On the Dirichlet Problem for a Three-dimensional Equation of Mixed Type with Two Planes of Type Change in a Parallelepiped

The first boundary value problem for the three-dimensional modelequation with two planes of type change in a parallelepiped isinvestigated. A criterion for the uniqueness of its solution isestablished. The solution of problem is constructed as the sum ofthe series in the biorthogonal system of the correspondingspectral problem for an ordinary differential operator. Whenproving the existence of a solution, a problem of smalldenominators arises. Therefore, are obtained estimates on theseparation of these denominators from zero with the correspondingasymptotics. This allows us to establish the existence of asolution to the problem from the required class.