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Lipschitz regularity for elliptic equations with random coefficients

Authors
  • Armstrong, Scott N.
  • Mourrat, Jean-Christophe
Type
Preprint
Publication Date
Jun 29, 2015
Submission Date
Nov 13, 2014
Identifiers
DOI: 10.1007/s00205-015-0908-4
Source
arXiv
License
Yellow
External links

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.

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