Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

Scott Armstrong, Jessica Lin

Research output: Contribution to journalArticle

Abstract

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto (Ann Probab 39(3):779–856, 2011; Ann Appl Probab 22(1):1–28, 2012) for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green’s functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a C1,1 regularity theory down to microscopic scale, which is of independent interest and is inspired by the C0,1 theory introduced in the divergence form case by the first author and Smart (Ann Sci Éc Norm Supér (4) 49(2):423–481, 2016).

Original languageEnglish (US)
Pages (from-to)1-55
Number of pages55
JournalArchive for Rational Mechanics and Analysis
DOIs
StateAccepted/In press - Apr 17 2017

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Stochastic Homogenization
Green's function
Elliptic Equations
Derivatives
Corrector
Estimate
Divergence
Concentration Inequalities
Regularity Theory
Exponential Type
Norm
Derivative
Form
Coefficient

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form. / Armstrong, Scott; Lin, Jessica.

In: Archive for Rational Mechanics and Analysis, 17.04.2017, p. 1-55.

Research output: Contribution to journalArticle

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