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A note on the Maxwell's eigenvalues on thin sets

Francesco Ferraresso, Luigi Provenzano

TL;DR

This work analyzes the Maxwell operator on thin tubular domains around a smooth surface and shows that Maxwell eigenvalues converge to Laplacian eigenvalues on the core surface as the tube radius $h$ vanishes. By reformulating Maxwell's curl-curl problem as a Hodge Laplacian eigenproblem on co-closed $1$-forms with relative boundary conditions, the authors derive explicit eigenvalue structures and eigenfunctions on product manifolds and establish spectral convergence via a quasi-isometry argument between pull-back and product metrics. The results reveal spectral instabilities under topology-changing perturbations, demonstrate the failure of Faber–Krahn-type inequalities for Maxwell eigenvalues, and provide explicit Maxwell eigenfunctions on product domains, thereby connecting thin-domain asymptotics with geometric spectral theory on surfaces. Overall, the paper extends known cylinder-based results to curved, embedded surfaces, highlighting rich interactions between geometry, topology, and Maxwell spectral theory.

Abstract

We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce the Maxwell's eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

A note on the Maxwell's eigenvalues on thin sets

TL;DR

This work analyzes the Maxwell operator on thin tubular domains around a smooth surface and shows that Maxwell eigenvalues converge to Laplacian eigenvalues on the core surface as the tube radius vanishes. By reformulating Maxwell's curl-curl problem as a Hodge Laplacian eigenproblem on co-closed -forms with relative boundary conditions, the authors derive explicit eigenvalue structures and eigenfunctions on product manifolds and establish spectral convergence via a quasi-isometry argument between pull-back and product metrics. The results reveal spectral instabilities under topology-changing perturbations, demonstrate the failure of Faber–Krahn-type inequalities for Maxwell eigenvalues, and provide explicit Maxwell eigenfunctions on product domains, thereby connecting thin-domain asymptotics with geometric spectral theory on surfaces. Overall, the paper extends known cylinder-based results to curved, embedded surfaces, highlighting rich interactions between geometry, topology, and Maxwell spectral theory.

Abstract

We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of . We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential -forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce the Maxwell's eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

Paper Structure

This paper contains 10 sections, 10 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction and statement of the main results
  2. Geometric preliminaries and the interpretation of problem \ref{['eq:curlcurl']} as an eigenvalue problem for the Hodge Laplacian
  3. Notation and functional spaces
  4. Hodge Laplacian and geometric functional setting
  5. Hodge Laplacian spectrum on co-closed $1$-forms with relative conditions on $3$-dimensional product manifolds
  6. Eigenvalues and eigenfunctions when $\partial\Sigma=\emptyset$
  7. Eigenvalues and eigenfunctions when $\partial\Sigma\ne\emptyset$
  8. Convergence of eigenvalues on tubes around embedded surfaces: proof of Theorem \ref{['main_convergence']}
  9. Convergence of eigenfunctions
  10. Full Hodge Laplacian spectrum with relative conditions on product manifolds

Key Result

Theorem 1.1

Let $\Sigma$ be a smooth, compact, embedded, orientable surface in $\mathbb R^3$, and let $\Omega_h$ be the tube of size $h$ around $\Sigma$ defined by tube. Let $\{\lambda_j(\Omega_h)\}_{j=1}^{\infty}$ be the sequence of Maxwell's eigenvalues. Then, for all $j\in\mathbb N$:

Figures (2)

  • Figure 1: Surface $\Sigma$ without boundary and domain $\Omega_h$
  • Figure 2: Surface $\Sigma$ with boundary and domain $\Omega_h$

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • ...and 5 more