Baur’s high-field disk conjecture for all higher eigenvalues
For $B \ge 0$ let $A_B(x) = \frac{B}{2}(-x_2, x_1)$, the vector potential of a constant magnetic field of strength $B$ in the plane. For a bounded open set $\Omega \subset \mathbb{R}^2$ let $\lambda_1(\Omega,B) \le \lambda_2(\Omega,B) \le \cdots$ be the eigenvalues, counted with multiplicity, of the magnetic Dirichlet Laplacian $(-i\nabla - A_B)^2$, defined by the quadratic form $\int_\Omega |(-i\nabla - A_B)u|^2\,dx$ on $H^1_0(\Omega; \mathbb{C})$. Let $D$ denote a disk with $|D| = |\Omega|$. Erdős proved $\lambda_1(\Omega,B) \ge \lambda_1(D,B)$ for every $B \ge 0$ (1996). Baur computed numerical minimizers of $\lambda_k(\cdot\,,B)$ at fixed area for $k \le 7$; once the flux $B|\Omega|/2\pi$ exceeds $k$, the minimizer he found was always a disk. Lotoreichik and Morin proved that for each $k$ there are $c_k, B_k > 0$ such that every bounded simply connected domain $\Omega$ of unit area with $\lambda_k(\Omega,B) \le \lambda_k(D,B)$ and $B \ge B_k$ has Fraenkel asymmetry at most $(\ln B / (c_k^2 B))^{1/3}$.
Problem. Is $$\lambda_k(\Omega,B) \ge \lambda_k(D,B), \qquad |D| = |\Omega|,$$ for every bounded simply connected domain $\Omega \subset \mathbb{R}^2$, every integer $k \ge 2$ and every $B$ with $B|\Omega| \ge 2\pi k$?
Source: Baur (2025), Conjecture 4.2, for all bounded open sets; for simply connected domains, Lotoreichik and Morin (2026), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, isoperimetric inequalities, magnetic Laplacians
References
@article{baur2025,
author = {Baur, Matthias},
title = {Numerical optimization of eigenvalues of the planar magnetic {D}irichlet {L}aplacian with constant magnetic field},
journal = {J. Math. Phys.},
volume = {66},
number = {7},
year = {2025},
pages = {072104},
doi = {10.1063/5.0253311},
eprint = {2412.06533},
archivePrefix = {arXiv},
}
@article{lotoreichik2026,
author = {Lotoreichik, Vladimir and Morin, L{\'e}o},
title = {On shape optimization with large magnetic fields in two dimensions},
journal = {J. Geom. Anal.},
volume = {36},
number = {4},
year = {2026},
pages = {120},
doi = {10.1007/s12220-026-02363-7},
eprint = {2509.08412},
archivePrefix = {arXiv},
}
@article{erdos1996,
author = {Erd{\H{o}}s, L{\'a}szl{\'o}},
title = {Rayleigh-type isoperimetric inequality with a homogeneous magnetic field},
journal = {Calc. Var. Partial Differential Equations},
volume = {4},
number = {3},
year = {1996},
pages = {283--292},
doi = {10.1007/BF01254348},
}
@misc{aim-baur-high-field-disk,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Baur’s high-field disk conjecture for all higher eigenvalues}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/105-baur-high-field-disk.md},
note = {Entry 105 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00331, "Baur’s high-field disk conjecture for all higher eigenvalues", clopen.solutions.