Advertisement

Mathematical Notes

, Volume 77, Issue 3–4, pp 461–470 | Cite as

Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk

  • V. P. Burskii
  • E. A. Buryachenko
Article

Abstract

In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order 2m with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases m=1, 2, 3 are studied separately. For the case m=2, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions.

Key words

homogeneous Dirichlet problem linear equation of even order with constant complex coefficients elliptic system polynomial solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    A. V. Bitsadze, “On the uniqueness of the solution of the Dirichlet problem for partial differential equations of elliptic type,” Uspekhi Mat. Nauk [Russian Math. Surveys], 3 (1948), no. 6, 211–212.Google Scholar
  2. 2.
    A. V. Bitsadze, Equations of Mixed Type [in Russian], Fizmatgiz, Moscow, 1959.Google Scholar
  3. 3.
    J.-L. Lions and E. Magenes, Problèmes aux limites nonhomogènes et applications, vol. 1, Dunod, Paris, 1968; Russian translation: Mir, Moscow, 1971.Google Scholar
  4. 4.
    A. I. Markovskii, “On the range of the elliptic polynomial,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 25 (1973), no. 2, 228–234.Google Scholar
  5. 5.
    M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb. [Math. USSR-Sb.], 29 (1951), no. 3, 615–676.Google Scholar
  6. 6.
    R. A. Aleksandryan, “On the Dirichlet problem for the string equation and on the completeness of a system of functions in the disk,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 73 (1950), no. 5.Google Scholar
  7. 7.
    R. A. Aleksandryan, “Spectral properties of operators generated by systems of S. L. Sobolev-type differential equations,” Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.], 9 (1960), 455–507.Google Scholar
  8. 8.
    V. P. Burskii, “Violation of the uniqueness of the solution to the Dirichlet problem for elliptic systems in the disk,” Mat. Zametki [Math. Notes], 48 (1990), no. 3, 32–36.Google Scholar
  9. 9.
    A. O. Babayan, “On unique solvability of Dirichlet problem for fourth-order properly elliptic equation, ” Izv. Nat. AN Armenii. Matem., 34 (1999), no. 5, 5–18.Google Scholar
  10. 10.
    E. A. Buryachenko, “Uniqueness of the solutions of the Dirichlet problem in the disk for fourth-order differential equations in degenerate cases,” Nonlinear Boundary-Value Problems, 10 (2000), 44–49.Google Scholar
  11. 11.
    V. P. Burskii, “Uniqueness of the solution of some boundary-value problems for differential equations in a domain with algebraic boundary,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 45 (1993), no. 7, 898–906.Google Scholar
  12. 12.
    I. N. Vekua, New Methods for Solving Elliptic Equations [in Russian], OGIZ, Moscow, 1948.Google Scholar
  13. 13.
    A. V. Bitsadze, Several Classes of Partial Differential Equations [in Russian], Mir, Moscow, 1981.Google Scholar
  14. 14.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York-Toronto-London, 1953; Russian translation: Nauka, Moscow, 1974.Google Scholar
  15. 15.
    E. A. Buryachenko, “To the problem of violation of the uniqueness of the solution of the Dirichlet problem for fourth-order partial differential equations,” Trudy Inst. Prikl. Mat. Mekh., Ukrainian National Academy of Sciences, 4 (1999), 15–21.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. P. Burskii
    • 1
  • E. A. Buryachenko
    • 1
  1. 1.Institute for Applied Mathematics and MechanicsUkrainian National Academy of SciencesUkraine

Personalised recommendations