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arXiv · 2502.13067

Spectral geometry of the curl operator on smoothly bounded domains

Josef Greilhuber·Willi Kepplinger
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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.

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Josef Greilhuber, Willi Kepplinger. 2025-02-18. Spectral geometry of the curl operator on smoothly bounded domains. https://arxiv.org/abs/2502.13067

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