Skip to main content
SpringerLink
Log in

L p -estimates for solutions of second-order elliptic equation Dirichlet problem

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

For solutions of the Dirichlet problem for a second-order elliptic equation, we establish an analogue of the Carleson theorem on Lp-estimates. Under the same conditions on the coefficients for which the unique solvability of the considered problem is known, we prove this criterion for the validity of estimate of the solution norm in the space Lp with a measure. We require their Dini continuity on the boundary, but we assume only their measurability and boundedness in the domain under consideration.

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. L. Carleson, Amer. J. Math., 80, 921–930 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Carleson, Ann. Math., 76, 547–559 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Hörmander,, Math. Scand., 20, 65–78 (1967).

    MathSciNet  MATH  Google Scholar 

  4. N. K. Nikol’skii, Lectures on the Shift Operator [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  5. J. B. Garnett, Bounded Analytic Functions (Grad. Texts Math., Vol. 236), Springer, New York (2007).

    Google Scholar 

  6. A. K. Gushchin, Math. USSR-Sb., 65, 19–66 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1988); English transl. prev. ed. (Pure Appl. Math., Vol. 3, A. Jeffrey, ed.), Marcel Dekker, New York (1971).

    MATH  Google Scholar 

  8. V. P. Mikhailov, Differential Equations, 12, 1877–1891, 1919. (1976).

    Google Scholar 

  9. O. I. Bogoyavlenskii, V. S. Vladimirov, I. V. Volovich, A. K. Gushchin, Yu. N. Drozhzhinov, V. V. Zharinov, and V. P. Mikhailov, Proc. Steklov Inst. Math., 175, 65–105 (1988).

    Google Scholar 

  10. I. M. Petrushko, Math. USSR-Sb., 48, 565–585 (1984).

    Article  MATH  Google Scholar 

  11. V. Zh. Dumanyan, Sb. Math., 202, 1001–1020 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. K. Gushchin and V. P. Mikhailov, Russian Acad. Sci. Sb. Math., 81, 101–136 (1995).

    MathSciNet  Google Scholar 

  13. A. K. Gushchin and V. P. Mikhailov, Sb. Math., 186, 197–219 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. K. Gushchin and V. P. Mikhailov, Dokl. Math., 54, 815–816 (1996).

    MATH  Google Scholar 

  15. A. K. Gushchin, Dokl. Math., 62, 32–34 (2000).

    Google Scholar 

  16. A. K. Gushchin, Sb. Math., 193, 649–668 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  17. E. De Giorgi, Mem. Accad. Sci. Torino, 3, 25–43 (1957).

    Google Scholar 

  18. J. Nash, Amer. J. Math., 80, 931–954 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Moser, Commun. Pure Appl. Math., 13, 457–468 (1960).

    Article  MATH  Google Scholar 

  20. A. K. Gushchin, Sb. Math., 189, 1009–1045 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. K. Gushchin, Siberian Math. J., 46, 826–840 (2005).

    Article  MathSciNet  Google Scholar 

  22. A. K. Gushchin, Theor. Math. Phys., 157, 1655–1670 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. K. Gushchin and V. P. Mikhailov, Vestnik SamGTU, Natural Science Series, 8, No. 1(67), 76–94 (2008).

    Google Scholar 

  24. A. K. Gushchin and V. P. Mikhailov, Math. USSR-Sb., 36, 1–19 (1980).

    Article  MATH  Google Scholar 

  25. Yu. A. Mikhailov, Differential Equations, 19, 318–337, 367 (1983).

    Google Scholar 

  26. A. K. Gushchin, Sb. Math., 203, 1–27 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu. A. Alkhutov and V. A. Kondrat’ev, Differential Equations, 28, 650–662 (1992).

    MathSciNet  Google Scholar 

  28. Yu. A. Alkhutov, Sb. Math., 189, 1–17 (1998).

    Article  MathSciNet  Google Scholar 

  29. O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973); English transl. prev. ed. (Math in Sci. and Engin., Vol. 46), Acad. Press, New York (1968).

    MATH  Google Scholar 

  30. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order [in Russian], Nauka, Moscow (1989); English transl. prev. ed. (Grundlehren Math. Wiss., Vol. 224), Springer, Berlin (1983).

    MATH  Google Scholar 

  31. V. P. Mikhailov and A. K. Gushchin, Additional Chapters of the Course “Equations of Mathematical Physics” (Lekts. Kursy NOTs, Vol. 7) [in Russian], Steklov Math. Inst., RAS, Moscow (2007).

    Google Scholar 

  32. V. A. Kondrat’ev, J. Kopáček, and O. A. Oleinik, Math. USSR-Sb., 59, No. 1, 113–127 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. V. A. Kondratiev and E. M. Landis, “Qualitative theory of second-order linear partial differential equations,” in: Partial Differential Equations 3 [in Russian] (Sovrem. Probl. Mat. Fund. Naprav., Vol 32), VINITI, Moscow (1988), pp. 99–215.

    Google Scholar 

  34. A. K. Gushchin, Dokl. Math., 446, 487–489 (2012).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, RAS, Moscow, Russia

    A. K. Guschin

Authors
  1. A. K. Guschin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Guschin.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 174, No. 2, pp. 243–255, February, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guschin, A.K. L p -estimates for solutions of second-order elliptic equation Dirichlet problem. Theor Math Phys 174, 209–219 (2013). https://doi.org/10.1007/s11232-013-0018-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-013-0018-0

Keywords

Search

Navigation