Steklov Eigenvalues of Nearly Spherical Domains
Robert Viator, Braxton Osting · SIAM Journal on Control and Optimization · 2022
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-$j$ symbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if $\ell$ is a square integer, the volume-normalized $\ell$th Steklov eigenvalue is stationary for a ball.