The generalized Holmgren problem for elliptic equation with several singular coefficients

TL;DR

This paper derives explicit solutions for a generalized Holmgren problem involving a multidimensional singular elliptic equation with multiple singular coefficients, utilizing Lauricella hypergeometric functions and their properties.

Contribution

It provides a novel explicit solution to the generalized Holmgren problem for elliptic equations with several singular coefficients, expanding the analytical tools for such equations.

Findings

Explicit solution expressed via Lauricella hypergeometric functions

Utilization of decomposition formulas and adjacent relations

Advancement in solving multidimensional singular elliptic equations

Abstract

Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the generalized Holmgren problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables.