Random nonlinear elliptic equations homogenize at the optimal rate: order ε in d≥3 and order ε|log ε|^{1/2} in d=2, matching the sharp linear-elliptic rates.
Quantitative stochastic homogenization and large-scale regularity
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abstract
This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. There are several chapters containing new results, such as: quantitative estimates for the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion; optimal estimates for the homogenization of the parabolic and elliptic Green functions; and $W^{1,p}$-type estimates for two-scale expansions.
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Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems
Random nonlinear elliptic equations homogenize at the optimal rate: order ε in d≥3 and order ε|log ε|^{1/2} in d=2, matching the sharp linear-elliptic rates.