Global de Gennes bound for general planar domains

Establish whether the lowest eigenvalue of the magnetic Neumann Laplacian satisfies the strict bound \(\lambda(b)<\Theta_0 b\) for every positive magnetic-field strength on every smooth, bounded, simply connected planar domain, or at least on every convex planar domain.

Background

For the unit disk, the paper proves the strict global comparison λ(b)<Θ0b\lambda(b)<\Theta_0 b at every field strength, where Θ0\Theta_0 is the de Gennes constant. The authors explain that the corresponding inequality is known for large fields on the broader class of smooth planar domains through two-term asymptotics, but explicitly identify its validity at every field strength for such domains as unresolved.

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? Does the bound \lambda(b)<\Theta_0b hold for such domains at every field strength?

— 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”