Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients

TL;DR

This paper constructs fundamental solutions for a class of multidimensional elliptic equations with multiple singular coefficients, using hypergeometric functions and a new decomposition formula, aiding boundary value problem solutions.

Contribution

It introduces a new decomposition formula for multivariable hypergeometric functions, enabling explicit fundamental solutions for elliptic equations with singular coefficients.

Findings

Fundamental solutions expressed via multiple hypergeometric functions.

Decomposition formula for multivariable hypergeometric functions proved.

Properties of solutions related to singularity order determined.

Abstract

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined. Key words: multidimensional elliptic equation with several singular coefficients; fundamental solutions; multiple hypergeometric functions; decomposition formula; order of the singularity.