Elliptic homogenization with almost translation-invariant coefficients

Abstract

We consider an homogenization problem for the second order elliptic equation $-div\mathrm{(}\mathit{a}\mathrm{(}·\mathrm{/}\mathit{\epsilon }\mathrm{)}\nabla {\mathit{u}}^{\mathit{\epsilon }}\mathrm{)}=\mathit{f}$ 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 ${\mathit{L}}^{\mathit{p}}\mathrm{(}{\mathbb{R}}^{\mathit{d}}\mathrm{)}$ for $\mathit{p}\in [1\mathrm{,}+\infty [$ . When $\mathit{p}\mathrm{<}\mathit{d}$ , 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 ${\mathit{u}}^{\mathit{\epsilon }}$ . When $\mathit{p}⩾\mathit{d}$ , we exhibit admissible coefficients a such that ${\mathit{u}}^{\mathit{\epsilon }}$ possesses different subsequences that converge to different limits in ${\mathit{L}}^2$ .