Stochastic homogenization of linear elliptic equations: Higher-order error estimates in weak norms via second-order correctors

摘要

We are concerned with the homogenization of second-order linear ellipticequations with random coefficient fields. For symmetric coefficient fields withonly short-range correlations, quantified through a logarithmic Sobolevinequality for the ensemble, we prove that when measured in weak spatial norms,the solution to the homogenized equation provides a higher-order approximationof the solution to the equation with oscillating coefficients. In the case ofnonsymmetric coefficient fields, we provide a higher-order approximation (inweak spatial norms) of the solution to the equation with oscillatingcoefficients in terms of solutions to constant-coefficient equations. In bothsettings, we also provide optimal error estimates for the two-scale expansiontruncated at second order. Our results rely on novel estimates on thesecond-order homogenization corrector, which we establish via sensitivityestimates for the second-order corrector and a large-scale $L^p$ theory forelliptic equations with random coefficients. Our results also cover the case ofelliptic systems.