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Shape sensitivity analysis of Neumann-Poincaré eigenvalues

Matteo Dalla Riva, Pier Domenico Lamberti, Paolo Luzzini, Paolo Musolino

TL;DR

This paper establishes that Neumann-Poincaré eigenvalues depend real-analytically on boundary shape perturbations and derives the explicit first-order derivative of symmetric functions of eigenvalues under infinite-dimensional perturbations. The approach hinges on pulling back to a fixed boundary, employing Riesz projectors for symmetrizable operators, and expressing derivatives through a matrix representation on the eigen-subspace, with the derivative formula ultimately expressed via plasmonic eigenfunctions on the boundary. Applications include a Rellich-Pohožaev-type identity for dilations and a sphere-critically-for-sums result for NP-eigenvalues in any dimension $n\ge 3$, supporting conjectural optimality statements and generalizing prior Grieser and Zolésio findings. The work thus provides a rigorous perturbation-theory framework for NP-spectrum optimization, with potential implications for plasmonic design and boundary-shape optimization. Overall, the results extend analytic perturbation theory to symmetrizable, infinite-dimensional perturbations and establish concrete tools for understanding how geometry governs spectral behavior in boundary integral operators.

Abstract

This paper concerns the eigenvalues of the Neumann-Poincaré operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.

Shape sensitivity analysis of Neumann-Poincaré eigenvalues

TL;DR

This paper establishes that Neumann-Poincaré eigenvalues depend real-analytically on boundary shape perturbations and derives the explicit first-order derivative of symmetric functions of eigenvalues under infinite-dimensional perturbations. The approach hinges on pulling back to a fixed boundary, employing Riesz projectors for symmetrizable operators, and expressing derivatives through a matrix representation on the eigen-subspace, with the derivative formula ultimately expressed via plasmonic eigenfunctions on the boundary. Applications include a Rellich-Pohožaev-type identity for dilations and a sphere-critically-for-sums result for NP-eigenvalues in any dimension , supporting conjectural optimality statements and generalizing prior Grieser and Zolésio findings. The work thus provides a rigorous perturbation-theory framework for NP-spectrum optimization, with potential implications for plasmonic design and boundary-shape optimization. Overall, the results extend analytic perturbation theory to symmetrizable, infinite-dimensional perturbations and establish concrete tools for understanding how geometry governs spectral behavior in boundary integral operators.

Abstract

This paper concerns the eigenvalues of the Neumann-Poincaré operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.

Paper Structure

This paper contains 23 sections, 20 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Some notions of potential theory
  3. The double-layer potential.
  4. The single-layer potential.
  5. The plasmonic problem.
  6. The Calderón identity.
  7. Riesz projectors and symmetrizable operators
  8. Shape analyticity of the symmetric functions of the NP-eigenvalues
  9. Shape analyticity of the Riesz projector
  10. Shape analyticity of the symmetric functions
  11. First derivative of the symmetric functions of the NP-eigenvalues
  12. From $d_\phi \Lambda^m_h(\lambda_1[\phi_0],\ldots,\lambda_m[\phi_0])$ to $d_\phi K^*_{\phi_0}$
  13. From $d_\phi K^*_{\phi_0}$ to $d_\phi K_{\phi_0}$
  14. Taking the derivative under the integral sign
  15. Computing $\frac{d}{dt}(K_{I+t\theta})_{|t=0}$
  16. ...and 8 more sections

Key Result

Lemma 3.1

Let $\left(H,\langle\cdot,\cdot\rangle\right)$ be a Hilbert space and $B$ a compact symmetrizable operator in $H$. Let $\lambda\neq 0$ be an (isolated) eigenvalue of $B$. Let $\gamma$ be a simple closed path around $\lambda$ with positive orientation (cf. gamma). Then

Theorems & Definitions (41)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 31 more