  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].
