H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

In this paper, we aim to study the asymptotic behavior (when ε → 0 ) of the solution of a quasilinear problem of the form - div ( A ε ( · , u ε ) ∇ u ε ) = f given in a perforated domain Ω \ T ε with a Neumann boundary condition on the holes T ε and a Dirichlet boundary condition on ∂ Ω . We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices ( x , d ) ↦ A ε ( x , d ) is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d , the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to A ε ( · , d ) in the perforated domain. Once the H 0 -limit A 0 ( · , d ) of the pair ( A ε , T ε ) is determined, in the second step, we deduce that the solution u ε converges in some sense to the unique solution u 0 in H 0 1 ( Ω ) of the quasilinear equation - div ( A 0 ( · , u 0 ) ∇ u ) = χ 0 f (where χ 0 is L ∞ weak ⋆ limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.