We study the equilibrium configuration of a nematic liquid crystal bounded by a rough surface. If ξ and η are, respectively, the nematic coherence length and the characteristic wavelength of the roughness, we show that in both limiting cases ξ η and η ξ the wrinkling of the surface induces a nontrivial structure in the equilibrium solution which may be interpreted in terms of an effective weak anchoring potential. We also determine how the effective surface extrapolation length is related to the microscopic anchoring parameters.Nematic liquid crystals are fluid aggregates of elongated molecules. When the nematic rods interact with an external surface, the elastic strain energy induces them to align parallel to the unit normal, even if the surface is not perfectly flat. It is however physically intuitive that nematic molecules will disorder if we force them to follow a rough surface, i.e., if we impose a rapidly varying boundary condition. This surface melting was first experimentally detected and then has been confirmed by approximated analytical solutions, numerical calculations, and molecular Monte Carlo simulations. The combined effect of a rapidly varying director anchoring and surface melting gives rise to an effective weak-anchoring effect that was first observed in [6]. The problem of relating the effective anchoring extrapolation length to the microscopic roughness parameters has been studied in several theoretical papers, all framed within the Frank theory [7][8][9][10]. They neglect the degree of orientation decrease which has been recognized by many authors as a crucial effect of surface roughness [2,4]. We therefore frame within the Landau-de Gennes order-tensor theory, to be able to detect the effects on both the director and the degree of orientation. The order tensor Q is defined as the trace-free part of the second moment of the probability distribution of molecular orientations [11].We study a nematic semi-infinite cell, in contact with a rough boundary. The roughness of the surface is modeled by an oscillating anchoring condition, characterized by an oscillation amplitude Δ and a wavelength η. On the limiting surface, a homeotropic condition is enforced. Let ξ, ω, s pr be respectively, the nematic coherence length, the depth of the potential well and the preferred degree of orientation [1].When ξ/η 1 we can perform a singular perturbation expansion of the associated Euler-Lagrange equations derived by minimization of the free energy functional. We find that the average solutions for both the degree of order s and the molecular tilt angle θ show a boundary-layer structure. In particular, the degree of order s exhibits an inner layer of thickness ξ and a mid layer of thickness η, while θ shows only a layer of thickness η. This is well evidenced in Fig. 1 and 2 [1]. Fig. 1 Boundary layers in the mean degree of orientation s(x, z) x when ξ = 0.25η, spr = 0.8, ω = 0.6, m = 0.1/η, and Δ = 1.5. The plot exhibits the presence of two boundary layers, the internal one being required by the free boundary ...