Quantitative stochastic homogenization: Local control of homogenization error through corrector

Abstract

This note addresses the homogenization error for linear elliptic equations in divergenceform with random stationary coefficients. The homogenization error is measured by comparing the quenched Green's function to the Green's function belonging to the homogenized coefficients, more precisely, by the (relative) spatial decay rate of the difference of their second mixed derivatives. The contribution of this note is purely deterministic: It uses the expanded notion of corrector, namely the couple of scalar and vector potentials (φ, σ), and shows that the rate of sublinear growth of (φ, σ) at the points of interest translates one-to-one into the decay rate.

Precised approximations in elliptic homogenization beyond the periodic setting

“…One interesting feature we emphasize in the present contribution is that the results of [1,24] of the periodic setting indeed carry over not only to the perturbed periodic setting, but also to a quite general abstract setting, which we make precise in Section 1.2 below. The latter observation about the generalization of the results of [1] and related works to non periodic setting is corroborated by the recent works [5,17]. Some of the necessary assumptions presented there (in the context of random homogenization) are quite close in spirit to our own formalization.…”

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

“…when d > 2, is an L 2 -off-diagonal bound for ∇∇G D and ∇G D , in both space variables x and y and depending only on the dimension and the ellipticity ratio. It is mainly obtained through a duality argument à la Avellaneda-Lin ( [3], Theorem 13) on standard energy estimates for solutions of −∇ · a∇u = ∇ · g, combined with an inner-regularity estimate for a-harmonic functions in the spirit of Lemma 4 of [5]. We stress here that this result is inspired by Lemma 2 of [5] and provides the new and pivotal ingredient for the first fundamental estimate for G D .…”

“…We start with two variants of Lemma 4 of [5]; while this last one is a statement for ensembles of locally a-harmonic functions, the following Lemma 2 takes into account the new perturbation term L n and the more general domain D. If d > 2, then Lemma 3 is a further generalisation to the case of functions solving −∇ · a∇u + εL n u = 0 on outer domains. We postpone the proofs of Lemma 2 and Lemma 3 to the Appendix.…”

“…Roughly speaking, Lemma 2 and Lemma 3 state that the previous inequalities remain true also if we exchange in the r.h.s. the order of the integration in μ and the supremum over the functionals F. We may refer to the result of Lemma 4 of [5], which corresponds to Lemma 2 with D = R d and ε = 0, as a compactness statement for ensembles of locally a-harmonic functions. Indeed, as we show in the appendix, inequality (35) actually follows by an inner regularity estimate which allows to control the energy of an a-harmonic function u in an interior domain by the L 2 -norm on {|x| < 2R} of (− N ) − l 2 u for any even l ∈ N. Here, − N denotes the Laplacian with Neumann boundary conditions.…”

Effective multipoles in random media

“…Loosely speaking, it can be seen as relating the quenched Green's function to the homogenized Green's function, on the level of gradients. In this sense, this work takes up the analysis started by the three authors in [15]. Going beyond [15], this paper provides a second-order error analysis.…”