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A Regularity Theory for Random Elliptic Operators

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Abstract

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic large-scale regularity of Avellaneda & Lin (namely, intrinsic large-scale Schauder and Calderón-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field r* which we call the minimal radius. This regularity theory is qualitative in the sense that r* is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is quantifiable in the sense thatr* has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing optimal moment bounds on r* for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.

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Acknowledgements

We thank Peter Bella, Mitia Duerinckx, and Julian Fischer for suggestions on the manuscript. AG acknowledges financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410). SN acknowledges support by the DFG in the context of TU Dresden’s Institutional Strategy “The Synergetic University”. AG and FO acknowledge the hospitality of the Mittag-Leffler Institute and the support of the Chaire Schlumberger at IHÉS.

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Authors and Affiliations

  1. Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005, Paris, France

    Antoine Gloria

  2. Université Libre de Bruxelles, Bruxelles, Belgium

    Antoine Gloria

  3. Faculty of Mathematics, TU Dresden, Germany

    Stefan Neukamm

  4. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Felix Otto

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  1. Antoine Gloria
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Correspondence to Antoine Gloria.

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Lezione Leonardesca Lecture (November 20, 2017) and 6th RISM Course (July 24–27, 2018), delivered by Felix Otto.

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Gloria, A., Neukamm, S. & Otto, F. A Regularity Theory for Random Elliptic Operators. Milan J. Math. 88, 99–170 (2020). https://doi.org/10.1007/s00032-020-00309-4

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