# A criterion for the unique solvability of the spectral Dirichlet problem in a cylindrical domain for a class of multidimensional elliptic equations

## Keywords:

multidimensional elliptic equation, Dirichlet spectral problem, multidimensional cylindrical domain, solvability, criterion## Abstract

Correctness of boundary problems in the plane for elliptic equations is well analyzed by analitic

function theory of complex variable. There appear principal difficulties in similar problems when

the number of independent variables is more than two. An attractive and suitable method of

singular integral equations is less strong because of lock of any complete theory of multidimensional

singular integral equations. In the cylindrical domain of Euclidean space, for a single class of

multidimensional elliptic equations, the spectral Dirichlet problem with homogeneous boundary

conditions is considered. The solution is sought in the form of an expansion in multidimensional

spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions

for unique solvability of the problem are obtained, which essentially depend on the height of the

cylinder.

## References

P. 170.

[2] Aldashev S.A. On the Darboux problem for a class of multidimensional hyperbolic equations. Differentsial’nye uravneniia

]Differential Equations], 1998, Vol. 34, No. 1, pp. 64-68

[3] Aldashev S.A. Correctness of the Dirichlet problem in a cylindrical domain for the multidimensional Laplace equation //

Izv. Saratov. un-ta. New Ser., Ser.mat., Fur., Inf., - 2012, Vol.12, Vol. 3.- P. 3 - 7.

[4] Aldashev S.A. Correctness of Dirichlet’s Problem in a Cylindric Domain for a Single Class of Many-dimensional Elliptic

Equations // Vestnik of NGU. Series Mathematics, Mechanics, Informatics, -2012. Vоl. 12, issue 1. P. 7 - 13.

[5] Aldashev S.A. A criterion for the unique solvability of the spectral Dirichlet problem for the multidimensional hyperbolo-

parabolic equation // Second Int. Russian-Uzbek symposium "Equations of mixed type, related problems of analysis and

informatics". Nalchik, Research Institute of PMA KBSC RAS, 2012.-p. 24-27.

[6] Baitman G., Erdei A. Higher transcendental functions, vol.2, M .: The science, 1974 - 295 p.

[7] Bitsadze A.B. Mixed-type Equations. Moscow: Akad. Nauk USSR, 1959. P. 164.

[8] Bitsadze A.B. Boundary value problems for elliptic equations of the second order Moscow: Nauka. 1966. P. 203.

[9] Bitsadze A.B. Some classes of equations in partial derivatives Moscow: Nauka. 1981. P. 448.

[10] Kamke E. Handbook of Ordinary Differential Equations, Moscow: The science, 1965 - 703 p.

[11] Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis, Moscow: The science, 1976 -

543 p.

[12] Mikhlin S.G. Multidimensional singular integrals and integral equations, M .: Fizmatgiz, 1962 - 254 p.

[13] Tikhonov A.N., Samarsky А.А. Equations of mathematical physics. М., Nauka, 1966, 724 p.

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*Journal of Mathematics, Mechanics and Computer Science*,

*96*(4), 23–30. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/566