The Tricomi–Neumann Problem for a Three-Dimensional Mixed-Type Equation with Singular Coefficients

Abstract

Under study is the Tricomi–Neumann problem for a three-dimensional mixed-type equation with three singular coefficients in a mixed domain consisting of a quarter of a cylinder and a triangular straight prism. We prove the unique solvability of the problem in the class of regular solutions by using the separation of variables in the hyperbolic part of the mixed domain, which yields the eigenvalue problems for one-dimensional and two-dimensional equations. Finding the eigenfunctions of the problems, we use the formula of the solution of the Cauchy–Goursat problem to construct a solution to the two-dimensional problem. In result, we find the solutions to eigenvalue problems for the three-dimensional equation in the hyperbolic part. Using the eigenfunctions and the gluing condition, we derive a nonlocal problem in the elliptic part of the mixed domain. To solve the problem in the elliptic part, we reformulate the problem in the cylindrical coordinate system and separating the variables leads to the eigenvalue problems for two ordinary differential equations. We prove a uniqueness theorem by using the completeness property of the systems of eigenfunctions of these problems and construct the solution to the problem as the sum of a double series. Justifying the uniform convergence of the series relies on some asymptotic estimates for the Bessel functions of the real and imaginary arguments. These estimates for each summand of the series made it possible to prove the convergence of the series and its derivatives up to the second order, as well as establish the existence theorem in the class of regular solutions.