Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems
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We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on R^d with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale ε>0, we establish homogenization error estimates of the order ε in case d≥ 3, respectively of the order ε |ε|^1/2 in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence ε^δ. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/ε)^-d/2 for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^1,α regularity theory is available.
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