Spectral geometry of the curl operator on smoothly bounded domains

TL;DR

This paper investigates the spectral properties of the curl operator on smoothly bounded domains, proving eigenvalue simplicity for generic domains and deriving conditions for extremizing eigenvalues using a novel variational formula.

Contribution

It introduces a new variational formula for curl eigenvalues and applies it to establish simplicity and extremal conditions for the spectrum on smooth domains.

Findings

Eigenvalues of the curl operator are generically simple.

A Hadamard-like formula for curl eigenvalues is derived.

Necessary conditions for domain extremality of eigenvalues are established.

Abstract

We show that the spectrum of the curl operator on a generic smoothly bounded domain in three-dimensional Euclidean space consists of simple eigenvalues. The main new ingredient in our proof is a formula for the variation of curl eigenvalues under a perturbation of the domain, reminiscent of Hadamard's formula for the variation of Laplace eigenvalues under Dirichlet boundary conditions. As another application of this variational formula, we simplify the derivation of a well-known necessary condition for a domain to minimize the first curl eigenvalue functional among domains of a given volume and derive similar necessary conditions for a domain extremizing higher eigenvalue functionals.