Journal
NETWORKS AND HETEROGENEOUS MEDIA
Volume 13, Issue 1, Pages 155-176Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/nhm.2018007
Keywords
Stochastic homogenization; large-scale regularity; Green's function estimates
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Funding
- German Science Foundation DFG [BE 5922/1-1]
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We study estimates of the Green's function in R-d with d >= 2, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case d >= 3, we obtain estimates on the Green's function, its gradient, and the second mixed derivatives which scale optimally in space, in terms of the minimal radius r(*) introduced in [Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators; ArXiv e-prints (2014)]. As an application, our result implies optimal stochastic Gaussian bounds on the Green's function and its derivatives in the realm of homogenization of equations with random coefficient fields with finite range of dependence. In two dimensions, where in general the Green's function does not exist, we construct its gradient and show the corresponding estimates on the gradient and mixed second derivatives. Since we do not use any scalar methods in the argument, the result holds in the case of uniformly elliptic systems as well.
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