For an oscillating boundary of period and amplitude ", it is known that the asymptotic behavior when " tends to zero of a three-dimensional viscous°uid satisfying slip boundary conditions is the same as if we assume no-slip (adherence) boundary conditions. Here we consider the case where the period is still " but the amplitude is " with " =" converging to zero. We show that if " =" 3 2 tends to in¯nity, the equivalence between the slip and no-slip conditions still holds. If the limit of " =" 3 2 belongs to ð0; þ1Þ (critical size), then we still have the slip boundary conditions in the limit but with a bigger friction coe±cient. In the case where " =" 3 2 tends to zero the boundary behaves as a plane boundary. Besides the limit equation, we also obtain an approximation (corrector result) of the pressure and the velocity in the strong topology of L 2 and H 1 respectively.

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In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F n , n ∈ N, defined in L 2 (Ω), for a bounded open subset Ω of R 2 . We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of F n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.

In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L 1 combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two. More precisely, in the two-dimensional case we extend the Murat-Tartar H-convergence which holds for uniformly bounded and equi-coercive conductivity sequences, as well as the compactness result which holds for bounded and equiintegrable conductivity sequences in L 1 . Our homogenization results are based on extensions of the classical div-curl lemma, which are also specific to the dimension two.

The aim of the present paper is to adapt the method of two-scale convergence to the homogenization of a pseudomonotone Dirichlet problem in perforated domains with periodic structure. The limit problem and a corrector result are obtained.