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Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

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Abstract

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (C k, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities), which yields, by a new “multiscale” Poincaré inequality, quantitative estimates on the sublinearity of the corrector.

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Authors and Affiliations

  1. Ceremade (UMR CNRS 7534), Université Paris-Dauphine, Paris, France

    Scott Armstrong

  2. Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

    Tuomo Kuusi

  3. Ecole normale supérieure de Lyon, CNRS, Lyon, France

    Jean-Christophe Mourrat

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  1. Scott Armstrong
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  2. Tuomo Kuusi
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  3. Jean-Christophe Mourrat
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Correspondence to Scott Armstrong.

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Communicated by L. Erdös

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Armstrong, S., Kuusi, T. & Mourrat, JC. Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization. Commun. Math. Phys. 347, 315–361 (2016). https://doi.org/10.1007/s00220-016-2663-2

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  • DOI: https://doi.org/10.1007/s00220-016-2663-2

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