Periodic unfolding and homogenization for the Ginzburg-Landau Equation

Abstract

We investigate, on a bounded domain $\Omega$ of $\R^2$ with fixed $S^1$-valued boundary condition $g$ of degree $d>0$, the asymptotic behaviour of solutions $u_{\varepsilon,\delta}$ of a class of Ginzburg-Landau equations driven by two parameter : the usual Ginzburg-Landau parameter, denoted $\varepsilon$, and the scale parameter $\delta$ of a geometry provided by a field of $2\times 2$ positive definite matrices $x\to A(\frac{x}{\delta})$. The field $\R^2\ni x\to A(x)$ is of class $W^{2,\infty}$ and periodic. We show, for a suitable choice of the $\varepsilon$'s depending on $\delta$, the existence of a limit configuration $u_\infty\in H^1_g(\Omega,S^1)$, which, out of a finite set of singular points, is a weak solution of the equation of $S^1$-valued harmonic functions for the geometry related to the usual homogenized matrix $A^0$.