Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Networks and Heterogeneous Media Année : 2008

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

Résumé

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. 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).
Fichier principal
Vignette du fichier
prehom.pdf (296.58 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00358338 , version 1 (03-02-2009)

Identifiants

  • HAL Id : hal-00358338 , version 1

Citer

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks and Heterogeneous Media, 2008, 3 (3), pp.461-487. ⟨hal-00358338⟩
383 Consultations
144 Téléchargements

Partager

Gmail Mastodon Facebook X LinkedIn More