Why is $$\frac{1}{n\alpha(n)}\int_{\partial B(0,1)}\frac{1-|x|^2}{|x-y|^n}dS(y)=\int_{B(0,1)}G(x,y)dy$$ where $G$ is the Green function and defined as(in the unit ball) $G(x,y):=\Phi(y-x)-\Phi(|x|(y-\frac{x}{|x|^2}))$ and $\Phi$ is the fundamental solution of the Poisson equation for the dimension $n\ge 3$, i.e. $\Phi(x)=\frac{1}{n(n-2)\alpha(n)}\frac{1}{|x|^{n-2}}$
I think I have to use first Green formula,
$\int_B \Delta f=\int_{\partial B}Df\cdot n\ dS$
what is $n$ here ?, $\frac{y}{|y|}$ ?
If $Df\cdot n=\frac{1-|x|^2}{|x-y|^n}$ then I have to find $Df$ and apply the Laplace operator to verify the formula but how do I get $Df$ ?