Strong convergence to the homogenized limit of elliptic equations with random coefficients

Conlon, Joseph; Spencer, Thomas

doi:10.1090/S0002-9947-2013-05762-5

Strong convergence to the homogenized limit of elliptic equations with random coefficients
HTML articles powered by AMS MathViewer

by Joseph G. Conlon and Thomas Spencer PDF

Trans. Amer. Math. Soc. 366 (2014), 1257-1288 Request permission

Abstract:

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\mathbf {Z}^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green’s function, together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It has also been shown that if the random environment is ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized elliptic PDE on $\mathbf {R}^d$. In this paper point-wise estimates are obtained on the difference between the averaged Green’s function and the homogenized Green’s function for certain random environments which are strongly mixing.

References

Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
Leo Breiman, Probability, Classics in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. Corrected reprint of the 1968 original. MR 1163370, DOI 10.1137/1.9781611971286
Luis A. Caffarelli and Panagiotis E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 (2010), no. 2, 301–360. MR 2609244, DOI 10.1007/s00222-009-0230-6
Joseph G. Conlon, PDE with random coefficients and Euclidean field theory, J. Statist. Phys. 116 (2004), no. 1-4, 933–958. MR 2082201, DOI 10.1023/B:JOSS.0000037204.93858.f2
Joseph G. Conlon and Ali Naddaf, On homogenization of elliptic equations with random coefficients, Electron. J. Probab. 5 (2000), no. 9, 58. MR 1768843, DOI 10.1214/EJP.v5-65
Joseph G. Conlon and Ali Naddaf, Green’s functions for elliptic and parabolic equations with random coefficients, New York J. Math. 6 (2000), 153–225. MR 1781430
T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla \phi$ interface model, Probab. Theory Related Fields 133 (2005), no. 3, 358–390. MR 2198017, DOI 10.1007/s00440-005-0430-y
Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau $\nabla \phi$ interface model, Comm. Math. Phys. 185 (1997), no. 1, 1–36. MR 1463032, DOI 10.1007/s002200050080
Antoine Gloria and Felix Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab. 39 (2011), no. 3, 779–856. MR 2789576, DOI 10.1214/10-AOP571
Antoine Gloria and Felix Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab. 22 (2012), no. 1, 1–28. MR 2932541, DOI 10.1214/10-AAP745
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Bernard Helffer and Johannes Sjöstrand, On the correlation for Kac-like models in the convex case, J. Statist. Phys. 74 (1994), no. 1-2, 349–409. MR 1257821, DOI 10.1007/BF02186817
S. M. Kozlov, Averaging of random structures, Dokl. Akad. Nauk SSSR 241 (1978), no. 5, 1016–1019 (Russian). MR 510894
Norman G. Meyers, An $L^{p}$e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR 159110
Per-Gunnar Martinsson and Gregory J. Rodin, Asymptotic expansions of lattice Green’s functions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2027, 2609–2622. MR 1942800, DOI 10.1098/rspa.2002.0985
Ali Naddaf and Thomas Spencer, On homogenization and scaling limit of some gradient perturbations of a massless free field, Comm. Math. Phys. 183 (1997), no. 1, 55–84. MR 1461951, DOI 10.1007/BF02509796
A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems, 1998 preprint.
Hirofumi Osada and Herbert Spohn, Gibbs measures relative to Brownian motion, Ann. Probab. 27 (1999), no. 3, 1183–1207. MR 1733145, DOI 10.1214/aop/1022677444
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, pp. 835–873. MR 712714
William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge, 2004. Reprint of the 1981 original. MR 2140546
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh. 27 (1986), no. 4, 167–180, 215 (Russian). MR 867870
V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546, DOI 10.1007/978-3-642-84659-5