The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: the non-zero degree case

TL;DR

This paper analyzes the behavior of vortices in a Ginzburg-Landau model with rapidly oscillating impurities, showing they are pinned by impurities and their locations are influenced by impurity size and interactions.

Contribution

It extends the understanding of vortex pinning and location in Ginzburg-Landau models with discontinuous, oscillating impurities, including effects of impurity size.

Findings

Vortices are exactly d in number and pinned by impurities.

Vortex locations are governed by repelling effects and minimize a renormalized energy.

Vortices are pinned by the largest impurities when sizes vary.

Abstract

We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree $d > 0$. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and H\'elein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. In some special cases we prove that their macroscopic location tends to minimize the renormalized energy of Bethuel-Brezis-H\'elein. In addition, impurities affect the microscopic location of vortices. Our technics allows us to work with impurities having different size. In this situation we prove that vortices are pinned by the largest impurities.