Ginzburg-Landau model with small pinning domains

Mickaël Dos Santos; Oleksandr Misiats

doi:10.3934/nhm.2011.6.715

Article Contents

2011, Volume 6, Issue 4: 715-753. Doi: 10.3934/nhm.2011.6.715

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Ginzburg-Landau model with small pinning domains

Mickaël Dos Santos^1, and
Oleksandr Misiats^2,

1.
Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne
2.
Department of Mathematics, The Pennsylvania State University, University Park PA 16802

Received: February 28, 2011

Revised: September 30, 2011

Published: November 2011

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Abstract

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.

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Mathematics Subject Classification: Primary: 49K20, 35J66, 35J50; Secondary: 47H11.

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