Quantitative estimates in stochastic homogenization for correlated coefficient fields

• DOI: 10.2140/apde.2021.14.2497
• Publication Date: 2022-02-09T06:50:11Z
• Authors: Antoine Gloria; Stefan Neukamm; Felix Otto
• Keywords: Nabla symbol; Homogenization; Elliptic operator; Exponent

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations.It deduces optimal estimates error from growth (extended) corrector.In line heuristics, there are transitions at dimension d = 2, and for a correlation-decay exponent β 2; we capture correct power logarithms coming these two sources criticality.The decay correlations sharply encoded terms multiscale logarithmic Sobolev inequality (LSI) ensemble under consideration -the results would fail if correlation were an α-mixing condition.Among other ensembles popular modelling media, this class includes coefficient fields local transformations Gaussian fields.The corrector φ derived bounding size spatial averages F ´g • ∇φ its gradient.This turn done by (deterministic) sensitivity estimate , is, estimating functional derivative ∂F ∂a w. r. t. field a. Appealing to LSI concentration measure yields stochastic on .The argument relies large-scale Schauder theory heterogeneous operator -∇ a∇.The treatment allows non-symmetric systems like elasticity.