Skip to main content
SpringerLink
Log in

Bounded Correctors in Almost Periodic Homogenization

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

Abstract

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Armstrong, S., Kuusi, T., Mourrat, J.-C.: Mesoscopic higher regularity and subadditivity in elliptic homogenization. Comm. Math. Phys. to appear, arXiv:1507.06935.

  2. Armstrong S.N., Cardaliaguet P., Souganidis P.E.: Error estimates and convergence rates for the stochastic homogenization of Hamilton–Jacobi equations. J. Am. Math. Soc. 27(2), 479–540 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Armstrong S.N., Mourrat J.-C.: Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219(1), 255–348 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Armstrong, S.N., Shen, Z.: Lipschitz estimates in almost-periodic homogenization. Comm. Pure Appl. Math. (2016) (to appear). arXiv:1409.2094

  5. Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (2016) (to appear). arXiv:1406.0996

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. AMS Chelsea Publishing, Providence, RI, 2011. Corrected reprint of the 1978 original [MR0503330].

  9. Besicovitch, A.S.: Almost periodic functions. Dover Publications, Inc., New York (1955)

  10. Drmota, M., Tichy, R.F.: Sequences, discrepancies and applications, vol. 1651 of Lecture Notes in Mathematics. Springer, Berlin (1997)

  11. Duerinckx M., Gloria A.: Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas. Arch. Ration. Mech. Anal. 220(1), 297–361 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gérard-Varet D., Masmoudi N.: Homogenization in polygonal domains. J. Eur. Math. Soc. (JEMS) 13(5), 1477–1503 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

  14. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc., (2016) (in press). arXiv:1409.0801

  17. Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of differential operators and integral functionals. Springer, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan].

  18. Kozlov, S.M.: Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.) 107 (149)(2), 199–217, 317, (1978)

  19. Marahrens D., Otto F.: Annealed estimates on the Green function. Probab. Theory Related Fields, 163(3–4), 527–573 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shen Z.: Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems. Anal. PDE, 8(7), 1565–1601 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, 75016, Paris, France

    Scott Armstrong

  2. Département de mathématique, Université Libre de Bruxelles, Belgium and project-team MEPHYSTO, Inria Lille-Nord Europe, Villeneuve d’Ascq, France

    Antoine Gloria

  3. Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

    Tuomo Kuusi

Authors
  1. Scott Armstrong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoine Gloria
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tuomo Kuusi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott Armstrong.

Additional information

Communicated by F. Lin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Armstrong, S., Gloria, A. & Kuusi, T. Bounded Correctors in Almost Periodic Homogenization. Arch Rational Mech Anal 222, 393–426 (2016). https://doi.org/10.1007/s00205-016-1004-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-1004-0

Keywords

Search

Navigation