The disk maximizes the first magnetic Neumann eigenvalue
For $B \ge 0$ let $A_B(x) = \frac{B}{2}(-x_2, x_1)$, a vector potential for the constant magnetic field $B$. For a bounded simply connected domain $\Omega \subset \mathbb{R}^2$ with $C^\infty$ boundary let $$\mu_1(\Omega, B) = \inf_{0 \ne u \in H^1(\Omega; \mathbb{C})} \frac{\int_\Omega |(-i\nabla - A_B) u|^2 \, dx}{\int_\Omega |u|^2 \, dx},$$ the lowest eigenvalue of the magnetic Neumann Laplacian. For the Dirichlet realization, Erdős proved the magnetic Faber–Krahn inequality: for every $B \ge 0$ the lowest Dirichlet eigenvalue is smallest, among domains of the same area, for the disk. Fournais and Helffer proved that if $\Omega$ is not a disk, then $\mu_1(\Omega, B) < \mu_1(D, B)$ for the disk $D$ of the same area and for all sufficiently small and all sufficiently large $B > 0$ (with thresholds depending on $\Omega$), and showed that the inequality below can fail for domains that are not simply connected. Colbois, Léna, Provenzano and Savo proved it, with equality only for disks, whenever $B|\Omega| \le \beta_* \pi$, where $\beta_* \ge 1$ is the largest number such that, for every $\beta \in [0, \beta_*]$, the lowest magnetic Neumann eigenvalue of the unit disk with field $\beta$ has a radial eigenfunction (numerically $\beta_* \approx 3.85$).
Problem. Let $D$ be a disk with $|D| = |\Omega|$. Is $\mu_1(\Omega, B) \le \mu_1(D, B)$ for every $B > 0$ and every such $\Omega$?
Source: Fournais and Helffer (2019), Section 1.4, inequality (1.8), and Section 5.1; stated as a conjecture, with equality only for disks, by Colbois, Léna, Provenzano and Savo (2024), Conjecture 1.5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, isoperimetric inequalities, magnetic Laplacians, Neumann eigenvalues
References
@article{fournais2019,
author = {Fournais, S{\o}ren and Helffer, Bernard},
title = {Inequalities for the lowest magnetic {N}eumann eigenvalue},
journal = {Lett. Math. Phys.},
volume = {109},
number = {7},
year = {2019},
pages = {1683--1700},
doi = {10.1007/s11005-018-01154-8},
eprint = {1706.01950},
archivePrefix = {arXiv},
}
@article{colbois2024,
author = {Colbois, Bruno and L{\'e}na, Corentin and Provenzano, Luigi and Savo, Alessandro},
title = {A reverse {F}aber--{K}rahn inequality for the magnetic {L}aplacian},
journal = {J. Math. Pures Appl. (9)},
volume = {192},
year = {2024},
pages = {103632},
doi = {10.1016/j.matpur.2024.103632},
eprint = {2403.11336},
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-magnetic-neumann-disk,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The disk maximizes the first magnetic Neumann eigenvalue}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/103-magnetic-neumann-disk.md},
note = {Entry 103 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-00360, "The disk maximizes the first magnetic Neumann eigenvalue", clopen.solutions.