Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

Abstract

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\epsilon )$ or $O({\epsilon }^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\epsilon )$ is open and dense in the set of $C^{2,\hat{I}\pm }$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\epsilon )$.

15 pages; some typos and imprecise points are corrected