Homogenization of the Ginzburg-landau Equation in a Domain with Oscillating Boundary

We study the asymptotic behaviour, as h tends to +∞, of the nonlinear system:    −∆uh − uh + |uh|uh = f in Ωh, Duh · ν = 0 on ∂Ωh, uh : Ωh → R2, in a varying domain Ωh in R2. The boundary ∂Ωh contains an oscillating part like a comb with fine teeth periodically distributed in the first direction 0x1 with period h−1 and thickness λh−1, 0 < λ < 1. We identify the limit problem where the operator −∆ is reduced to − ∂ 2 ∂x2 in the domain corresponding to the oscillating boundary. AMS (MOS) Subject Classification. 35B27, 35Q55, 58E50.