Capacity of a multiply-connected domain and nonexistence of Ginzburg-Landau minimizers with prescribed degrees on the boundary

Suppose that ω ⊂ Ω ⊂ R. In the annular domain A = Ω \ ω̄ we consider the class J of complex valued maps having degree 1 on ∂Ω and ∂ω. It was conjectured in [5] that the existence of minimizers of the Ginzburg-Landau energy Eκ in J is completely determined by the value of the H-capacity cap(A) of the domain and the value of the Ginzburg-Landau parameter κ. The existence of minimizers of Eκ for all κ when cap(A) ≥ π (domain A is “thin”) and for small κ when cap(A) < π (domain A is “thick”) was demonstrated in [5]. Here we provide the answer for the case that was left open in [5]. We prove that, when cap(A) < π, there exists a finite threshold value κ1 of the Ginzburg-Landau parameter κ such that the minimum of the Ginzburg-Landau energy Eκ not attained in J when κ > κ1 while it is attained when κ < κ1.