On a mixed problem for a multidimensional elliptic equation with singular coefficients

Abstract

At present the fundamental solutions of the multidimensional singular elliptic equation are known and they are expressed through the well-known Lauricella function $F_A^{(n)}$, the number of variables of which is equal to the number of singular coefficients of the equation under consideration. On the other hand, in applications of any hypergeometric function of many variables, expansion formulas are very important, allowing one to represent this hypergeometric function as an infinite sum of products of one-dimensional hypergeometric Gaussian functions for each variable of the studied function of many variables. In this paper we study one mixed problem for an elliptic equation with many singular coefficients in the first hyperoctant of the unit ball, the uniqueness of its solution is proved by the method of energy integrals, and its existence by the Green's function method. When finding the desired solution, expansion and summation formulas are used, as well as the limit relation for the Lauricella function.