The Steklov Spectrum of Spherical Cylinders

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.

💡 Research Summary

The paper studies the Steklov eigenvalue problem on a “spherical cylinder”, i.e. the Cartesian product of an (n‑1)-dimensional Euclidean ball of radius R with an interval (−L, L) in ℝⁿ (n ≥ 3). The Steklov problem asks for harmonic functions u in the interior of a compact Lipschitz domain M such that the outward normal derivative on the boundary satisfies ∂_ν u = σ u, where σ is the eigenvalue. The spectrum is discrete, accumulates only at infinity, and the counting function N(σ)=#{σ_j<σ} is the main object of interest.

The authors prove that for the spherical cylinder Ω the counting function admits a two‑term Weyl asymptotic expansion as σ→∞:

 N(σ)=C_{n,1}|∂Ω| σ^{n‑1}+C_{n,2}∫{∂Ω}κ dS σ^{n‑2}+C{n,3}|∂²Ω| σ^{n‑2}+O(σ^{n‑2‑1/4}).

Here |∂Ω| is the total (n‑1)-dimensional surface area, κ denotes the mean curvature of the smooth part of the boundary, and |∂²Ω| is the (n‑2)-dimensional measure of the edges where the cylindrical side meets the two flat caps. The constants C_{n,1}, C_{n,2}, C_{n,3} are given explicitly in terms of volumes of unit balls and elementary beta‑function integrals. In particular C_{n,3}>0, showing that the edges contribute a genuinely positive term independent of curvature.

The proof proceeds in three major steps.

1. Separation of variables and transcendental equations.
   In cylindrical coordinates (r, x∈S^{n‑2}, z) the Laplacian separates, and eigenfunctions are products of radial Bessel functions (J_ν or modified I_ν), spherical harmonics Y_k(x), and one‑dimensional functions (cosh, sinh, cos, sin) in the z‑direction. The eigenvalues σ are characterized as the positive solutions of four families of transcendental equations of the form
   α J’{ν}(αR)+β J{ν}(αR)=α e’(αL)/e(αL)
   where (e, e’) stands for (cosh, sinh) or (cos, sin), and ν=k+β with β=(n‑3)/2. When L=kR for an integer k, a special eigenvalue σ=L appears with multiplicity ω_{n‑2,k}.

2. Classification into transversely and radially localised modes.
   The eigenvalues split naturally into two groups. “Transversely localised” modes decay exponentially in the axial direction (z) and oscillate radially; their eigenvalues are determined by the intersection of the monotone function α tanh(αL) (or α coth(αL)) with the Bessel‑ratio function α J’{ν}(αR)/J{ν}(αR). “Radially localised” modes decay radially and oscillate axially; they are governed by the intersection of α tan(αL) (or α cot(αL)) with the modified‑Bessel ratio α I’{ν}(αR)/I{ν}(αR). For each fixed angular momentum k the set of solutions can be indexed by a positive integer ℓ, leading to a double sum over (k,ℓ).

3. Lattice point counting and asymptotic analysis.
   The double sums are rewritten as weighted lattice point counts in the (k,ℓ)‑plane. The authors introduce phase functions θ(ν,x) and modulus m(ν,x) for Bessel functions, together with auxiliary functions ψ, η₁, η₂ that approximate the transcendental equations up to errors of order σ^{-1/3} (for the J‑type equations) or σ^{-1} (for the I‑type equations). The counting problem then reduces to estimating sums of the form Σ_{k} ω_{n‑2,k}⌊f(k)⌋ where f(k)≈σ·(1−(k+β)/σ)+corrections.

   To pass from sums to integrals, the paper employs the Euler–Maclaurin formula together with Van der Corput’s second‑derivative estimate. The latter requires the derivative of f to be monotone and bounded away from zero; this is guaranteed by the monotonicity of the Bessel phase and the hyperbolic/trigonometric functions involved. The rounding error ρ₁(x) (the sawtooth function) contributes an O(σ^{n‑2‑1/4}) term after careful bounding of its variation.

   The leading integral yields the surface‑area term C_{n,1}|∂Ω|σ^{n‑1}. The next order term splits into two contributions: one coming from the curvature integral ∫{∂Ω}κ dS, reproducing the known second‑order term for smooth boundaries, and another coming from the edge measure |∂²Ω|, whose coefficient C{n,3} is expressed as an explicit beta‑function integral G’{n‑1,1}=∫₀¹(1+x)^{-½}(1−x)^{n/2−1}dx / B(½,n/2). The positivity of C{n,3} is shown by a direct estimate of the integral, confirming that edges always add a non‑trivial positive contribution.

The paper also discusses several ancillary results: the weak Weyl law N(σ)=C_{n,1}|∂M|σ^{n‑1}+o(σ^{n‑1}) holds for any Lipschitz domain; sharper results with O(σ^{n‑2}) remainders require C¹,¹ or C²,α regularity; for domains satisfying a dynamical non‑periodicity condition the second term involves only curvature. The spherical cylinder, being piecewise smooth with right‑angle edges, provides a concrete example where both curvature and edge terms appear simultaneously.

Finally, the authors remark that the error term O(σ^{n‑2‑1/4}) is unlikely to be optimal; a more refined analysis of the lattice sums could improve it to O(σ^{n‑2‑1/3}). They also note that the full Steklov spectrum uniquely determines the geometric parameters (R, L, n), suggesting potential inverse‑spectral applications.

In summary, the work delivers a precise two‑term Weyl law for a class of piecewise smooth domains, elucidates how curvature and edge geometry each contribute to the second asymptotic coefficient, and showcases a blend of separation of variables, Bessel‑function asymptotics, and sophisticated lattice‑point counting techniques to achieve the result.