Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

Scott Armstrong*, Tuomo Kuusi, Jean Christophe Mourrat

*Tämän työn vastaava kirjoittaja

Tutkimustuotos: LehtiartikkeliArticleScientificvertaisarvioitu

20 Sitaatiot (Scopus)

Abstrakti

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.

AlkuperäiskieliEnglanti
Sivut315-361
Sivumäärä47
JulkaisuCOMMUNICATIONS IN MATHEMATICAL PHYSICS
Vuosikerta347
Numero2
DOI - pysyväislinkit
TilaJulkaistu - 1 lokakuuta 2016
OKM-julkaisutyyppiA1 Julkaistu artikkeli, soviteltu

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