Skip to main content
SpringerLink
Log in

On the dirichlet problem for an improperly elliptic equation

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider the problem of solvability of an inhomogeneous Dirichlet problem for a scalar improperly elliptic differential equation with complex coefficients in a bounded domain. A model case where the unit disk is chosen as the domain and the equation does not contain lower terms is studied. We prove that the classes of Dirichlet data for which the problem has a unique solution in the Sobolev space are spaces of functions with exponentially decreasing Fourier coefficients.

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. J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris (1968).

    MATH  Google Scholar 

  2. Ya. B. Lopatinskii, “On one method for the reduction of boundary-value problems for systems of differential equations of elliptic type to regular integral equations,” Ukr. Mat. Zh., 5, No. 2, 123–151 (1953).

    MathSciNet  Google Scholar 

  3. A. V. Bitsadze, “On the uniqueness of a solution of the Dirichlet problem for elliptic equations with partial derivatives,” Usp. Mat. Nauk, 3, No. 6, 211–212 (1948).

    MATH  Google Scholar 

  4. A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  5. V. P. Burskii, “On the violation of the uniqueness of a solution of the Dirichlet problem for elliptic systems in a disk,” Mat. Zametki, 48, No. 3, 32–36 (1990).

    MathSciNet  Google Scholar 

  6. V. P. Burskii, “Solutions of the Dirichlet problem for elliptic systems in a disk,” Ukr. Mat. Zh., 44, No. 10, 1307–1313 (1992); English translation : Ukr. Math. J., 44, No. 10, 1197–1203 (1992).

    Article  MathSciNet  Google Scholar 

  7. V. P. Burskii, Methods for Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).

    Google Scholar 

  8. V. P. Burskii, “Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem,” Ukr. Mat. Zh., 45, No. 11, 1476–1483 (1993); English translation : Ukr. Math. J., 45, No. 11, 1659–1668 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Mandelbrojt, Quasianalytic Classes of Functions [Russian translation], ONTI NKTR SSSR, Leningrad (1937).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine

    V. P. Burskii & E. V. Kirichenko

Authors
  1. V. P. Burskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. V. Kirichenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 2, pp. 156–164, February, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burskii, V.P., Kirichenko, E.V. On the dirichlet problem for an improperly elliptic equation. Ukr Math J 63, 187–195 (2011). https://doi.org/10.1007/s11253-011-0497-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0497-9

Keywords

Search

Navigation