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Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

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Date
2016
Link to item file
https://hal-ens-lyon.archives-ouvertes.fr/ensl-01401898
Dewey
Probabilités et mathématiques appliquées
Sujet
stochastic homogenization
Journal issue
Communications in Mathematical Physics
Volume
347
Number
2
Publication date
2016
Article pages
315-361
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s00220-016-2663-2
URI
https://basepub.dauphine.fr/handle/123456789/16351
Collections
  • CEREMADE : Publications
Metadata
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Author
Armstrong, Scott N.
Kuusi, Tuomo
Mourrat, Jean-Christophe
Type
Article accepté pour publication ou publié
Abstract (EN)
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 (Ck, 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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