The Ginzburg-Landau energy with a pinning term oscillating faster than the coherence length

TL;DR

This paper investigates the Ginzburg-Landau energy with rapidly oscillating pinning terms, demonstrating that the problem simplifies to an averaged version in both periodic and random cases, with convergence results for minimizers.

Contribution

It introduces a method to reduce the complex oscillating pinning term problem to an averaged problem using decoupling techniques, applicable to both periodic and random cases.

Findings

Reduction of the oscillating problem to an averaged problem.

Uniform convergence of minimizers to the mean pinning value.

Application of blow-up and Liouville techniques for convergence analysis.

Abstract

The aim of this article is to study the magnetic Ginzburg-Landau functional with an oscillating pinning term. We consider here oscillations of the pinning term that are much faster than the coherence length \(\varepsilon>0\) which is also the inverse of the Ginzburg-Landau parameter. We study both the case of a periodic potential and of a random stationary ergodic one. We prove that we can reduce the study of the problem to the case where the pinning term is replaced by its average, in the periodic case, and by its expectation with respect to the random parameter in the random case. In order to do that we use a decoupling of the energy due to Lassoued-Mironescu. This leads us to the study of the convergence of a scalar positive minimizer of the Ginzburg-Landau energy with pinning term and with homogeneous Neumann boundary conditions. We prove uniform convergence of this minimizer towards the mean value of the pinning term by using a blow-up argument and a Liouville type result for non-vanishing entire solutions of the real Ginzburg-Landau/Allen-Cahn equation, due to Farina.