Abstract
The Dirichlet problem for a class of properly elliptic sixth-order equations in the unit disk is considered. Formulas for determining the defect numbers of this problem are obtained. Linearly independent solutions of the homogeneous problem and conditions for the solvability of the inhomogeneous problem are given explicitly.
Similar content being viewed by others
References
J.–L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications, in Travaux et Recherches Mathématiques, V. 1 (Dunod, Paris, 1968), Vol. 17.
N. E. Tovmasyan, Non–Regular Differential Equations and Calculations of Electromagnetic Fields (World Sci. Publ., River Edge, NJ, 1998).
A. V. Bitsadze, Boundary Value Problems for Second–Order Elliptic Equations (Nauka, Moscow, 1966; North–Holland Publ. Co., Amsterdam, 1968).
N. E. Tovmasyan, “New formulations and investigations of the first, second and third boundary value problems for strongly connected elliptic systems of two second order differential equations with constant coefficients,” Izv. Akad. Armyan. SSR Ser. Mat. 3 (6), 497–521 (1968) [in Russian].
A. O. Babayan, “The unique solvability of a Dirichlet problem for fourth order properly elliptic equation,” Izv. Nats. Akad. Nauk Armen., Mat. 34 (5), 5–18 (1999) [J. Contemp. Math. Anal., Armen. Acad. Sci. 34 (5), 1–15 (1999)].
A. H. Babayan and M. H. Mohammadi, “On a Dirichlet Problem for One Properly Elliptic Equation in the Unit Disk,” Reports of the National Academy of Sciences of Armenia 117 (3), 192–200 (2017).
A. O. Babayan, “The Dirichlet problem for a fourth–order partial differential equation in the case of double roots of the characteristic equation,” Math. Montisnigri 32, 66–80 (2015).
A. O. Babayan, “Dirichlet problem for an improperly elliptic equation of fourth order,” in Nonclassical Equations of Mathematical Physics, Proceedings of the International Conference “Differential Equations, Theory of Functions and Applications" dedicated to the Centennial of Academician I. N. Vekua (Novosibirsk, 2007), pp. 56–69 [in Russian].
N. E. Tovmasyan and V. S. Zakaryan, “The Dirichlet problem for properly elliptic equations in multiply connected domains,” Izv. Nats. Akad. Nauk Armen., Mat. 37 (6), 5–40 (2002) [J. Contemp. Math. Anal., Armen. Acad. Sci. 37 (6), 2–34 (2002)].
A. O. Babayan, “On unique solvability of the Dirichlet problem for one class of properly elliptic equations,” in Topics in Analysis and Its Applications, NATO Sci. Ser. II Math. Phys. Chem. (Kluwer Acad. Publ., Doderecht, 2004), Vol. 147, pp. 287–293.
A. O. Babayan, “Dirichlet problem for a properly elliptic equation in a unit disk,” Izv. Nats. Akad. Nauk Armen.,Mat. 38 (6), 39–48 (2003) [J. Contemp.Math. Anal., Armen. Acad. Sci. 38 (6), 19–28 (2003)].
V. P. Burskii, Investigation Methods of Boundary Value Problems for General Differential Equations (Naukova Dumka, Kiev, 2002) [in Russian].
V. P. Burskii and K. 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), 498–508 (2005) [Math. Notes 77 (4), 461–470 (2005)].
A. H. Babayan and V. A. Babayan, “Defect numbers of the Dirichlet problem for higher order partial differential equations in the unit disc,” Caspian J. Comp. Math. Engineering, No. 1, 4–19 (2016).
S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, in Grad. Texts in Math. (Springer, New–York, 2001), Vol. 137.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Nikolai Karapetovich Karapetyants, a wonderful mathematician and an amazing person, with love and appreciation from the authors.
Original Russian Text © A. O. Babayan, S. O. Abelyan, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 3, pp. 345–355.
Rights and permissions
About this article
Cite this article
Babayan, A.O., Abelyan, S.O. Defect Numbers of the Dirichlet Problem for a Properly Elliptic Sixth-Order Equation. Math Notes 104, 339–347 (2018). https://doi.org/10.1134/S0001434618090031
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618090031