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Spherical caps do not always maximize Neumann eigenvalues on the sphere

Dorin Bucur, Richard S. Laugesen, Eloi Martinet, Mickaël Nahon

TL;DR

The paper shows that Neumann eigenvalues on spheres can contradict the intuitive maximality of geodesic balls for large area when topological constraints are relaxed. By constructing a multiply connected perturbation of a spherical cap—specifically four small elliptical holes aligned with hot spots—and performing a rigorous multi-hole asymptotic analysis, the authors derive explicit second-order corrections $\mu_i(\Omega^\epsilon)=\mu+\epsilon^2\kappa_i+o(\epsilon^2)$, where the $\kappa_i$ are eigenvalues of a finite matrix determined by boundary data. For apertures $\theta>\Theta$, they prove the first eigenvalue of the perturbed domain exceeds that of the corresponding cap, providing a rigorous counterexample to disk maximality under area constraint. They supplement the analytic construction with numerical helmet-domain experiments that extend counterexamples to broader parameter ranges, while also proving that single-hole perturbations cannot yield such counterexamples. The results highlight the necessity of topological or geometric restrictions to guarantee maximality of spherical caps for Neumann eigenvalues on the sphere.

Abstract

We prove the existence of an open set $Ω\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.

Spherical caps do not always maximize Neumann eigenvalues on the sphere

TL;DR

The paper shows that Neumann eigenvalues on spheres can contradict the intuitive maximality of geodesic balls for large area when topological constraints are relaxed. By constructing a multiply connected perturbation of a spherical cap—specifically four small elliptical holes aligned with hot spots—and performing a rigorous multi-hole asymptotic analysis, the authors derive explicit second-order corrections , where the are eigenvalues of a finite matrix determined by boundary data. For apertures , they prove the first eigenvalue of the perturbed domain exceeds that of the corresponding cap, providing a rigorous counterexample to disk maximality under area constraint. They supplement the analytic construction with numerical helmet-domain experiments that extend counterexamples to broader parameter ranges, while also proving that single-hole perturbations cannot yield such counterexamples. The results highlight the necessity of topological or geometric restrictions to guarantee maximality of spherical caps for Neumann eigenvalues on the sphere.

Abstract

We prove the existence of an open set for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.

Paper Structure

This paper contains 10 sections, 6 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure of the first and second eigenfunctions on the sphere
  3. Multiple Neumann eigenvalues on perturbed domains
  4. Statement of the result
  5. Definition of the ansatz
  6. Proof of Proposition \ref{['Prop_limholes']}
  7. Proof of the main result, Theorem \ref{['main_th']}
  8. Numerical and heuristic analysis of a new class of domains
  9. No asymptotic counterexample with a single hole
  10. Proof of formulas \ref{['W_n_M_n']}, \ref{['formula_shapederivative']}, \ref{['eq_Fni']}, and Lemma \ref{['lem_estansatz']}, and harmonic extension to the holes

Key Result

Theorem 1

There exists $\delta>0$ such that for each $\theta\in (\Theta-\delta,\pi)$, there exists an open set $\Omega_\theta \subseteq {\mathbb{S}^2}$ with $|\Omega_\theta|=|B_\theta|$ and

Figures (4)

  • Figure 1: Different views of one punctured domain used in the proof of Theorem \ref{['main_th']}.
  • Figure 2: Sets $\Omega_\varepsilon(\theta)$ for several values of the aperture $\theta$ and for strip thickness $\varepsilon=0.05$ (top) and $\varepsilon=0.1$ (bottom). Here $e_3$ is oriented vertically.
  • Figure 3: Value of $\mu_1(\Omega_\varepsilon(\theta))$ as a function of $\theta/\pi$, for $\varepsilon=0.05$ and $\varepsilon=0.1$. The red curve $\varepsilon=0$ corresponds to the spherical cap $B_\theta$. These graphs provide numerical counterexamples to maximality of $\mu_1$ at the cap, for cap apertures $\theta \in (0.623\pi, 0.925\pi)$ for $\varepsilon=0.05$ and $\theta \in (0.646\pi, 0.912\pi)$ for $\varepsilon=0.1$. The vertical dotted line is at $\Theta \simeq 0.7\pi$; recall that Theorem \ref{['main_th']} provides rigorous counterexamples to the right of that aperture.
  • Figure 4: Eigenfunctions associated to $\mu_1$ for $\varepsilon=0.05$, for $\theta = 1.96$ (left) and $\theta=1.97$ (right). The red color corresponds to the value $1$ while deep blue corresponds to $-1$.

Theorems & Definitions (9)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Remark 5
  • Proposition 6
  • proof
  • Lemma 7: Harmonic extension across a hole
  • proof