MINIMIZERS OF THE MAGNETIC GINZBURG–LANDAU FUNCTIONAL IN SIMPLY CONNECTED DOMAIN WITH PRESCRIBED DEGREE ON THE BOUNDARY

Abstract

We study the minimizers of the Ginzburg–Landau free energy functional in the class (u, A) ∈ H1(Ω; ℂ) × H1(Ω; ℝ2) with |u| = 1 on ∂Ω, where Ω is a bounded simply connected domain in ℝ2. 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 ([Formula: see text] is the Ginzburg–Landau 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).