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The Steklov Spectrum of Spherical Cylinders

Spencer Bullent

TL;DR

This paper analyzes the Steklov spectrum for spherical cylinders $\Omega = \mathbb{B}_{n-1}(0,R) \times (-L,L)$ and establishes a sharp two-term Weyl law for the spectral counting function $N(\sigma)$ as $\sigma \to \infty$, revealing both curvature and edge contributions in the second term. The authors reduce the problem via separation of variables to transcendental equations involving Bessel and modified Bessel functions, partition eigenvalues into transversely and radially localised families, and recast the counts as weighted lattice-point problems. They develop precise asymptotics for these lattice counts using Euler–Maclaurin and Van der Corput techniques, supplemented by uniform asymptotics for Airy and Bessel functions, and obtain explicit coefficients in terms of volumes of unit balls and Beta functions. A key finding is that edge effects contribute nontrivially to the second-term correction, illustrating how curvature and edges interact in piecewise smooth domains. The results advance understanding of Steklov-type problems on domains with curvature along smooth pieces and edges, and suggest potential universality of the edge term in similar geometries.

Abstract

The Steklov problem on a compact Lipschitz domain is to find harmonic functions on the interior whose outward normal derivative on the boundary is some multiple (eigenvalue) of its trace on the boundary. These eigenvalues form the Steklov spectrum of the domain. This article considers the Steklov spectrum of spherical cylinders (Euclidean ball times interval). It is shown that the spectral counting function admits a two term asymptotic expansion. The coefficient of the second term consists of a contribution from the curvature of the boundary and a contribution from the edges.

The Steklov Spectrum of Spherical Cylinders

TL;DR

This paper analyzes the Steklov spectrum for spherical cylinders and establishes a sharp two-term Weyl law for the spectral counting function as , revealing both curvature and edge contributions in the second term. The authors reduce the problem via separation of variables to transcendental equations involving Bessel and modified Bessel functions, partition eigenvalues into transversely and radially localised families, and recast the counts as weighted lattice-point problems. They develop precise asymptotics for these lattice counts using Euler–Maclaurin and Van der Corput techniques, supplemented by uniform asymptotics for Airy and Bessel functions, and obtain explicit coefficients in terms of volumes of unit balls and Beta functions. A key finding is that edge effects contribute nontrivially to the second-term correction, illustrating how curvature and edges interact in piecewise smooth domains. The results advance understanding of Steklov-type problems on domains with curvature along smooth pieces and edges, and suggest potential universality of the edge term in similar geometries.

Abstract

The Steklov problem on a compact Lipschitz domain is to find harmonic functions on the interior whose outward normal derivative on the boundary is some multiple (eigenvalue) of its trace on the boundary. These eigenvalues form the Steklov spectrum of the domain. This article considers the Steklov spectrum of spherical cylinders (Euclidean ball times interval). It is shown that the spectral counting function admits a two term asymptotic expansion. The coefficient of the second term consists of a contribution from the curvature of the boundary and a contribution from the edges.

Paper Structure

This paper contains 13 sections, 25 theorems, 200 equations, 1 figure.

Table of Contents

  1. Introduction and Main Result
  2. Outline of the Proof of Main Theorem
  3. Further Notation and Definitions
  4. Steklov Eigenfunctions and Transcendental Equations Governing the Associated Eigenvalues
  5. Collected properties of Airy and Bessel functions
  6. Properties of Airy functions
  7. Properties of Bessel Functions
  8. Properties of Modified Bessel Function
  9. Lattice Point Count Formulation Over Transversely Localised Waves
  10. Lattice Point Count Formulation Over Radially Localised Waves
  11. Lattice Point Count Approximation
  12. Equivalence of Edge Term In [The Steklov Spectrum of Cuboids, Theorem 1.1]
  13. Proof of Integral Dimension Reduction Proposition

Key Result

Theorem 1.1

The spectral counting function of the Steklov spectrum of $\Omega$ admits a two term asymptotic expansion as $\sigma \to \infty$ Here, $\mathbb{B}_n \coloneq \Set*{x \in \mathbb{R}^n \nonscript\:\delimsize\vert\nonscript\: \lVert*\rVert{x} < 1}$, $\lvert*\rvert{.}$ denotes the Lebesgue measure of implicit dimension, $\kappa$ denotes the mean curvature form, $\partial^j$ denotes the $j$-th co-bound

Figures (1)

  • Figure 1: $x \frac{J_{}_1'(x)}{J_{}_1(x)}$ - solid, $x \coth(\tfrac{x}{4})$ - dashed, $x \tanh(\tfrac{x}{4})$ - dot dashed

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark
  • Remark
  • Remark
  • Proposition 2.1: Proved in section:Eigenfunctions-Eigenvalues
  • Proposition 2.2: Proved in section:lattice-point-bessel
  • Proposition 2.3: Proved in section:lattice-point-bessel
  • Proposition 2.4: Proved in section:lattice-point-mod-bessel
  • Proposition 2.5: Proved in section:lattice-point-mod-bessel
  • Proposition 2.6: van der Corput's method, krätzel2000analytische
  • ...and 45 more