Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients

TL;DR

This paper constructs explicit fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients, expressed via Lauricella hypergeometric functions, and analyzes their singularity behavior.

Contribution

It provides the first explicit construction of fundamental solutions for this class of elliptic equations with singular coefficients using Lauricella hypergeometric functions.

Findings

Eight fundamental solutions explicitly constructed

Solutions expressed through Lauricella hypergeometric functions

Solutions exhibit a $1/r$ singularity at the origin

Abstract

We consider an equation $$ L_{\alpha ,\beta ,\gamma} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x + \displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0 $$ in a domain ${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$. Here $\alpha ,\beta ,\gamma$ are constants, moreover $0 < 2\alpha, 2\beta, 2\gamma < 1$. Main result of this paper is a construction of eight fundamental solutions for above-given equation in an explicit form. They are expressed by Lauricella's hypergeometric functions with three variables. Using expansion of Lauricella's hypergeometric function by products of Gauss's hypergeometric functions, it is proved that the found solutions have a singularity of the order $1/r$ at $r \to 0$.