Elliptic homogenization with almost translation-invariant coefficients

Abstract

We consider an homogenization problem for the second order elliptic equation − div ( a ( · / ε ) ∇ u ε ) = 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 , + ∞ [. 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 ε . When p ⩾ d, we exhibit admissible coefficients a such that u ε possesses different subsequences that converge to different limits in L 2 .