Periodic unfolding and homogenization for the Ginzburg-Landau Equation

2009/04/11 by Myrto Sauvageot, Sauvageot, Myrto
Computer Science · Engineering · Mathematics · #35J25 #35J60 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #math.AP #msc:35J25 #msc:35J60

paper · pdf · doi:10.48550/arxiv.0904.1828

25 pages

arxiv created 2009/04/11 · openalex publication_date 2009/04/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate, on a bounded domain Ω of \R² with fixed S¹-valued boundary condition g of degree d>0, the asymptotic behaviour of solutions u_ε,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(\fracxδ). The field \R²\ni 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_∞∈ H¹_g(Ω,S¹), which, out of a finite set of singular points, is a weak solution of the equation of S¹-valued harmonic functions for the geometry related to the usual homogenized matrix A⁰.

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