The Neumann and Dirichlet Problems for One Four-Dimensional Degenerate Elliptic Equation

Abstract

In the present work we study the Neumann and Dirichlet boundary value problems for a four-dimensional (4D) degenerate elliptic equation by means the fundamental solutions, which were constructed earlier. Fundamental solutions are expressed by Lauricella hypergeometric functions. To prove the uniqueness of the solutions to the problems under consideration, the energy-integral method is used. In the course of proving the existence of solutions of the problems, differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. To express problem’s solution in explicit form the Gauss–Ostrogradsky formula is used.