I just have found out these information in my presentation work:
a) $V\subset\mathbb R^n $ open, $D:=\{(x,x)|x\in\mathbb R^n\}$ and a function $\alpha:(V\times V)\backslash D\rightarrow \mathbb R$ with:
b) $\alpha(x,y)\in C^1(V\backslash\{y\})\cap C^2(V\backslash\{y\})$ and
c) $\triangle_x\alpha(x,y)\in L^1(V)$ and
d) $\forall u\in C^1(V)\cap C^2(V): u(y)=-\int\limits_V\alpha(x,y)\triangle u(x)d^nx-\int\limits_{\partial V }u(x)\partial_{\nu_x}\alpha(x,y)d\sigma_x $
Now I want to check if $\alpha:V\backslash\{y\}\rightarrow\mathbb R$ is solving the following problem:
$\triangle\alpha=0$ in $V\backslash\{y\}$ with $\alpha=0$ on the boundary of $V$.
I have absolutely no idea how to prove this ;( I will be very happy about some help! Thank you guys!