Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions

Abstract

This work is a continuation of [3]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In [3] we studied a priori estimates in this setting; here we fully develop very weak estimates à la Nečas [17] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [3]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [13, 18].