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 and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
References:
[1]

S. ArmstrongT. 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.

[2]

_______, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.

[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.

[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.

[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.

[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).

[7]

P. Bella, A. Giunti and F. Otto, Effective multipoles in random media, arXiv e-prints (2017).

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[16]

A. Gloria, S. Neukamm and F. Otto, A regularity theory for random elliptic operators, arXiv e-prints (2014).

[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.

[18]

A. Gloria and F. Otto, The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, arXiv e-prints (2015).

[19]

_______, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (JEMS), 19 (2017), 3489-3548. doi: 10.4171/JEMS/745.

[20]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188-202,327. 

[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.

[22]

D. Marahrens and F. Otto, On annealed elliptic Green's function estimates, Math. Bohem., 140 (2015), 489-506. 

[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.

show all references

References:
[1]

S. ArmstrongT. 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.

[2]

_______, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.

[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.

[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.

[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.

[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).

[7]

P. Bella, A. Giunti and F. Otto, Effective multipoles in random media, arXiv e-prints (2017).

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[16]

A. Gloria, S. Neukamm and F. Otto, A regularity theory for random elliptic operators, arXiv e-prints (2014).

[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.

[18]

A. Gloria and F. Otto, The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations, arXiv e-prints (2015).

[19]

_______, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (JEMS), 19 (2017), 3489-3548. doi: 10.4171/JEMS/745.

[20]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188-202,327. 

[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.

[22]

D. Marahrens and F. Otto, On annealed elliptic Green's function estimates, Math. Bohem., 140 (2015), 489-506. 

[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.

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