### 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 C^{1,1} regularity theory down to microscopic scale, which is of independent interest and is inspired by the C^{0,1} 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 language | English (US) |
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Pages (from-to) | 1-55 |

Number of pages | 55 |

Journal | Archive for Rational Mechanics and Analysis |

DOIs | |

State | Accepted/In press - Apr 17 2017 |

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### ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

*Archive for Rational Mechanics and Analysis*, 1-55. https://doi.org/10.1007/s00205-017-1118-z

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

AU - Armstrong, Scott

AU - Lin, Jessica

PY - 2017/4/17

Y1 - 2017/4/17

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85017502450&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017502450&partnerID=8YFLogxK

U2 - 10.1007/s00205-017-1118-z

DO - 10.1007/s00205-017-1118-z

M3 - Article

SP - 1

EP - 55

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

ER -