一类具有奇异系数的三维椭圆型方程的基本解
Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients
摘要：
考虑区域${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$中的方程$$ L_{α,β,γ} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{2α}{x}u_x + \displaystyle \frac{2β}{y}u_y + \displaystyle \frac{2γ}{z}u_z = 0 $$.这里$α,β,γ$是常数，而$0 < 2α, 2β, 2γ< 1$是常数。本文的主要结果是以显式形式构造了上述方程的八个基本解。它们用劳里切拉的三元超几何函数表示。利用Gauss超几何函数的乘积展开Lauricella超几何函数，证明了所得到的解在$r \to 0$处具有$1/r$阶奇异性.
英文摘要：
We consider an equation $$ L_{α,β,γ} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{2α}{x}u_x + \displaystyle \frac{2β}{y}u_y + \displaystyle \frac{2γ}{z}u_z = 0 $$ in a domain ${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$. Here $α,β,γ$ are constants, moreover $0 < 2α, 2β, 2γ< 1$. 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 $r \to 0$.
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