Is the optimal magnetic rectangle a square?

TL;DR

This paper investigates whether the square minimizes the lowest eigenvalue of the magnetic Dirichlet Laplacian among rectangles, providing partial proofs for weak magnetic fields and exploring geometric and spectral properties.

Contribution

It conjectures the square as the optimal shape for the eigenvalue under area or perimeter constraints and establishes results for weak magnetic fields.

Findings

The conjecture holds for weak magnetic fields.

Lower and upper bounds for the eigenvalue are established.

The problem relates to eigenvalue simplicity and symmetry of minimizers.

Abstract

We are concerned with the dependence of the lowest eigenvalue of the magnetic Dirichlet Laplacian on the geometry of rectangles, subject to homogeneous fields. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to the well-known magnetic-free analogue, the present spectral problem does not admit explicit solutions. By establishing lower and upper bound to the eigenvalue, we establish the conjecture for weak magnetic fields. Moreover, we relate the validity of the conjecture to the simplicity of the eigenvalue and symmetries of minimisers of a non-convex minimisation problem.