Boundary Layer Estimates in Stochastic Homogenization

Abstract

We prove quantitative decay estimates for the boundary layer corrector in stochastic homogenization in the case of a half-space boundary. Our estimates are of optimal order and show that the gradient of the boundary layer corrector features nearly fluctuation-order decay; its expected value decays even one order faster. As a corollary, we deduce estimates on the accuracy of the representative volume element (RVE) method for the computation of effective coefficients: in $d\geq 3$ dimensions our understanding of the decay of boundary layers enables us to justify an improved formula for the RVE method, based on a combination of oversampling with the Hill-Mandel condition.