Is the optimal magnetic rectangle always a square?
For $B \ge 0$ put $A_B(x) = \frac{B}{2}(-x_2, x_1)$ and $q_{\Omega,B}[u] = \int_\Omega |(-i\nabla - A_B)u|^2 \, dx$. For $a > 0$ let $R_a = (0, a) \times (0, a^{-1})$, and let $$\lambda_1(R_a, B) = \inf_{0 \ne u \in H_0^1(R_a; \mathbb{C})} \frac{q_{R_a,B}[u]}{\|u\|_2^2}$$ be the lowest eigenvalue of the magnetic Dirichlet Laplacian on $R_a$. For $B = 0$ separation of variables shows that the square minimises $\lambda_1(R_a, 0)$; for $B > 0$ the problem does not separate. Erdős proved that among all planar domains of given area the disk minimises the lowest magnetic Dirichlet eigenvalue, for every constant field. Krejčiřík proved the inequality below for all $a > 0$ when $B$ is smaller than a constant independent of $a$, and Lotoreichik and Morin proved that as $B \to \infty$ any minimising rectangle converges to the square.
Problem. Is $\lambda_1(R_a, B) \ge \lambda_1(R_1, B)$ for every $a > 0$ and every $B > 0$? The area and the field are fixed while the aspect ratio varies.
Source: Krejčiřík (2026), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, magnetic Laplacians
References
@article{krejcirik2026,
author = {Krej{\v{c}}i{\v{r}}{\'\i}k, David},
title = {Is the optimal magnetic rectangle a square?},
journal = {J. Geom. Anal.},
volume = {36},
number = {3},
year = {2026},
pages = {Paper No. 116},
doi = {10.1007/s12220-026-02356-6},
eprint = {2508.16152},
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},
}
@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 = {Paper No. 120},
doi = {10.1007/s12220-026-02363-7},
eprint = {2509.08412},
archivePrefix = {arXiv},
}
@misc{aim-optimal-magnetic-rectangle,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Is the optimal magnetic rectangle always a square?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/106-optimal-magnetic-rectangle.md},
note = {Entry 106 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-00366, "Is the optimal magnetic rectangle always a square?", clopen.solutions.