Optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form

Abstract

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a $C^{1,1}$ regularity theory down to microscopic scale, which is of independent interest and is inspired by the $C^{0,1}$ theory introduced in the divergence form case by the first author and Smart \cite{AS2}.

49 pages, revised version to appear in Arch Ration Mech Anal

References in corpus (4)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• The additive structure of elliptic homogenization
• Mesoscopic higher regularity and subadditivity in elliptic homogenization

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• Absence of weak disorder for directed polymers on supercritical percolation clusters
• Optimal convergence rates in stochastic homogenization in a balanced random environment