Spectral geometry of the curl operator on smoothly bounded domains
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913864795291648 |
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| author | Greilhuber, Josef Kepplinger, Willi |
| author_facet | Greilhuber, Josef Kepplinger, Willi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral geometry of the curl operator on smoothly bounded domains Greilhuber, Josef Kepplinger, Willi Spectral Theory Analysis of PDEs 58J50 (Primary), 58C40, 58J37 (Secondary) 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. |
| title | Spectral geometry of the curl operator on smoothly bounded domains |
| topic | Spectral Theory Analysis of PDEs 58J50 (Primary), 58C40, 58J37 (Secondary) |
| url | https://arxiv.org/abs/2502.13067 |