Define a fundamental solution of a constant-coefficient linear partial differential operator, and prove that the distribution defined by the function $N: \mathbb{R}^{3} \rightarrow \mathbb{R}$

$$N(x)=(4 \pi|x|)^{-1}$$

is a fundamental solution of the operator $-\Delta$ on $\mathbb{R}^{3}$.

State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$ and deduce that any two smooth solutions of

$$-\Delta u=f, \quad f \in C^{\infty}\left(\mathbb{R}^{3}\right)$$

which satisfy the condition

$$\lim _{|x| \rightarrow \infty} u(x)=0$$

are in fact equal