Boundary problem for Lavrent'ev–Bitsadze equation with two internal lines of change of a type

We study the problem with boundary conditions of the first and second kind on the boundary of the rectangular area for an equation with two internal perpendicular lines of change of a type. With the use of spectral method we prove the uniqueness and the existence of a solution. Obtained in the process of separation of variables, the eigenvalue problem for an ordinary differential equation is not self-adjoint, and the system of root functions is not orthogonal. We construct corresponding biorthogonal system of functions and prove its completeness, based on which we establish a criterion for the uniqueness of the problem. A solution to the problem is constructed as a sum of biorthogonal series.