# American Institute of Mathematical Sciences

March  2018, 13(1): 155-176. doi: 10.3934/nhm.2018007

## Green's function for elliptic systems: Moment bounds

 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  December 2016 Revised  April 2017 Published  March 2018

Fund Project: The first author was supported by the German Science Foundation DFG in the context of the Emmy Noether junior research group BE 5922/1-1.

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.

Citation: Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
##### References:
 [1] S. Armstrong, T. Kuusi and J.-C. Mourrat, Mesoscopic higher regularity and subadditivity in elliptic homogenization, Comm. Math. Phys., 347 (2016), 315-361.  doi: 10.1007/s00220-016-2663-2.  Google Scholar [2] _______, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.  Google Scholar [3] S. N. Armstrong and J.-C. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., 219 (2016), 255-348.  doi: 10.1007/s00205-015-0908-4.  Google Scholar [4] S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 423-481.  doi: 10.24033/asens.2287.  Google Scholar [5] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar [6] P. Bella, B. Fehrman and F. Otto, A Liouville theorem for elliptic systems with degenerate ergodic coefficients, To appear in Annals of App. Probabiliy, arXiv e-prints (2016). Google Scholar [7] P. Bella, A. Giunti and F. Otto, Effective multipoles in random media, arXiv e-prints (2017). Google Scholar [8] P. Bella, A. Giunti and F. Otto, Quantitative stochastic homogenization: Local control of homogenization error through corrector, Mathematics and Materials, IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI, 23 (2017), 301-327.  Google Scholar [9] P. Bella and F. Otto, Corrector estimates for elliptic systems with random periodic coefficients, Multiscale Model. Simul., 14 (2016), 1434-1462.  doi: 10.1137/15M1037147.  Google Scholar [10] J. G. Conlon, A. Giunti and F. Otto, Green's function for elliptic systems: Existence and Delmotte-Deuschel bounds, Calc. Var. Partial Differential Equations, 56 (2017), Art. 163, 51 pp. doi: 10.1007/s00526-017-1255-0.  Google Scholar [11] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (4), 1 (1968), 135-137.   Google Scholar [12] T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nablaφ$ interface model, Probab. Theory Related Fields, 133 (2005), 358-390.  doi: 10.1007/s00440-005-0430-y.  Google Scholar [13] J. Fischer and F. Otto, A higher-order large-scale regularity theory for random elliptic operators, Comm. Partial Differential Equations, 41 (2016), 1108-1148.  doi: 10.1080/03605302.2016.1179318.  Google Scholar [14] _______, Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations, Stoch. Partial Differ. Equ. Anal. Comput., 5(2017), 220-255. doi: 10.1007/s40072-016-0086-x.  Google Scholar [15] A. Gloria and D. Marahrens, Annealed estimates on the green functions and uncertainty quantification, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1153-1197.  doi: 10.1016/j.anihpc.2015.04.001.  Google Scholar [16] A. Gloria, S. Neukamm and F. Otto, A regularity theory for random elliptic operators, arXiv e-prints (2014). Google Scholar [17] _______, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515. doi: 10.1007/s00222-014-0518-z.  Google Scholar [18] A. Gloria and F. Otto, The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, arXiv e-prints (2015). Google Scholar [19] _______, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (JEMS), 19 (2017), 3489-3548. doi: 10.4171/JEMS/745.  Google Scholar [20] S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188-202,327.   Google Scholar [21] D. Marahrens and F. Otto, Annealed estimates on the Green function, Probab. Theory Related Fields, 163 (2015), 527-573.  doi: 10.1007/s00440-014-0598-0.  Google Scholar [22] D. Marahrens and F. Otto, On annealed elliptic Green's function estimates, Math. Bohem., 140 (2015), 489-506.   Google Scholar [23] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam-New York, 1981,835-873.  Google Scholar

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##### References:
 [1] S. Armstrong, T. Kuusi and J.-C. Mourrat, Mesoscopic higher regularity and subadditivity in elliptic homogenization, Comm. Math. Phys., 347 (2016), 315-361.  doi: 10.1007/s00220-016-2663-2.  Google Scholar [2] _______, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.  Google Scholar [3] S. N. Armstrong and J.-C. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., 219 (2016), 255-348.  doi: 10.1007/s00205-015-0908-4.  Google Scholar [4] S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 423-481.  doi: 10.24033/asens.2287.  Google Scholar [5] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar [6] P. Bella, B. Fehrman and F. Otto, A Liouville theorem for elliptic systems with degenerate ergodic coefficients, To appear in Annals of App. Probabiliy, arXiv e-prints (2016). Google Scholar [7] P. Bella, A. Giunti and F. Otto, Effective multipoles in random media, arXiv e-prints (2017). Google Scholar [8] P. Bella, A. Giunti and F. Otto, Quantitative stochastic homogenization: Local control of homogenization error through corrector, Mathematics and Materials, IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI, 23 (2017), 301-327.  Google Scholar [9] P. Bella and F. Otto, Corrector estimates for elliptic systems with random periodic coefficients, Multiscale Model. Simul., 14 (2016), 1434-1462.  doi: 10.1137/15M1037147.  Google Scholar [10] J. G. Conlon, A. Giunti and F. Otto, Green's function for elliptic systems: Existence and Delmotte-Deuschel bounds, Calc. Var. Partial Differential Equations, 56 (2017), Art. 163, 51 pp. doi: 10.1007/s00526-017-1255-0.  Google Scholar [11] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (4), 1 (1968), 135-137.   Google Scholar [12] T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nablaφ$ interface model, Probab. Theory Related Fields, 133 (2005), 358-390.  doi: 10.1007/s00440-005-0430-y.  Google Scholar [13] J. Fischer and F. Otto, A higher-order large-scale regularity theory for random elliptic operators, Comm. Partial Differential Equations, 41 (2016), 1108-1148.  doi: 10.1080/03605302.2016.1179318.  Google Scholar [14] _______, Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations, Stoch. Partial Differ. Equ. Anal. Comput., 5(2017), 220-255. doi: 10.1007/s40072-016-0086-x.  Google Scholar [15] A. Gloria and D. Marahrens, Annealed estimates on the green functions and uncertainty quantification, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1153-1197.  doi: 10.1016/j.anihpc.2015.04.001.  Google Scholar [16] A. Gloria, S. Neukamm and F. Otto, A regularity theory for random elliptic operators, arXiv e-prints (2014). Google Scholar [17] _______, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515. doi: 10.1007/s00222-014-0518-z.  Google Scholar [18] A. Gloria and F. Otto, The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, arXiv e-prints (2015). Google Scholar [19] _______, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (JEMS), 19 (2017), 3489-3548. doi: 10.4171/JEMS/745.  Google Scholar [20] S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188-202,327.   Google Scholar [21] D. Marahrens and F. Otto, Annealed estimates on the Green function, Probab. Theory Related Fields, 163 (2015), 527-573.  doi: 10.1007/s00440-014-0598-0.  Google Scholar [22] D. Marahrens and F. Otto, On annealed elliptic Green's function estimates, Math. Bohem., 140 (2015), 489-506.   Google Scholar [23] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam-New York, 1981,835-873.  Google Scholar
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