Minimizers of the magnetic Ginzburg-Landau functional in simply connected domain with prescribed degree on the boundary

We study the minimizers of the GinzburgLandau free energy functional in the class (u, A) ∈ H¹(Ω; ) × H¹(Ω; ²) with |u| = 1 on ∂Ω, where Ω is a bounded simply connected domain in ². We consider the connected components of this class defined by the prescribed topological degree d of u on the boundary ∂Ω. We show that for d ≠ 0 the minimizers exist if 0 < λ ≤ 1 and do not exist if λ > 1, where λ is the coupling constant ($\sqrt{\lambda/2}$ is the GinzburgLandau parameter). We also establish the asymptotic locations of vortices for λ → 1 - 0 (the critical value λ = 1 is known as the Bogomol'nyi integrable case).