Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

Scott Armstrong*, Tuomo Kuusi, Jean Christophe Mourrat

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

31 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)315-361
Number of pages47
Issue number2
Publication statusPublished - 1 Oct 2016
MoE publication typeA1 Journal article-refereed


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