The additive structure of elliptic homogenization

Abstract

Abstract. One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in [2]: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

“…We conclude by showing first in Section 6, by a deterministic argument resembling a numerical analysis exercise, that the convergence of the subadditive quantities implies control of the error in homogenization for the Dirichlet problem. This concludes the proof of our first main result and gives us the harmonic approximation we need to run the arguments of [4,6,7,19] to obtain the quantitative C k;1 regularity theory and, in particular, the Liouville results. The latter is summarized in Section 7.…”

“…Following [4,5], we next show that Theorem 1.1 and Lemma 7.2 imply a form of higher regularity for coarsenings of a-harmonic functions on mesoscopic scales. The following lemma can be compared to [ PROOF.…”

“…Following [4,5], we next show that Theorem 1 and Lemma 7.2 imply a form of higher regularity for coarsenings of a-harmonic functions on mesoscopic scales. The following lemma can be compared to [ and, for each k ∈ N, a constant C(k, d, p, λ) < ∞ such that, for every R ≥ 2X , u ∈ A(C ∞ ∩ B R ) and r ∈ X , 1 2 R , (7.18) inf…”

“…We next introduce the subadditive quantities. These are based on similar quantities introduced in the continuum, uniformly elliptic setting in [7] and variants of the quantities which have been recently used to obtain optimal estimates and scaling limits in stochastic homogenization of uniformly elliptic equations [4,5].…”

“…With Theorem 1.1 now proved, the second main result of the paper, Theorem 1.2, essentially follows from the arguments introduced in the uniformly elliptic case in [7] and elaborated in [4,5]. The main idea is that an appropriate quantitative homogenization result, like Theorem 1.1, can be thought of as a result about harmonic approximation: it implies that an arbitrary solution of the heterogeneous equation can be well-approximated by an x a-harmonic function.…”