Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems
Julian Fischer, Stefan Neukamm · Archive for Rational Mechanics and Analysis · 2021
Abstract 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 $$\mathbb {R}^d$$ Rd with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varepsilon >0$$ ε>0 , we establish homogenization error estimates of the order $$\varepsilon $$ ε in case $$d\geqq 3$$ d≧3 , and of the order $$\varepsilon |\log \varepsilon |^{1/2}$$ ε|logε|1/2 in case $$d=2$$ d=2 . Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $$\varepsilon ^\delta $$ εδ . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $$(L/\varepsilon )^{-d/2}$$ (L/ε)-d/2 for a representative volume of sizeL. Our results also hold in the case of systems for which a (small-scale) $$C^{1,\alpha }$$ C1,α regularity theory is available.