Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions
Vuk Milišić · Methods and Applications of Analysis · 2009
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].