Periodic Unfolding and Homogenization for the Ginzburg-landau Equation Preliminary Draft

We investigate, on a bounded domain Ω of R with fixed S-valued boundary condition g of degree d > 0, the asymptotic behaviour of solutions uε,δ of a class of Ginzburg-Landau equations driven by two parameter : the usual Ginzburg-Landau parameter, denoted ε, and the scale parameter δ of a geometry provided by a field of 2 × 2 positive definite matrices x → A( δ ). The field R ∋ x → A(x) is of class W 2,∞ and periodic. We show, for a suitable choice of the ε’s depending on δ, the existence of a limit configuration u∞ ∈ H g (Ω, S), which, out of a finite set of singular points, is a weak solution of the equation of S-valued harmonic functions for the geometry related to the usual homogenized matrix A.