Boundary Value Problems in a Semi-infinite Parallelepiped for an Elliptic Equation with Three Singular Coefficients

The Dirichlet and the Keldysh problems for a three-dimensional elliptic equation with three singular coefficients are investigated in a semi-infinite parallelepiped. The uniqueness and the existence theorems are proved using the spectral analysis method. For the problem posed, using the Fourier method, one dimensional spectral problems are obtained. Based on the completeness property of systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution of the problem under study is constructed in the form of the sum of the Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.