The problem of finding eigenvalues and eigenfunctions of boundary value problems for mixed-type equations

In this work, eigenvalues and eigenfunctions of the boundary value problem with the Frankl condition for an elliptic-hyperbolic type equation in a special domain are considered. In the elliptic part of the domain, using polar coordinates and the method of separation of variables, we derive the spectral problems for the ordinary differential equations. By solving these problems, we obtain the eigenvalues and eigenfunctions of the formulated problem. Furthermore, we prove that the system of eigenfunctions is incomplete in the L 2 space, which means that not every square-integrable function in the domain can be represented as a series expansion in terms of these eigenfunctions. This incompleteness is demonstrated by constructing a specific function orthogonal to the entire system of eigenfunctions. By exploring the spectral properties of mixed-type equations, this paper contributes to a broader understanding of how solutions behave in domains with varying types of differential operators. The study highlights the challenges posed by the change in operator type, emphasizing the difficulties in obtaining a complete and comprehensive eigenfunction system. The research expands on previous works in the field of spectral analysis for mixedtype equations, particularly with respect to the role of spectral parameters and their impact on the completeness of the solution space. This research provides valuable insights into the mathematical and physical implications of mixed-type boundary value problems.