Ginzburg-Landau model with small pinning domains

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 {it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $vto0$; here, $v$ 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 $d O$, with topological degree ${rm deg}_{d O} (g) = d >0$. Our main result is that, for small $v$, 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 {it local renormalized energy} which does not depend on the external boundary conditions.