Γ-limit of the Ginzburg–Landau energy in a thin domain with a large magnetic field

Abstract

A one-dimensional Ginzburg-Landau model that describes a superconducting closed thin wire with an arbitrary cross-section subject to a large applied magnetic field is derived from the three-dimensional Ginzburg-Landau energy in the spirit of Γ -convergence. Our result proves the validity of the formal result of Richardson and Rubinstein, which reveals the double limit of a large field and a thin domain. An additional magnetic potential related to the applied field is found in the limiting functional, which yields a parabolic background for the oscillatory phase transition curve between the normal and superconducting states.

Thin Film Limits for Ginzburg–Landau with Strong Applied Magnetic Fields

“…The proof of the compactness and Γ-convergence results will be presented in section 3. We note that a similar phenomenon, whereby inhomogeneities in a thin domain lead to a curious dependence on the direction of an applied field, has been observed by Richardson and Rubinstein [RR99] and proved by Shieh [Shi08] in the context of thin three-dimensional domains which shrink as ε → 0 to closed space curves. Shieh also considers Γ-limits with applied fields on the order of ε −1 .…”