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- ItemMoment bounds for the corrector in stochastic homogenization of a percolation model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Lamacz, Agnes; Neukamm, Stefan; Otto, FelixWe study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on Zd, d > 2. The model is obtained from the classical {0,1}-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result in [8], where uniformly elliptic conductances are treated, to the degenerate case. Our argument is based on estimates on the gradient of the elliptic Green's function.
- ItemMoment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Ben-Artzi, Jonathan; Marahrens, Daniel; Neukamm, StefanWe 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.