# 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.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

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