Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

Scott Armstrong, Tuomo Kuusi, Jean Christophe Mourrat

Research output: Contribution to journalArticle

Abstract

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 (US)
Pages (from-to)315-361
Number of pages47
JournalCommunications in Mathematical Physics
Volume347
Issue number2
DOIs
StatePublished - Oct 1 2016

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Subadditivity
homogenizing
regularity
Homogenization
Regularity Theory
Regularity
Stochastic Homogenization
Energy
Concentration Inequalities
Logarithmic Sobolev Inequality
Corrector
Probability Space
Weak Convergence
Length Scale
Bootstrap
Elliptic Equations
Accelerate
energy
Linear equation
Divergence

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization. / Armstrong, Scott; Kuusi, Tuomo; Mourrat, Jean Christophe.

In: Communications in Mathematical Physics, Vol. 347, No. 2, 01.10.2016, p. 315-361.

Research output: Contribution to journalArticle

Armstrong, Scott ; Kuusi, Tuomo ; Mourrat, Jean Christophe. / Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization. In: Communications in Mathematical Physics. 2016 ; Vol. 347, No. 2. pp. 315-361.
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