  We study the magnetic Laplacian in a two-dimensional exterior domain with
Neumann boundary condition and uniform magnetic field. For the exterior of the
disk we establish accurate asymptotics of the low-lying eigenvalues in the weak
magnetic field limit. For the exterior of a star-shaped domain, we obtain an
asymptotic upper bound on the lowest eigenvalue in the weak field limit,
involving the $4$-moment, and optimal for the case of the disk. Moreover, we
prove that, for moderate magnetic fields, the exterior of the disk is a local
maximizer for the lowest eigenvalue under a $p$-moment constraint.
