Quantitative homogenization of elliptic equations with infinitely many scales

In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\epsilon =({\epsilon }_1,{\epsilon }_2,\cdots )\in (0,1{)}^{\mathrm{\infty }}$, with ${\epsilon }_1>{\epsilon }_2>\cdots$ and ${\epsilon }_n\to 0$ as $n\to \mathrm{\infty }$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\epsilon$.