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

International audienceLet $A$ 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. Let $J$ be 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 $E_\lambda (u)$ among all maps $u\in J$. It turns out that, under appropriate assumptions on the large parameter $\lambda=\lambda(\delta)$, 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, by the same authors (J. Funct. Anal. 2006), and reduces homogenization of the Ginzburg-Landau functional to the one of harmonic maps, already performed by the first author and Khruslov (SIAM J. Appl. Math. 1999). 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 by the first author, Golovaty and Rybalko (C. R. Acad. Sci. Paris 2006)