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

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 $\epsilon >0$, 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 (detailed in Lassoued and Mironescu’s 1999 paper) that 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 blowup argument and a Liouville-type result for non-vanishing entire solutions of the real Ginzburg–Landau/ Allen–Cahn equation, due to the results of Farina (2003).