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.

  • $\begingroup$ Where this question came from? Maybe, the theory contained in the book can be useful to solve the problem. $\endgroup$
    – DiegoMath
    Mar 5, 2021 at 22:51
  • $\begingroup$ @DiegoMath that is from our calculus of variations course (the lecturer doesn't use a specific book for it). I honestly can't imagine how can variations help me as it looks like an exercise in anything but calculus of variations $\endgroup$
    – dnes
    Mar 6, 2021 at 10:37
  • $\begingroup$ Now I see here that the integral is differentiable by the parameter y, so straight calculation of $\Delta \phi$ yields the necessary result. Silly me. $\endgroup$
    – dnes
    Jul 27, 2021 at 11:49


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