Newtonian potential

Remarks

Example: for $n = 3$ and $\mu = m \, \delta_a$ a point mass, $U^\mu(x) = m / \abs{x - a}$. Non-example: in $\R^3$ the function $u(x) = \abs{x}^2$ is not the Newtonian potential of any such $\mu$, since a Newtonian potential tends to $0$ at infinity. The function $U^\mu$ is lower semicontinuous, harmonic on $\R^n \setminus \operatorname{supp} \mu$, and $U^\mu(x) \abs{x}^{n-2} \to \mu(\R^n)$ as $\abs{x} \to \infty$. If $\mu = f \, dx$ with $f \ge 0$ twice continuously differentiable with compact support, then $U^\mu$ is twice continuously differentiable and $\Delta U^\mu = -(n-2)\sigma_n f$, where $\sigma_n$ is the area of the unit sphere. Equivalently, $U^\mu = (n-2)\sigma_n \, \Phi * \mu$, where $\Phi$ is the fundamental solution of the Laplacian. Some texts call $\Phi * f$ itself the Newtonian potential of $f$, and use the same name in the plane with the logarithmic kernel.