Skip to main content
SpringerLink
Log in

Lipschitz Regularity for Elliptic Equations with Random Coefficients

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L ∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (for example finite range of dependence). We also prove a quenched L 2 estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65

  2. Armstrong, S.N., Shen, Z.: Lipschitz Estimates in Almost-Periodic Homogenization. Preprint, arXiv:1409.2094

  3. Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. arXiv:1406.0996

  4. Avellaneda M., Lin F.-H.:: Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40(6), 803–847 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Avellaneda M., Lin F.-H.:: L p bounds on singular integrals in homogenization. Comm. Pure Appl. Math. 44(8–9), 897–910 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bauschke H.H., Wang X.:: The kernel average for two convex functions and its application to the extension and representation of monotone operators. Trans. Amer. Math. Soc. 361(11), 5947–5965 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bella, P., Otto, F.: Corrector estimates for elliptic systems with random periodic coefficients. Preprint, arXiv:1409.5271

  8. Ben-Artzi, J., Marahrens, D., Neukamm, S.: Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations. Preprint, arXiv:1407.6984

  9. Brézis, H., Ekeland, I.: Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282(17), A971–A974 (1976)

  10. Conlon J.G., Naddaf A.:: On homogenization of elliptic equations with random coefficients. Electron. J. Probab. 5(9), 58 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. 144(4), 347–389 (1986)

  12. Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)

  13. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French, Studies in Mathematics and its Applications, Vol. 1.

  14. Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), vol. 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., pp. 59–65. Austral. Nat. Univ., Canberra (1988)

  15. Geng J., Shen Z.:: The Neumann problem and Helmholtz decomposition in convex domains. J. Funct. Anal. 259(8), 2147–2164 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer Monographs in Mathematics. Springer, New York (2009)

  17. Ghoussoub, N., Moameni, A., Sáiz, R.Z.: Homogenization of maximal monotone vector fields via selfdual variational calculus. Adv. Nonlinear Stud. 11(2), 323–360 (2011)

  18. Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  19. Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311/312, 145–169 (1979)

  20. Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge (2003)

  21. Gloria, A., Marahrens, D.: Annealed estimates on the Green functions and uncertainty quantification. Preprint, arXiv:1409.0569

  22. Gloria, A., Neukamm, S., Otto, F.: An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal. 48(2), 325–346 (2014)

  23. Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math. 199(2), 455–515 (2015)

  24. Gloria, A., Neukamm, S., Otto, F.: A regularity theory for random elliptic operators. Preprint, arXiv:1409.2678

  25. Gloria A., Otto F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39(3), 779–856 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gloria A., Otto F.:: An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22(1), 1–28 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. Preprint, arXiv:1409.0801

  28. Kenig, C., sc Lin, F., Shen, Z.: Convergence rates in L 2 for elliptic homogenization problems. Arch. Ration. Mech. Anal. 203(3), 1009–1036 (2012)

  29. Kenig, C., Lin, F., Shen, Z.: Homogenization of elliptic systems with Neumann boundary conditions. J. Am. Math. Soc., 26, 901–937 (2013)

  30. Lamacz, A., Neukamm, S., Otto, F.: Moment bounds for the corrector in stochastic homogenization of a percolation model. Preprint arXiv:1406.5723

  31. Marahrens, D., Otto, F.: Annealed estimates on the Green’s function. Preprint arXiv:1304.4408

  32. Mourrat J.-C.: Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47(1), 294–327 (2011)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. Naddaf, A., Spencer, T.: Estimates on the variance of some homogenization problems (1998) (Unpublished preprint).

  34. Nayroles, B.: Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282(17), A1035–A1038 (1976)

  35. Nolen, J.: Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields (2014) (to appear)

  36. Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8), 855–871 (2004)

  37. Rockafellar, R.T., Wets, R. J.-B. : Variational analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1998)

  38. Visintin A.:: Scale-transformations and homogenization of maximal monotone relations with applications. Asymptot. Anal. 82(3–4), 233–270 (2013)

    MATH  MathSciNet  Google Scholar 

  39. Visintin A.:: Variational formulation and structural stability of monotone equations. Calc. Var. Partial Differ. Equ. 47(1–2), 273–317 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ceremade (UMR CNRS 7534), Université Paris-Dauphine, Paris, France

    Scott N. Armstrong

  2. Ecole normale supérieure de Lyon, CNRS, Lyon, France

    Jean-Christophe Mourrat

Authors
  1. Scott N. Armstrong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Christophe Mourrat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott N. Armstrong.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Armstrong, S.N., Mourrat, JC. Lipschitz Regularity for Elliptic Equations with Random Coefficients. Arch Rational Mech Anal 219, 255–348 (2016). https://doi.org/10.1007/s00205-015-0908-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0908-4

Keywords

Search

Navigation