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Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

Ben-Artzi, Jonathan, Marahrens, Daniel and Neukamm, Stefan 2017. Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations. Communications in Partial Differential Equation 42 (2) , pp. 179-234.

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We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d≥2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green’s function in weighted spaces. In the critical case d = 2, our argument for the estimate on the gradient of the elliptic Green’s function uses a Calderón–Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coefficients.

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: Corrector equation, stochastic homogenization, two-scale expansion, variance estimate
Publisher: Taylor & Francis
Date of Acceptance: 6 January 2017
Last Modified: 07 Jun 2017 14:32

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