Elliptic homogenization with almost translation-invariant coefficients

We consider an homogenization problem for the second order elliptic equation - div( a(center dot /e)Delta u(e)) = f when the coefficient a is almost translation-invariant at infinity and models a geometry close to a periodic geometry. This geometry is characterized by a particular discrete gradient of the coefficient a that belongs to a Lebesgue space L-p(R-d) for p. [1,+infinity[. When p < d, we establish a discrete adaptation of the Gagliardo-Nirenberg-Sobolev inequality in order to show that the coefficient a actually belongs to a certain class of periodic coefficients perturbed by a local defect. We next prove the existence of a corrector and we identify the homogenized limit of u(e). When p >= d, we exhibit admissible coefficients a such that ue possesses different subsequences that converge to different limits in L-2.