Methods and Applications of Analysis

Volume 16 (2009)

Number 2

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

Pages: 157 – 186



Vuk Milišić


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].


Wall-laws, rough boundary, Laplace equation, multi-scale modelling, boundary layers, error estimates, natural boundary conditions, vertical boundary correctors

2010 Mathematics Subject Classification

35B27, 65Mxx, 76D05, 76Mxx

Published 1 January 2009