The Steklov spectrum of cuboids

TL;DR

This paper derives a detailed asymptotic formula for Steklov eigenvalues on cuboids, revealing contributions from facets and addressing complexities unique to the Steklov problem.

Contribution

It provides the first two-term asymptotic expansion for Steklov eigenvalues on cuboids, including explicit second term calculation and novel methods for eigenfunction completeness.

Findings

Two-term asymptotic formula for Steklov eigenvalues

Explicit calculation of the second asymptotic term

Applications to spectral determination of cuboids

Abstract

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.