Dezin Problem for an Equation of the Mixed Type with a Power-Law Degeneracy

Abstract

We study a boundary value problem with periodicity conditions and with a nonlocal Dezin condition for a mixed elliptic-hyperbolic equation in a rectangular domain with power-law degeneracy on the transition line. Necessary and sufficient conditions for the uniqueness of the solution are established, the uniqueness of the solution being proved based on the completeness of the system of eigenfunctions of a one-dimensional eigenvalue problem. The solution is constructed in the form of a series. The problem of small denominators occurs when justifying the convergence of the series. Under some conditions imposed on the given parameters and functions, the convergence of the series is proved in the class of regular solutions.