Lipschitz Regularity for Elliptic Equations with Random Coefficients

Armstrong, Scott N.; Mourrat, Jean-Christophe

Armstrong, Scott N.; Mourrat, Jean-Christophe (2016), Lipschitz Regularity for Elliptic Equations with Random Coefficients, Archive for Rational Mechanics and Analysis, 219, 1, p. 255-348. 10.1007/s00205-015-0908-4

Type

Article accepté pour publication ou publié

External document link

https://arxiv.org/abs/1411.3668v3

Date

2016

Journal name

Archive for Rational Mechanics and Analysis

Volume

219

Number

Publisher

Springer

Pages

255-348

Abstract (EN)

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∞-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 (for example finite range of dependence). We also prove a quenched L2 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.

Subjects / Keywords

quasilinear elliptic equations