Fundamental solutions for a class of four-dimensional degenerate elliptic equation

ABSTRACT

We consider the generalized Gellerstedt equation $y^m{z}^k{t}^l{u}_{xx}+x^n{z}^k{t}^l{u}_{yy}+x^n{y}^m{t}^l{u}_{zz}+x^n{y}^m{z}^k{u}_{tt}=0$ in a domain $R_{+}^4=\left\{\left(x,y,z,t\right):x>0,y>0,z>0,t>0\right\}$. Here m,n,k,l>0 are constants. The goal of the present paper is finding the fundamental solutions for the generalized four-dimensional Gellerstedt equation in an explicit form. Solutions are expressed by $F_A^{\left(4\right)}$ Lauricella hypergeometric functions of four variables. By means of expansion of the hypergeometric Lauricella function by products of Gauss hypergeometric functions, it is proved that the found solutions have a singularity of the order $1/r^2$ at $r\to 0$. These fundamental solutions are important for solving a number of boundary value problems for the aforementioned degenerate elliptic equation. In addition, some properties of these solutions are shown.

COMMUNICATED BY:

• J. Shi

KEYWORDS:

• Fundamental solutions
• multidimensional hypergeometric functions
• elliptic differential equation with singular coefficients
• Lauricella's hypergeometric functions of four variables

AMS SUBJECT CLASSIFICATIONS:

• 35J25
• 35E05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by a grant no. AP05131026 from the Ministry of Science and Education of the Republic of Kazakhstan.