The Steklov Spectrum of Spherical Cylinders

TL;DR

This paper analyzes the Steklov spectrum of spherical cylinders, deriving a two-term asymptotic expansion for the spectral counting function that accounts for boundary curvature and edges.

Contribution

It provides the first detailed asymptotic expansion of the Steklov spectrum for spherical cylinders, including boundary and edge effects.

Findings

Spectral counting function admits a two-term asymptotic expansion.

Boundary curvature contributes to the second term.

Edges also contribute to the second term.

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.