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

Let $A$ be an annular type domain in $\mathbb{R}^2$. Let $A_\delta$ be a perforated domain obtained by punching periodic holes of size $\delta$ in $A$; here, $\delta$ is sufficiently small. Suppose that $\J$ is the class of complex-valued maps in $A_\delta$, of modulus $1$ on $\partial A_\delta$ and of degrees $1$ on the components of $\partial A$, respectively $0$ on the boundaries of the holes.

We consider the existence of a minimizer of the Ginzburg-Landau energy

$E_\lambda(u)=\frac 1\2_[\int_{A_\delta}](|\nabla u|^2+\frac\lambda 2(1-|u|^2)^2)$
among all maps in $u\in\J$.

It turns out that, under appropriate assumptions on $\lambda=\lambda(\delta)$, existence is governed by the asymptotic behavior of the $H^1$-capacity of $A_\delta$. When the limit of the capacities is $>\pi$, we show that minimizers exist and that they are, when $\delta\to 0$, equivalent to minimizers of the same problem in the subclass of $\J$ formed by the $\mathbb{S}^1$-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 $<\pi$, we prove that, for small $\delta$, the minimum is not attained, and that minimizing sequences develop vortices. In the case of a fixed domain, this was proved in [1].