Dirichlet problem for an elliptic equation with three singular coefficients
We prove the unique solvability of the first boundary-value problem for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of a solution to the problem. We prove the existence of a solution by the spectral Fourier method based on the separation of variables. A solution to the problem is constructed in the form of a double Fourier-Bessel series. The justification of the uniform convergence of this series is based on asymptotic methods. We obtain an estimate, which allows one to prove the convergence of the series and its derivatives up to the second order and the existence theorem for the class of regular solutions of the equation considered.