Asymptotic analysis of a class of Ginzburg-Landau equations in thin multidomains

摘要

This paper addresses the asymptotic analysis of minimizers of the classical Ginzburg-Landau energy in a thin multidomain. More precisely we are interested in the minimization of the energy over all maps u : Ω_n → C, satisfying a partial boundary Dirichlet condition u = g on a specific subset of δ?_n. Here Ω_n c K~2 is a thin bounded multidomain, and g : C → S~1 is a given smooth map. The analysis is performed in the case where Ω_n is made of a thin horizontal plate of vanishing thickness with a "forest" of vertical cylinders on the top of it of vanishing width as n → ∞. The main issue addressed here is to determine the behavior of minimizers as n → ∞ and then ε → 0, and conversely.