Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary

Abstract

We study the asymptotic behaviour of the solution of Laplace equation in a domain with very rapidly oscillating boundary. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rugose wall. The rugose wall is a plane covered with periodic asperities which size depends on a small parameter $ϵ>0$. The assumption of sharp asperities is made, that is the height of the asperities does not vanish as $ϵ\to 0$. We prove that, up to an exponentially decreasing error, the solution of Laplace equation can be approximated, outside a layer of width 2$ϵ$, by a non-oscillating explicit function.