Spectral geometry of the curl operator on smoothly bounded domains
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.
Keywords
Cite
@article{arxiv.2502.13067,
title = {Spectral geometry of the curl operator on smoothly bounded domains},
author = {Josef Greilhuber and Willi Kepplinger},
journal= {arXiv preprint arXiv:2502.13067},
year = {2025}
}
Comments
For version 2, the proof of the Hadamard rule (Theorem 4.1) has been significantly shortened and streamlined, mostly by switching from singular/simplicial to de Rham cohomology. Additionally more care was taken in spelling out the meaning of using complex versus real Lagrangean boundary conditions. Otherwise many typos were fixed. 25 pages, 1 figure