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Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

Armstrong, Scott N.; Kuusi, Tuomo; Mourrat, Jean-Christophe (2016), Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization, Communications in Mathematical Physics, 347, 2, p. 315-361. 10.1007/s00220-016-2663-2

Type
Article accepté pour publication ou publié
External document link
https://hal-ens-lyon.archives-ouvertes.fr/ensl-01401898
Date
2016
Journal name
Communications in Mathematical Physics
Volume
347
Number
2
Publisher
Springer
Pages
315-361
Publication identifier
10.1007/s00220-016-2663-2
Metadata
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Author(s)
Armstrong, Scott N.

Kuusi, Tuomo

Mourrat, Jean-Christophe
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.
Subjects / Keywords
stochastic homogenization

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