Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

Abstract

We characterize diffusion matrices that yield a L ∞ convergence rate of O(ε 2 ) in the theory of periodic homogenization of linear elliptic equations in nondivergence-form. Such type-ε 2 diffusion matrices are of particular interest as the optimal rate of convergence in the generic case is only O(ε). First, we provide a new class of type-ε 2 diffusion matrices, confirming a conjecture posed in [15]. Then, we give a complete characterization of diagonal diffusion matrices in two dimensions and a systematic study in higher dimensions.

Quantitative homogenization in a balanced random environment

“…Thus, for ≥ 5, we obtain the optimal rate of convergence for the homogenization of the Dirichlet problem, which is generically of scale −1 . This is consistent with the generically optimal rate −1 for the periodic setting (see the classical books [8,36] for the derivation, and [32,45,30] for discussions on the optimality of the rates.) It is not clear to us what the optimal rates are when 2 ≤ ≤ 4, which deserve further analysis.…”

“…Theorem 14). Note that in the discrete setting, we will need (30) as well because of discretization. It would be also clear later in Section 3 that the log factor in the bound of 2 will help us achieve the log factor in Theorem 5.…”

“…In what follows we will apply the classical method of two-scale expansions to quantify the rate of the homogenization for the Dirichlet problem (19). The proof is similar to that in the periodic setting (see [32,45,30] for example). The only difference is that we use the approximate corrector here instead of the actual corrector in the periodic setting.…”

Stochastic integrability of heat-kernel bounds for random walks in a balanced random environment

“…In the classical periodic environment setting, it is well-known that the existence of a stationary corrector implies that the optimal homogenization error of problem (2) is generically of scale −1 . Readers may refer to the classical books [4,23] for the derivation of the rate in the periodic setting, and [21,20] for discussions on the optimality of the rates.…”