Maximizer among bounded planar domains with fixed area for the magnetic Neumann Laplacian

Determine whether, for a fixed uniform magnetic field, the disk maximizes the lowest eigenvalue of the two-dimensional magnetic Laplace operator with Neumann boundary condition among all bounded simply-connected planar domains having a prescribed area. Specifically, for each b > 0, ascertain if λ1(b,Ω) attains its maximum when Ω is a disk among all simply-connected bounded Ω ⊂ ℝ^2 with |Ω| fixed.

Background

In spectral geometry, sharp geometric optimization of eigenvalues is classical for the non-magnetic Laplacian (e.g., Faber–Krahn for Dirichlet). For the magnetic Laplacian, Erdős established a Dirichlet analogue, but much less is known under Neumann conditions. In bounded interior domains with a uniform magnetic field, a prominent conjecture posits that the planar disk should maximize the lowest magnetic Neumann eigenvalue under a fixed-area constraint.

The paper focuses primarily on exterior domains, yet it situates its contributions in the broader context of magnetic Neumann optimization problems. The quoted passage highlights that, even for interior domains, the conjectural disk optimality remains unresolved despite recent progress, underscoring a central open direction in the field.