Asymptotic Approximation of the Solution of the Laplace Equation in a Domain with Highly Oscillating Boundary

We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whose size depends on a small parameter $\varepsilon >0$. The assumption of sharp asperities is made; that is, the height of the asperities is fixed. Using a boundary layer corrector, we derive and analyze a nonoscillating approximation of the solution at order $O(\varepsilon^\frac{3}{2})$ for the $H^1$-norm.