Lauricella hypergeometric function and its application to the solution of the Neumann problem for a multidimensional elliptic equation with several singular coefficients in an infinite domain

TL;DR

This paper investigates the Neumann problem for a multidimensional elliptic equation with multiple singular coefficients, utilizing Lauricella hypergeometric functions to establish explicit solutions and prove uniqueness in an infinite domain.

Contribution

It introduces a method to explicitly solve the Neumann problem for elliptic equations with singular coefficients using Lauricella hypergeometric functions, including new differentiation formulas and integral relations.

Findings

Proved the uniqueness of the solution using the integral energy method.

Derived explicit solutions involving Lauricella hypergeometric functions.

Established new differentiation and limiting relations for these functions.

Abstract

At present, the fundamental solutions of the multidimensional elliptic equation with the several singular coefficients are known and they are expressed in terms of the Lauricella hypergeometric function of many variables. In this paper we study the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain. Using the method of the integral energy the uniqueness of solution has been proved. In the course of proving the existence of the explicit solution of the Neumann problem, a differentiation formula, some adjacent and limiting relations for the Lauricella hypergeometric functions and the values of some multidimensional improper integrals are used.