Dirichlet type boundary value problem for an elliptic equation with three singular coefficients in the first octant

Abstract

The paper investigates a Dirichlet-type boundary value problem for a three-dimensional elliptic equation with three singular coefficients in the first octant. The uniqueness of the solution within the class of regular solutions is established using the energy integral method. To prove the existence of a solution, the Hankel integral transform method is employed. The use of the Hankel transform is particularly appropriate when the variables in the equation range from zero to infinity. This transform is an effective method for obtaining solutions to such problems. In three-dimensional space, to derive the image equation, the Hankel integral transform is applied to the original equation with respect to the variables x and y. As a result, a boundary value problem for an ordinary differential equation in the variable z arises. By solving this problem, a solution to the original boundary value problem is constructed in the form of a double improper integral involving Bessel functions of the first kind and Macdonald functions. To justify the uniform convergence of the improper integrals, asymptotic estimates of the Bessel functions of the first kind and Macdonald functions are utilized. Based on these estimates, bounds for the integrands are obtained, which ensure the convergence of the resulting double improper integral, that is, the solution to the original boundary value problem and its derivatives up to second order, inclusively, as well as the theorem of existence within the class of regular solutions.