Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

This work is a continuation of [E. Bonnetier, D.Bresch, V. Milisic, submitted]; 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 [E. Bonnetier, D.Bresch, V. Milisic, submitted] we studied a priori estimates in this setting; here we fully develop very weak estimates a la Necas [J. Necas. Les m\'ethodes directes en th\'eorie des \'equations elliptiques] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [E. Bonnetier, D.Bresch, V. Milisic, submitted]. 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 [W. J\"ager and A. Mikelic. J. Diff. Equa., N. Neus, M. Neus, A. Mikelic, Appl. Anal. 2006].