Solution of the Neumann problem for one four-dimensional elliptic equation

In this article we investigate the Neumann problem for a degenerate elliptic equation in four variables. A fundamental solution is used to construct a solution to the problem. The fundamental solutions are written by using the Lauricella's hypergeometric functions. The energyintegral method is used to prove the uniqueness of the solution to the problem under consideration. In the course of proving the existence of the problem solution, differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. The Gauss–Ostrogradsky formula is used to express problem's solution in an explicit form.