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Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

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Abstract

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto (Ann Probab 39(3):779–856, 2011; Ann Appl Probab 22(1):1–28, 2012) for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green’s functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a C 1,1 regularity theory down to microscopic scale, which is of independent interest and is inspired by the C 0,1 theory introduced in the divergence form case by the first author and Smart (Ann Sci Éc Norm Supér (4) 49(2):423–481, 2016).

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  1. Scott Armstrong

    Present address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

Authors and Affiliations

  1. CNRS, UMR [7534], CEREMADE, Université Paris-Dauphine, PSL Research University, Paris, France

    Scott Armstrong

  2. Department of Mathematics, University of Wisconsin, Madison, MI, USA

    Jessica Lin

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  1. Scott Armstrong
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  2. Jessica Lin
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Correspondence to Scott Armstrong.

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Communicated by A. Braides

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Armstrong, S., Lin, J. Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form. Arch Rational Mech Anal 225, 937–991 (2017). https://doi.org/10.1007/s00205-017-1118-z

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  • DOI: https://doi.org/10.1007/s00205-017-1118-z

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