension-specific refinements and explicit spherical-hole formulas. The results illuminate how hole location relative to nodal structure influences the sign and rate of spectral variation, and provide detailed 2D/3D expansions, including an explicit interface in the spherical-hole case. These findings advance precise spectral stability understanding in perforated domains and have implications for shape optimization and numerical analysis of cavities.

Abstract

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are "small" in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation.
Paper Structure (9 sections, 23 theorems, 286 equations, 3 figures)

This paper contains 9 sections, 23 theorems, 286 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Preliminaries
  4. Asymptotics of simple eigenvalues
  5. Blow-up analysis
  6. The case of a spherical hole
  7. The case $N\geq3$
  8. The case N=2
  9. Appendix

Key Result

Theorem 2.4

Let $n\geq 1$ be such that eq:simple is satisfied. Let $\{\Sigma_\varepsilon\}_{\varepsilon\in(0,\varepsilon_0)}$ satisfy assumptions (H) and eq:cap-to-0. Then In addition, if $U_\varepsilon:=U_{\Omega,\Sigma_\varepsilon,\partial_{\bm{\nu}}\varphi_n}\in H^1(\Omega_\varepsilon)$ denotes the function achieving $\mathcal{T}_{\overline{\Omega}_\varepsilon}(\partial\Sigma_\varepsilon,\partial_{\bm{\n

Figures (3)

  • Figure 1: The case $\lambda_{0,0,1}$
  • Figure 2: The case $\lambda_{1,1,1}$
  • Figure 3: Interface $\Gamma$ and nodal lines of the eigenfunction for the cases $\alpha_{01}$ (left) and $\alpha_{02}$ (right).

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 49 more