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

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.