Periodic unfolding and homogenization for the Ginzburg-Landau Equation
Abstract
Description
We investigate, on a bounded domain $Ω$ of $\R^2$ with fixed $S^1$-valued boundary condition $g$ of degree $d>0$, the asymptotic behaviour of solutions $u_{\varepsilon,δ}$ of a class of Ginzburg-Landau equations driven by two parameter : the usual Ginzburg-Landau parameter, denoted $\varepsilon$, and the scale parameter $δ$ of a geometry provided by a field of $2\times 2$ positive definite matrices $x\to A(\frac{x}δ)$. 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 $δ$, the existence of a limit configuration $u_\infty\in H^1_g(Ω,S^1)$, 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 $A^0$.
25 pages
25 pages