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 which 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 which size depends on a small parameter ε > 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(ε) for the H-norm. Classification MSC : 35B40, 35C20, 35J05 ∗This work is partially supported by the C.N.R. Project “Agenzia 2000”. †Laboratoire de Mathématiques Appliquées, CNRS (UMR 6620), Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex,France ([email protected]). ‡Laboratoire de Mathématiques Appliquées, CNRS (UMR 6620), Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex,France ([email protected]). §Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy ([email protected]). ¶Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, Università di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy, ([email protected] ).