Asymmetry of curl eigenfields solving Woltjer's variational problem

2025/06/26 by Daniel Peralta‐Salas, Peralta-Salas, Daniel, David Perrella +3
Engineering · #35P15 #35Q31 #37C10 (Secondary) #53C80 #53Z05 #58E30 #76W05 (Primary) #Analysis of PDEs (math.AP) #Elasticity and Material Modeling #Elasticity and Wave Propagation #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2506.21243

openalex publication_date 2025/06/26 · openalex created_date 2025/10/15 · openalex updated_date 2026/08/02

Abstract

We construct families of rotationally symmetric toroidal domains in \mathbb R³ for which the eigenfields associated to the first (positive) Ampèrian curl eigenvalue are symmetric, and others for which no first eigenfield is symmetric. This implies, in particular, that minimizers of the celebrated Woltjer's variational principle do not need to inherit the rotational symmetry of the domain. This disproves the folk wisdom that the eigenfields corresponding to the lowest curl eigenvalue must be symmetric if the domain is.

Citations

• On the shape of the first fractional eigenfunction
• Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

Related

• Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media
• Spectral geometry of the curl operator on smoothly bounded domains
• Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue
• Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime
• Steklov eigenvalues of nearly circular area-normalized domains