Summary: We study the equilibrium configuration of a nematic liquid crystal bounded by a rough surface. The wrinkling of the surface induces a partial melting in the degree of orientation. This softened region penetrates the bulk up to a length scale which turns out to coincide with the characteristic wavelength of the corrugation. Within the boundary layer where the nematic degree of orientation decreases, the tilt angle steepens and gives rise to a nontrivial structure, which may be interpreted in terms of an effective weak anchoring potential. We determine how the effective surface extrapolation length is related to microscopic anchoring parameters. We also analyze the crucial role played by the boundary conditions assumed on the degree of orientation. Quite different features emerge depending on whether they are Neumann- or Dirichlet-like. These features may be useful to ascertain experimentally how the degree of orientation interacts with an external boundary.