Let $\Omega \subseteq \mathbb R^3$ be a connected, open, and bounded set, and let $\Gamma = \partial \Omega$. Let $\sigma \in C^1(\mathbb R^3)$, $\sigma \geq 0$. Consider the integral $$\phi(x) = \int_\Gamma \frac{\sigma(y) dS(y)}{|x-y|}.$$ Show that $\Delta\phi=0$ for all $x \in \Omega$.
The only thing that came to my mind is that $\Delta f = f''_{rr} + f'_r \cdot \frac{2}{r}$ if $f = f(r)$ where $r = \sqrt{x^2+y^2+z^2}$, but that does not seem to be the case as there is an arbitrary function $\sigma$. The Divergence theorem which comes at handy when dealing with closed manifolds also doesn't seem to help.
