Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain

Let A be an annular type domain in ℝ². Let Aδ be a perforated domain obtained by punching periodic holes of size δ in A; here, δ is sufficiently small. Suppose that J is the class of complex-valued maps in Aδ, of modulus 1 on ∂Aδ and of degrees 1 on the components of ∂A, respectively 0 on the boundaries of the holes. We consider the existence of a minimizer of the Ginzburg-Landau energy Eλ(u) = 1/2 Aδ∫ ( 2▽u ² + λ/2 (1- u ²)²) among all maps in u ∈ J. It turns out that, under appropriate assumptions on λ = λ(δ), existence is governed by the asymptotic behavior of the H¹-capacity of Aδ. When the limit of the capacities is > π, we show that minimizers exist and that they are, when δ → 0, equivalent to minimizers of the same problem in the subclass of J formed by the S¹-valued maps. This result parallels the one obtained, for a fixed domain, in [3], and reduces homogenization of the Ginzburg-Landau functional to the one of harmonic maps, already known from [2]. When the limit is < π, we prove that, for small δ, the minimum is not attained, and that minimizing sequences develop vortices. In the case of a fixed domain, this was proved in [1].