Advertisement

Ukrainian Mathematical Journal

, Volume 62, Issue 5, pp 676–690 | Cite as

Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles

  • E. A. Buryachenko
Article

We consider the homogeneous Dirichlet problem in the unit disk KR 2 for a general typeless differential equation of any even order 2m, m ≥ 2, with constant complex coefficients whose characteristic equation has multiple roots ± i. For each value of multiplicity of the roots i and – i, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order.

Keywords

Characteristic Equation Dirichlet Problem Slope Angle Nontrivial Solution Trivial Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 solution of the Dirichlet problem for partial elliptic differential equations,” Usp. Mat. Nauk, 3, Issue 6, 211–212 (1948).zbMATHGoogle Scholar
  2. 2.
    V. P. Burskii, Methods for the Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).Google Scholar
  3. 3.
    E. A. Buryachenko, “On the uniqueness of solutions of the Dirichlet problem in a disk for differential equations of the fourth order in degenerate cases,” Nelinein. Granich. Zad., 10, 44–49 (2000).zbMATHGoogle Scholar
  4. 4.
    A. O. Babayan, “On the unique solvability of Dirichlet problem for a fourth-order property elliptic equation,” Izv. Nats. Akad. Nauk Armen., 34, No. 5, 5–18 (1999).MathSciNetGoogle Scholar
  5. 5.
    Ya. B. Lopatinskii, “On one method for the reduction of boundary-value problems for systems of elliptic differential equations to regular integral equations,” Ukr. Mat. Zh., 5, No. 2, 123–151 (1953).MathSciNetGoogle Scholar
  6. 6.
    V. P. Burskii and E. A. Buryachenko, “Some problems of nontrivial solvability of the homogeneous Dirichlet problem for linear equations of any even order in the disk,” Mat. Zametki, 74, No. 4, 1032–1043 (2005).Google Scholar
  7. 7.
    E. A. Buryachenko, “Solvability of the homogeneous Dirichlet problem in a disk for equations of order 2m in the case of multiple characteristics with slope angles,” Mat. Met. Fiz.-Mekh. Polya, 51, No. 1, 33–41 (2008).zbMATHGoogle Scholar
  8. 8.
    E. A. Buryachenko, “Nontrivial solvability of the homogeneous Dirichlet problem in a disk for equations of order 2m in the case of characteristics without slope angles,” Visn. Donets. Nats. Univ., Ser. Fiz.-Mat. Nauk, 50, No. 2, 10–21 (2007).Google Scholar
  9. 9.
    I. N. Vekua, New Methods for the Solution of Elliptic Equations [in Russian], OGIZ, Moscow (1948).Google Scholar
  10. 10.
    A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • E. A. Buryachenko
    • 1
  1. 1.Donetsk National UniversityDonetskUkraine

Personalised recommendations