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Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media

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Abstract

We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Luis A. Caffarelli

  2. Department of Mathematics, The University of Chicago, Chicago, IL, 60637, USA

    Panagiotis E. Souganidis

Authors
  1. Luis A. Caffarelli
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  2. Panagiotis E. Souganidis
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Correspondence to Luis A. Caffarelli.

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Both authors were partially supported by the National Science Foundation.

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Caffarelli, L.A., Souganidis, P.E. Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. math. 180, 301–360 (2010). https://doi.org/10.1007/s00222-009-0230-6

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