Periodic unfolding and homogenization for the Ginzburg-Landau Equation

We investigate, on a bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^2$ with fixed $S^1$-valued boundary condition $g$ of degree $d>0$, the asymptotic behaviour of solutions $u_{\epsilon ,\delta }$ of a class of Ginzburg-Landau equations driven by two parameter : the usual Ginzburg-Landau parameter, denoted $\epsilon$, 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 ${\mathbb{R}}^2\ni x\to A(x)$ is of class $W^{2,\mathrm{\infty }}$ and periodic. We show, for a suitable choice of the $\epsilon$’s depending on $\delta$, the existence of a limit configuration $u_{\mathrm{\infty }}\in H_g^1(\mathrm{\Omega },S^1)$, 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 $A^0$.