Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk
We consider the lowest eigenvalue $λ(b)$ of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength $b>0$. We prove that $λ$ is strictly increasing on $(0,+\infty)$. This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound $λ(b)<Θ_0 b$, where $Θ_0$ is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field $H_{C_3}^{\mathrm{loc}}$ of Ginzburg--Landau theory is, in the disk, uniquely determined for every value of the Ginzburg--Landau parameter, and not only for large ones. We also give a second proof of the bound $λ(b)<Θ_0 b$, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for $λ(b)$, valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.