Large vorticity stable solutions to the Ginzburg-Landau equations

TL;DR

This paper constructs stable solutions to the Ginzburg-Landau equations with a large, prescribed number of vortices that arrange uniformly and tend to minimize a Coulombian energy, advancing understanding of vortex patterns in superconductivity.

Contribution

It introduces a method to construct stable vortex solutions with a large number of vortices, extending previous asymptotic analysis and employing a novel energy minimization approach.

Findings

Vortices arrange with uniform density in a subregion

Vortices asymptotically minimize Coulombian energy

Solutions are stable with prescribed large vortex number

Abstract

We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices N is prescribed and blows up as the parameter epsilon, inverse of the Ginzburg-Landau parameter kappa, tends to zero. We treat the case of N as large as log epsilon, and a wide range of intensity of external magnetic field. The vortices of our solutions arrange themselves with uniform density over a subregion of the domain bounded by a "free boundary" determined via an obstacle problem, and asymptotically tend to minimize the "Coulombian renormalized energy" W introduced in [14]. The method, inspired by [22], consists in minimizing the energy over a suitable subset of the functional space, and in showing that the minimum is achieved in the interior of the subset. It also relies heavily on refined asymptotic estimates for the Ginzburg-Landau energy obtained in [14].