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 Mat. Zametki, 2005, Volume 77, Issue 4, Pages 498–508 (Mi mz2508)

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

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

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.

DOI: https://doi.org/10.4213/mzm2508

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English version:
Mathematical Notes, 2005, 77:4, 461–470

Bibliographic databases:

UDC: 517.946
Revised: 09.02.2004

Citation: V. P. Burskii, E. A. Buryachenko, “Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk”, Mat. Zametki, 77:4 (2005), 498–508; Math. Notes, 77:4 (2005), 461–470

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz/v77/i4/p498

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Buryachenko E.A., “Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles”, Ukrainian Math. J., 62:5 (2010), 676–690
2. Render H., “Cauchy, Goursat and Dirichlet Problems for Holomorphic Partial Differential Equations”, Comput. Methods Funct. Theory, 10:2 (2010), 519–554
3. A. O. Babayan, S. O. Abelyan, “Defect Numbers of the Dirichlet Problem for a Properly Elliptic Sixth-Order Equation”, Math. Notes, 104:3 (2018), 339–347
4. Baranetskij Ya.O. Ivasiuk I.Ya. Kalenyuk I P. Solomko V A., “The Nonlocal Boundary Problem With Perturbations of Antiperiodicity Conditions For the Eliptic Equation With Constant Coefficients”, Carpathian Math. Publ., 10:2 (2018), 215–234
5. Babayan A.H., “On a Dirichlet Problem For One Improperly Elliptic Equation”, Complex Var. Elliptic Equ., 64:5 (2019), 825–837
6. V. P. Burskii, “Equation–Domain Duality in the Dirichlet Problem for General Differential Equations in the Space $L_2$”, Proc. Steklov Inst. Math., 306 (2019), 33–42
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