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On the splitting of Neumann eigenvalues in perforated domains

Veronica Felli, Lorenzo Liverani, Roberto Ognibene

TL;DR

This work analyzes the Neumann Laplacian in a bounded domain under a singular perturbation created by removing a small interior hole that shrinks to a point. It develops a novel asymptotic framework for perturbed eigenvalues, establishing that generic splitting occurs for holes centered outside a set of Hausdorff dimension at most $N-1$, and provides explicit leading-order expansions in terms of a limit bilinear form $\mathcal{L}_{x_0,\Sigma}$ (or $\mathcal{B}_{x_0}$ in 2D). In particular, multiple eigenvalues split into branches of lower multiplicity; for double eigenvalues this implies simplicity of the perturbed eigenvalues for almost every hole center when the hole is spherical. The theory relies on a Colin de Verdière small-eigenvalue approach, a blow-up analysis of boundary torsion functions, and a careful study of the associated limit forms, with a Dirichlet extension discussed in the Appendix. Numerical experiments in 2D and 3D, including non-spherical holes, corroborate the splitting phenomenon and illustrate behavior beyond the proven setting.

Abstract

We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension $N-1$ and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension $N\geq 3$, it is valid for holes of arbitrary shape.

On the splitting of Neumann eigenvalues in perforated domains

TL;DR

This work analyzes the Neumann Laplacian in a bounded domain under a singular perturbation created by removing a small interior hole that shrinks to a point. It develops a novel asymptotic framework for perturbed eigenvalues, establishing that generic splitting occurs for holes centered outside a set of Hausdorff dimension at most , and provides explicit leading-order expansions in terms of a limit bilinear form (or in 2D). In particular, multiple eigenvalues split into branches of lower multiplicity; for double eigenvalues this implies simplicity of the perturbed eigenvalues for almost every hole center when the hole is spherical. The theory relies on a Colin de Verdière small-eigenvalue approach, a blow-up analysis of boundary torsion functions, and a careful study of the associated limit forms, with a Dirichlet extension discussed in the Appendix. Numerical experiments in 2D and 3D, including non-spherical holes, corroborate the splitting phenomenon and illustrate behavior beyond the proven setting.

Abstract

We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension , it is valid for holes of arbitrary shape.
Paper Structure (12 sections, 20 theorems, 162 equations, 5 figures)

This paper contains 12 sections, 20 theorems, 162 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Strategy of Proof
  4. Notation
  5. Preliminaries
  6. Asymptotics of multiple eigenvalues
  7. Splitting of double eigenvalues
  8. Numerical exploration of the general case
  9. Validation Experiment
  10. The 2D Non-spherical Case
  11. The 3D Non-spherical Case
  12. The Dirichlet Setting

Key Result

Theorem 1.1

Let $N\geq 2$ and $m>1$. There exists a relatively closed set $\Gamma\subseteq\Omega$ such that $\dim_{\mathcal{H}}\Gamma\leq N-1$ and the following holds (with $\dim_{\mathcal{H}}\Gamma$ denoting the Hausdorff dimension of $\Gamma$): for every $x_0\in\Omega\setminus\Gamma$ there exists $\varepsilon

Figures (5)

  • Figure 1: The computed eigenvalues for different values of $\varepsilon$ in the case $\Sigma=B_1$. For $\varepsilon=0$ we have the eigenvalues of the unperturbed problem. The multiple eigenvalues are colored in blue.
  • Figure 2: The setting of the second experiment. In this case, the hole is shaped as a five-pointed star. Here, we can see the FEM mesh used by the numerical solver in the case $\varepsilon = 1$.
  • Figure 3: The computed eigenvalues for different values of $\varepsilon$ in the case $\Sigma$ is a five-pointed star. For $\varepsilon=0$ we have the eigenvalues of the unperturbed problem. The multiple eigenvalues are colored in blue.
  • Figure 4: The perforated domain used in this numerical experiment. In this case $\varepsilon=0.25$.
  • Figure 5: The computed eigenvalues for different values of $\varepsilon$ in the case $\Sigma$ is a hollow cilinder. For $\varepsilon=0$ we have some multiple eigenvalues of the unperturbed problem. The double eigenvalues are colored in blue. The triple eigenvalue is colored in red.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 24 more