Problem 7. Assume $n \ge 3$ and let $c(n) = \frac{1}{(n-2)\omega_{n-1}}$. Let $\rho \in C_0(\mathbb{R}^n)$, and define \begin{equation*} U(x) = c(n) \int_{\mathbb{R}^n} \frac{\rho(y)}{|x - y|^{n-2}} dy. \end{equation*} (a) Using a known result: __________, prove that $\Delta U = -\rho$ in $\mathcal{D}'(\mathbb{R}^n)$.
(b) Prove that $\Delta U = 0$ in $\mathcal{D}'(\Omega)$, where $\Omega = \mathbb{R}^n \setminus K$, with $K = \text{supp } \rho$.
(c) You can take as a given that $U \in C(\Omega)$ (no need to prove this!). Prove that for any $x \in \Omega$ and any $r > 0$ such that $B(x, r) \subset \Omega$ one has
$$M_U(x, r) = U(x),$$where $M_U(x, r)$ indicates the spherical average of $U$ on the sphere $S(x, r)$. Deduce that $U \in C^\infty(\Omega)$ and it is a classical harmonic function in $\Omega$.
