Local strong magnetic fields and the Little-Parks effect

Starting from the Ginzburg–Landau model in a planar simply connected domain, with a local compactly supported applied magnetic field, we derive an effective model in the strong field limit, defined on a non-simply connected domain. The effective model features oscillations in the Little-Parks and Aharonov–Bohm spirit. We discuss also a similar question for the lowest eigenvalue of the magnetic Laplacian.