Global strong diamagnetism beyond the disk

Determine whether global strong diamagnetism holds for the magnetic Neumann ground-state energy on every smooth, bounded, simply connected planar domain, or at least on every convex planar domain.

Background

The paper proves that the lowest eigenvalue of the magnetic Neumann Laplacian in the unit disk is strictly increasing with respect to the magnetic-field strength, establishing global strong diamagnetism in that rotationally symmetric domain. The authors note that their proof relies substantially on the disk’s angular-momentum decomposition and explicitly leave open whether an analogous monotonicity result holds for all smooth, bounded, simply connected planar domains or for the narrower class of convex domains.

References

Three natural questions remain open. Does global strong diamagnetism hold for every smooth, bounded, simply connected planar domain, or at least for every convex one?

— Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk  (2609.08774 - Léna et al., 8 Sep 2026) in Section 1, subsection “Further questions”