Singular Limits for Thin Film Superconductors in Strong Magnetic Fields STAN ALAMA & LIA BRONSARD Department of Mathematics and Statistics McMaster University Hamilton, ON, Canada {alama,bronsard}@mcmaster.ca BERNARDO GALV ˜ AO-SOUSA Department of Mathematics University of Toronto Toronto, ON, Canada beni@math.toronto.edu Abstract. We consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain has characteristic thickness on the scale ε> 0, and we consider the simultaneous limit as the thickness ε → 0 and the Ginzburg-Landau parameter κ →∞. We assume that the applied field is strong (on the order of ε -1 in magnitude) in its components tangential to the film domain, and of order log κ in its dependence on κ. We prove that the Ginzburg-Landau energy Γ-converges to an energy associated with a two-obstacle problem, posed on the planar domain which supports the thin film. The same limit is obtained regardless of the relationship between ε and κ in the limit. Two illustrative examples are presented, each of which demonstrating how the curvature of the film can induce the presence of both (positively oriented) vortices and (negatively oriented) antivortices coexisting in a global minimizer of the energy. Keywords : Partial Differential Equations; Calculus of Variations; Ginzburg-Landau; superconductivity. Mathematics Subject Classification 2000: 35J50, 35Q56, 49J45. 1. Introduction In this paper we continue the study of thin-film superconductors begun in our previous paper [ABGS10]. The superconducting sample occupies a domain Ω ε ⊂ R 3 , Ω ε = {(x ′ ,x 3 ) ∈ R 3 : x ′ ∈ ω, εf (x ′ ) <x 3 <εg(x ′ )}, where ω ⊂ R 2 is a bounded regular domain in the plane, f,g : ω → R are smooth functions on ω with f (x ′ ) <g(x ′ ) for all x ′ ∈ ω, and ε> 0. We denote by a(x ′ )= g(x ′ ) − f (x ′ ), the thickness of the film for given x ′ ∈ ω. We study minimizers and Gamma-limits of the full three-dimensional Ginzburg–Landau model, for the superconductor Ω ε subjected to a spatially constant external magnetic field, h ex ∈ R 3 . The state of the superconductor is determined via a complex order parameter u :Ω ε → C and the magnetic vector potential A : R 3 → R 3 , which determines the magnetic field h = ∇× A. The energy of the configuration (u, A) is given by: I ε,κ (u, A) := 1 2  Ωε  |∇ A u| 2 + κ 2 2 ( 1 −|u| 2 ) 2  dx + 1 2  R 3 |h − h ex | 2 dx, 1 arXiv:1209.3696v1 [math.AP] 17 Sep 2012