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Steklov eigenvalues of nearly circular area-normalized domains

Lucas Alland, Robert Viator

TL;DR

The paper analyzes Steklov eigenvalues on area-normalized planar domains that are perturbations of the disk. It develops a rigorous perturbation framework to derive full asymptotic expansions in the deformation parameter $\varepsilon$, yielding explicit first- and second-order corrections linked to the Fourier content of the perturbation $\rho$. By constructing and analyzing the first- and second-order coupling matrices $M_n^{(1)}$ and $M_n^{(2)}$, the authors prove that for any $k\ge 2$, the disk is not a local maximizer of the $k$th nonzero Steklov eigenvalue, providing a quantitative stability result for isoperimetric questions in two dimensions. The combination of analytic perturbation theory, Fourier analysis on the circle, and explicit second-order calculations leads to a noballs theorem, complemented by motivating numerical experiments illustrating the eigenvalue behavior under non-symmetric perturbations.

Abstract

We consider Steklov eigenvalues of nearly circular domains in $\R^{2}$ of fixed unitary area. In \cite{viator2018}, the authors treated such domains as perturbations of the disk, and they computed the first-order term of the asymptotic expansions of the Steklov eigenvalues for reflection-symmetric perturbations; here, we expand these first-order results beyond reflection-symmetry. We also recover the second-order asymptotic expansions, which enable us to prove that no Steklov eigenvalue beyond the first positive one is locally shape-optimized by the disk.

Steklov eigenvalues of nearly circular area-normalized domains

TL;DR

The paper analyzes Steklov eigenvalues on area-normalized planar domains that are perturbations of the disk. It develops a rigorous perturbation framework to derive full asymptotic expansions in the deformation parameter , yielding explicit first- and second-order corrections linked to the Fourier content of the perturbation . By constructing and analyzing the first- and second-order coupling matrices and , the authors prove that for any , the disk is not a local maximizer of the th nonzero Steklov eigenvalue, providing a quantitative stability result for isoperimetric questions in two dimensions. The combination of analytic perturbation theory, Fourier analysis on the circle, and explicit second-order calculations leads to a noballs theorem, complemented by motivating numerical experiments illustrating the eigenvalue behavior under non-symmetric perturbations.

Abstract

We consider Steklov eigenvalues of nearly circular domains in of fixed unitary area. In \cite{viator2018}, the authors treated such domains as perturbations of the disk, and they computed the first-order term of the asymptotic expansions of the Steklov eigenvalues for reflection-symmetric perturbations; here, we expand these first-order results beyond reflection-symmetry. We also recover the second-order asymptotic expansions, which enable us to prove that no Steklov eigenvalue beyond the first positive one is locally shape-optimized by the disk.
Paper Structure (28 sections, 2 theorems, 105 equations, 2 figures)

This paper contains 28 sections, 2 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivating Numerics
  4. Outline
  5. Preliminaries
  6. Polar coordinates in $\mathbb{R}^2$
  7. Area-Normalized Domains
  8. Asymptotic expansions for geometric quantities
  9. Analytic Dependence on $\varepsilon >0$
  10. An Asymptotic Expansion for Steklov Eigenvalues of Nearly Circular Domains
  11. Left Hand Side ($\partial_{\hat{\boldsymbol{\nu}}}\ u$)
  12. Right Hand Side ($\lambda u$)
  13. Leading order terms
  14. First Order
  15. Forming $M_n^{(1)}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\sigma_k(\Omega)$ be the $k$th non-zero Steklov eigenvalue of the area-normalized domain $\Omega$. If $k\geq 2$, then $\sigma_k$ is never locally maximal when $\Omega$ is a disk.

Figures (2)

  • Figure 1: Eigenvalues on $\Omega_\varepsilon$ for $\rho(\theta) = 500\cos(3\theta)$
  • Figure 2: Eigenvalues on $\Omega_\varepsilon$ for $\rho(\theta) = 500\cos(12\theta)$

Theorems & Definitions (4)

  • Theorem 1.1
  • Proposition 5.1
  • proof
  • proof : Proof of Theorem \ref{['thm:noballs']}