Optimal Convergence Rates in Multiscale Elliptic Homogenization

ABSTRACT This paper is devoted to the quantitative homogenization of multiscale elliptic operator , where , , and . We assume that is 1‐periodic in each and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios . In the present paper, under the assumption of real analytic coefficients, we introduce the so‐called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to . This convergence rate is optimal in the sense that cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double‐log scale‐separation condition.