Lipschitz regularity for elliptic equations with random coefficients

Abstract

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^{\infty }$-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched $L^2$ estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

85 pages, minor revision

References in corpus (3)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

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