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 \(O(\varepsilon )\) or \(O(\varepsilon ^2)\) 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 \(O(\varepsilon )\) is open and dense in the set of \(C^2\) periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is \(O(\varepsilon )\).