On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects

Abstract

In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) $\check \Lambda(b,\nu)$ on the exterior of the disk with respect to a magnetic potential $A_{b, \nu}=A^b + A_\nu$ where, for $b\in \mathbb R$ and $\nu \in \mathbb R$, $A^b (x,y)= b\, (-y, x)$ and $A_\nu (x,y)$ is the Aharonov-Bohm potential centered at the origin of flux $2\pi \nu$. First, we show that the limit of $\check \Lambda(b,\nu)$ as $b\rightarrow 0$ is equal to the D-to-N map $\widehat \Lambda (\nu)$ on the interior of the disk associated with the potential $A_\nu (x,y)$. Secondly, we study the ground state energy of the D-to-N map $\check \Lambda(b,\nu)$ and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results obtained in the interior case for general domains.