Fundamental solutions for a class of four-dimensional degenerate elliptic equation
Abstract
We consider the generalized Gellerstedt equation ymzktluxx+xnzktluyy+xnymtluzz+xnymzkutt=0 in a domain R+4=x,y,z,t:x>0,y>0,z>0,t>0. 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 FA(4) 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/r2 at r→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.