Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices

Let Ω be a 2D simply connected domain, ω be a simply connected subdomain of Ω and set A = Ω\ω. In the annular type domain A, we consider the class J of complex valued maps having degrees 1 on ∂Ω and on ∂ω. We investigate whether the minimum of the Ginzburg-Landau energy Eλ is attained in J , as well as the asymptotic behavior of minimizers as the coherency length λ−1/2 tends to 0. We show that the answer to these questions is determined by the value of the H-capacity cap(A) of the domain A. This is due to the degree boundary conditions; by contrast, when Dirichlet conditions are prescribed, it is known that the behavior of minimizers does not depend on A. If cap(A) > π (A is a ”thin” or ”subcritical” domain), minimizers exist for each λ. As λ → ∞, they converge in H(A) (and even better) to an S-valued harmonic map we identify. Furthermore, these minimizers are vortexless for large λ. The same properties hold when cap(A) = π (”critical” domain), but the proof is more involved. When cap(A) < π (”thick” or ”supercritical” domain), we prove that either (i) minimizers cease to exist for large λ, or (ii) that they exist for each λ. For large λ, minimizing sequences (in case (i)) or minimizers (in case (ii)) develop exactly two vortices, one of degree 1 near ∂Ω, the other one of degree −1 near ∂ω. We conjecture that case (ii) never occurs.