Let $\Omega$ be a domain of a Riemannian manifold $M$. Let $\Omega_{D}$ be the diagonal of $\Omega$, that is,
$\Omega_{D} = \{(x,y)\in \Omega \times \Omega : x=y \}$.
According to my text, Green's function $G$ is defined as follows:
"A continuous function $G:(\bar{\Omega} \times \bar{\Omega}) \rightarrow \mathbb{R}$ is a Green's function of $\Omega$, if $G\mid \{(\Omega \times \Omega)\backslash\Omega_{D}\} \in C^{2}$,
$(\Delta _{y}G)(x,y)=0$
for all $x,y \in \Omega$.
$G\mid \Omega \times \Gamma =0$ (where $\Gamma=\partial \Omega$),
and, near $\Omega_{D}$, $G$ is given by
$G(x,y)=\psi(x,y) + h(x,y)$ ,
where $h\in C^{0}(\bar{\Omega}\times \bar{\Omega})\cap C^{2}(\Omega\times\Omega),$ and
$\psi(x,y)=\frac{1}{c_{n-1}}\biggl\{\begin{array}{11} d^{2-n}(x,y)/n-2,&n>2\\ -\log d(x,y),&n=2 \end{array} $
where $c_{n-1}$ is volume of $(n-1)$-sphere, $d(x,y)$ is distancefunction on $M$."
But according to many other books, Green's function is definde as
"$\Delta_{y}G(x,y)=\delta(x-y)$,
where $\delta$ is Dirac's delta function,
and satisfy appropriate boundary condition."
Is these same Green's function? I think the first one is defined as a standard function, the second one is defined as a distribution. But I have never heard that Green's function is a standard function, not a distribution.
I would appriciate your help.