We know that Green's third identity, is the following with $f$ a $C^2$ function on $D\subset\mathbb{R}^n$
\begin{equation}-4\pi f(r_0) = \int_D\frac{1}{|r-r_0|}\nabla^2f\,dV+\int_{\partial D}\left(\frac{1}{|r-r_0|}\nabla f-f\left(\nabla\frac{1}{|r-r_0|}\right)\cdot n \,dS\right),\end{equation}
where $n(r)$ is the unit vector that is (outward) normal to the surface $\partial D$ at the point $r\in\partial D$. Does anyone know a formula for the fractional laplacian $(-\Delta)^{a}$, $a\in(0,1)$?. I was looking for information, but I didn't find anything.
Anyway, the fractional laplacian can be written as a singular integral defined by $$(-\Delta)^a f(x)=c_{n,a}\int_{\mathbb{R}^n}{\frac{f(x)-f(y)}{|x-y|^{n+2a}}}\,dy$$ where $c_{n,d}=\frac{4^a\Gamma(n/2+a)}{\pi^{n/2}\Gamma(-a)}$ . Any idea is welcome.