Inventiones mathematicae

, Volume 180, Issue 2, pp 301–360 | Cite as

Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media



We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation.


Viscosity Solution Harnack Inequality Obstacle Problem Stochastic Homogenization Periodic Environment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cabre, X., Caffarelli, L.A.: Fully Nonlinear Elliptic Equations. Colloquium Publications, vol. 43. American Math. Soc., Providence (1995) MATHGoogle Scholar
  2. 2.
    Caffarelli, L.A.: A note on nonlinear homogenization. Commun. Pure Appl. Math. 52(7), 829–838 (1999) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Caffarelli, L.A., Kinderlehrer, D.: Potential methods in variational inequalities. J. Anal. Math. 37, 285–295 (1980) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Caffarelli, L.A., Souganidis, P.E.: A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pde. Commun. Pure Appl. Math. 61(1), 1–17 (2008) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Caffarelli, L.A., Souganidis, P.E., Wang, L.: Stochastic homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Commun. Pure Appl. Math. 30(1–3), 335–375 (2005) MathSciNetGoogle Scholar
  6. 6.
    Camilli, F., Marchi, C.: Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs. Nonlinearity 22, 1481–1498 (2009) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. AMS 27, 1–67 (1992) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986) MATHMathSciNetGoogle Scholar
  9. 9.
    Evans, L.C.: The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. Sect. A 111(3–4), 359–375 (1989) MATHGoogle Scholar
  10. 10.
    Evans, L.C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinburgh Sect. A 120(3–4), 245–265 (1992) MATHGoogle Scholar
  11. 11.
    Evans, L.C., Gomes, D.: Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157(1), 1–33 (2001) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fabes, E.B., Stroock, D.: The L p-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51(4), 997–1016 (1984) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ichihara, N.: A stochastic representation for fully nonlinear pdes and its application to homogenization. J. Math. Sci. Univ. Tokyo 12, 467–492 (2005) MATHMathSciNetGoogle Scholar
  14. 14.
    Ishii, H.: Almost periodic homogenization of Hamilton-Jacobi equations. In: International Conference on Differential Equations, vols. 1, 2, Berlin, 1999, pp. 600–605. World Sci. Publ., River Edge (2000) Google Scholar
  15. 15.
    Jensen, R., Lions, P.-L., Souganidis, P.E.: A uniqueness result for viscosity solutions of fully nonlinear second order partial differential equations. Proc. Am. Math. Soc. 4, 975–978 (1988) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994) Google Scholar
  17. 17.
    Kosygina, E., Varadhan, S.R.S.: Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. Commun. Pure Appl. Math. 61(6), 816–847 (2007) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kosygina, E., Rezankhanlou, F., Varadhan, S.R.S.: Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Commun. Pure Appl. Math. 59(10), 1489–1521 (2006) MATHCrossRefGoogle Scholar
  19. 19.
    Kozlov, S.M.: The method of averaging and walk in inhomogeneous environments. Russ. Math. Surv. 40, 73–145 (1985) MATHCrossRefGoogle Scholar
  20. 20.
    Lions, P.-L., Souganidis, P.E.: Correctors for the homogenization of Hamilton-Jacobi equations in a stationary ergodic setting. Commun. Pure Appl. Math. 56(10), 1501–1524 (2003) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lions, P.-L., Souganidis, P.E.: Homogenization for “viscous” Hamilton-Jacobi equations in stationary, ergodic media. Commun. Part. Differ. Equs. 30(1–3), 335–376 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lions, P.-L., Souganidis, P.E.: Homogenization of degenerate second-order pde in periodic and almost periodic environments and applications. Ann. Inst. H. Poincare, Anal. Non Lineaire 22(5), 667–677 (2005) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton-Jacobi equations. Preprint Google Scholar
  24. 24.
    Majda, A., Souganidis, P.E.: Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales. Nonlinearity 7(1), 1–30 (1994) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Papanicolaou, G., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Fritz, J., Lebaritz, J.L., Szasz, D. (eds.) Proceed. Colloq. on Random Fields, Rigorous Results in Statistical Mechanics and Quantum Field Theory. Colloquia Mathematica Soc. Janos Bolyai, vol. 10, pp. 835–873. (1979) Google Scholar
  26. 26.
    Papanicolaou, G., Varadhan, S.R.S.: Diffusion with random coefficients. In: Krishnaiah, P.R.: Essays in Statistics and Probability. North Holland, Amsterdam (1981) Google Scholar
  27. 27.
    Rezankhanlou, F., Tarver, J.: Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 151, 277–309 (2000) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Schwab, R.: Homogenization of Hamilton-Jacobi equations in space-time stationary ergodic medium. Indiana Univ. Math. J. 58(2), 537–582 (2009) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Souganidis, P.E.: Stochastic homogenization for Hamilton-Jacobi equations and applications. Asymptot. Anal. 20, 1–11 (1999) MATHMathSciNetGoogle Scholar
  30. 30.
    Yurinskii, V.V.: On the homogenization of boundary value problems with random coefficients. Sib. Mat. Zh. 21(3), 209–223 (1980). (English transl.: Sib. Math. J. 21, 470–482 (1981)) MathSciNetGoogle Scholar
  31. 31.
    Yurinskii, V.V.: On the homogenization of non-divergent second order equations with random coefficients. Sib. Mat. Zh. 23(2), 176–188 (1982). (English transl.: Sib. Mat. J. 23, 276–287 (1982)) MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Luis A. Caffarelli
    • 1
  • Panagiotis E. Souganidis
    • 2
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsThe University of ChicagoChicagoUSA

Personalised recommendations