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September  2008, 3(3): 461-487. doi: 10.3934/nhm.2008.3.461

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

Leonid Berlyand 1, and  Petru Mironescu 2,

1. 

Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802
2. 

Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 november 1918, F-69622 Villeurbanne, France

Received  April 2008 Published  June 2008

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].
Keywords: perforated domain, Ginzburg-Landau energy, vortices., homogenization.
Mathematics Subject Classification: Primary: 35B27; Secondary: 55M2.
Citation: Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461
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