Is the optimal magnetic rectangle a square?

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.