Steklov Eigenvalues of Nearly Spherical Domains

TL;DR

This paper studies how Steklov eigenvalues change for nearly spherical three-dimensional domains, deriving explicit formulas and proving that certain eigenvalues are stationary for a ball when specific conditions are met.

Contribution

It computes the first-order asymptotic expansion of Steklov eigenvalues for nearly spherical domains using Wigner 3j-symbols and proves a new isoperimetric stationarity result.

Findings

Explicit first-order asymptotic expansion in terms of Wigner 3j-symbols.

Stationarity of volume-normalized eigenvalues for balls when l is a perfect square.

Analytic dependence of eigenvalues on domain perturbations.

Abstract

We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion, which can explicitly be written in terms of the Wigner 3-jsymbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if l is a square integer, the volume-normalized l-th Steklov eigenvalue is stationary for a ball.