A criterion for the unique solvability of the spectral Dirichlet problem in a cylindrical domain for a class of multidimensional elliptic equations (Journal of Mathematics Mechanics and Computer Science, 2018)
Correctness of boundary problems in the plane for elliptic equations is well analyzed by analiticfunction theory of complex variable. There appear principal difficulties in similar problems whenthe number of independent variables is more than two. An attractive and suitable method ofsingular integral equations is less strong because of lock of any complete theory of multidimensionalsingular integral equations. In the cylindrical domain of Euclidean space, for a single class ofmultidimensional elliptic equations, the spectral Dirichlet problem with homogeneous boundaryconditions is considered. The solution is sought in the form of an expansion in multidimensionalspherical functions. The existence and uniqueness theorems of the solution are proved. Conditionsfor unique solvability of the problem are obtained, which essentially depend on the height of thecylinder.
Keywords: multidimensional elliptic equation; Dirichlet spectral problem; multidimensional cylindrical domain; solvability; criterion