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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 $\Omega$.

          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 $\Omega$.

Problem 7. Assume n ≥ 3 and let c(n) = (1)/((n-2)ωn-1). Let ρ∈ C0(ℝ^n), and define
U(x) = c(n) ∫ℝ^n(ρ(y))/(|x - y|^n-2) dy.
(a) Using a known result: , prove that Δ U = -ρ in 𝒟'(ℝ^n).
(b) Prove that Δ U = 0 in 𝒟'(Ω), where Ω = ℝ^n ∖ K, with K = supp ρ.
(c) You can take as a given that U ∈ C(Ω) (no need to prove this!). Prove that for any x ∈Ω and any r > 0 such that B(x, r) ⊂Ω one has

MU(x, r) = U(x),
where MU(x, r) indicates the spherical average of U on the sphere S(x, r). Deduce that U ∈ C^∞(Ω) and it is a classical harmonic function in Ω.

Added by Jimmy P.

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