Local optimality of the exterior disk under fixed area or fixed perimeter constraints

Establish whether the exterior of the disk is a local maximizer of the lowest eigenvalue of the two-dimensional magnetic Laplace operator with Neumann boundary condition and uniform magnetic field among exterior domains, under the geometric constraint of either fixed area or fixed perimeter of the interior complement. Precisely, for small deformations of the disk and a given b > 0, determine if λ1(b,Ωext) is locally maximized when Ω is a disk among star-shaped domains satisfying either |Ω| = const or Per(∂Ω) = const.

Background

The authors prove local optimality of the exterior disk under a p-moment constraint and derive sharp weak-field asymptotics. However, the most natural geometric constraints for exterior problems—fixed area or fixed perimeter of the interior domain—are not handled by their method.

The remark explicitly points out that local optimality under fixed area or perimeter remains open, noting technical challenges unique to the magnetic Neumann setting (e.g., non-radial ground states), and draws parallels with optimization of higher Robin eigenvalues in exterior domains.