Frankl’s problem for a three-dimensional equation of mixed type with a singular coefficient

The paper studies the Frankl problem for a three -dimensional equation of mixed type with a singular coefficient in a mixed domain consisting of a quarter of a cylinder and two rectangular prisms.The unique solvability of the problem in the class of regular solutions is proved.The Fourier method based on separation of variables is used.The eigenfunctions of the problem in elliptic and hyperbolic domains are constructed.The completeness of the system of eigenfunctions is investigated.Based on the completeness property of systems of eigenfunctions, the uniqueness theorem is proved, and the solution to the problem under study is constructed as the sum of a double series.When substantiating the uniform convergence of the constructed series, asymptotic estimates of the Bessel functions of the real and imaginary arguments were used.Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the obtained series and its derivatives up to the second order inclusive, as well as the existence theorem in the class of regular solutions.