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Rates of convergence for the homogenization of fully nonlinear uniformly elliptic PDE in random media. (English) Zbl 1192.35048
This paper is concerned with rates of convergence for the homogenization of general uniformly elliptic fully nonlinear second order PDEs in periodic, almost periodic and strongly mixing stationary random environments. In the case of strongly mixing stationary media the authors establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear PDEs of elliptic type. The proof can be divided into two main steps. Firstly, a rate for special quadratic data is derived and relies on the arguments from [L. A. Caffarelli, P. E. Souganidis and L. Wang, Commun. Pure Appl. Math. 58, No. 3, 319–361 (2005; Zbl 1063.35025)]. Secondly, the authors introduce a new notion of \(\delta\)-viscosity solution and show the rates already known for quadratic data can be extended to any general data. Next, as an application of this result, rates of convergence for homogenization in periodic and almost periodic media are obtained.

MSC:
35J60 Nonlinear elliptic equations
35J15 Second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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