Journal of Elliptic and Parabolic Equations

, Volume 5, Issue 2, pp 473–491 | Cite as

On correctors to elliptic problems in long cylinders

  • Adrien Ceccaldi
  • Sorin MardareEmail author


The last years have seen the development of the asymptotic study of partial differential equations in cylinders becoming unbounded in one or several directions, particularly under the impetus of Michel Chipot and his collaborators. In this paper, we aim to improve some results that have already been shown about the convergence to the solution of a linear elliptic problem on an infinite cylinder of the solutions of the same problem taken on larger and larger truncations of the cylinder. This aim will be realized by the construction of well-adjusted correctors. Thanks to our main results established in Sect. 2 of this paper, we conclude by an application in a particular case (by taking data that does not depend on the coordinate along the cylinder’s axis) that the convergence results that can be obtained using the methods introduced by Chipot and Yeressian (CR Acad Sci Paris Ser I 346:21–26, 2008) are optimal. The particularity here is that the optimality is taken in the sense of “the largest domain where the convergence takes place” instead of the classical optimality of the speed of convergence itself.


Elliptic equations Dirichlet problems Asymptotic analysis Cylinders 

Mathematics Subject Classification

35B40 35J20 35J25 


Compliance with ethical standards

Conflict of interest

Author Adrien Ceccaldi declares that he has no conflict of interest. Author Sorin Mardare declares that he has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Ceccaldi, A.: Elliptic problems in long cylinders revisited. Ricerche Mat 68, 1–16 (2018)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chipot, M.: \(\ell \) Goes to Plus Infinity. Birkäuser Verlag, Basel (2002)zbMATHGoogle Scholar
  3. 3.
    Chipot, M.: Asymptotic Issues for Some Partial Differential Equations. Imperial College Press, London (2016)CrossRefGoogle Scholar
  4. 4.
    Chipot, M.: On some elliptic problems in unbounded domains. Chin. Ann. Math. Ser. B 39(3), 597–606 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chipot, M., Guesmia, S.: Correctors for some asymptotic problems. Proc. Steklov Inst. Math. 270, 263–277 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chipot, M., Mardare, S.: On correctors for the Stokes problem in cylinders. In: Proceedings of the Conference on Nonlinear Phenomena with Energy Dissipation, Chiba, November 2007, Gakuto International Series, Mathematical Sciences and Applications, Vol. 29, Gakkotosho, pp. 37–52 (2008)Google Scholar
  7. 7.
    Chipot, M., Rougirel, A.: On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete Contin. Dyn. Syst. Ser. B 1(3), 319–338 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chipot, M., Roy, P., Shafrir, I.: Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity. Asympt. Anal. 85, 199–227 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chipot, M., Yeressian, K.: Exponential rates of convergence by an iteration technique. CR. Acad. Sci. Paris Ser. I 346, 21–26 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)CrossRefGoogle Scholar
  11. 11.
    Guesmia, S.: Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size. Nonlinear Anal. 70(9), 3320–3331 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    John, F.: Planes Waves and Spherical Means. Springer, New York (1981)CrossRefGoogle Scholar
  13. 13.
    Miranda, C.: Partial differential equations of elliptic type. Springer, Berlin (1970)CrossRefGoogle Scholar
  14. 14.
    Xie, Y.: Some convergence results for elliptic problems with periodic data, recent advances on elliptic and parabolic issues. In: Proceedings of the 2004 Swiss-Japanese Seminar, World Scientific, pp 265–282 (2006)Google Scholar

Copyright information

© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Rouen-Normandie, Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085Saint-Étienne-du-RouvrayFrance

Personalised recommendations