Green's function for elliptic systems: Moment bounds

Peter Bella; Arianna Giunti; Peter Bella; Arianna Giunti

doi:10.3934/nhm.2018007

Networks and Heterogeneous Media

2018, Volume 13, Issue 1: 155-176. doi: 10.3934/nhm.2018007

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Green's function for elliptic systems: Moment bounds

Peter Bella ^{1
,},
Arianna Giunti ^{2
,}

1.
Institute of Mathematics, Leipzig University, Augustusplatz 10, 04109 Leipzig, Germany
2.
Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received: 01 December 2016 Revised: 01 April 2017
Primary: 35J08, 35J15; Secondary: 74Q05

We study estimates of the Green's function in $\mathbb{R}^d$ 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.
- Stochastic homogenization,
- large-scale regularity,
- Green's function estimates
Citation: Peter Bella, Arianna Giunti. 2018: Green's function for elliptic systems: Moment bounds, Networks and Heterogeneous Media, 13(1): 155-176. doi: 10.3934/nhm.2018007

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Abstract

We study estimates of the Green's function in $\mathbb{R}^d$ 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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