Newtonian potential

Summary

The Newtonian potential of a mass distribution $\mu$ in $\R^n$, $n \ge 3$, is the function $U^\mu(x) = \int \abs{x - y}^{2-n} \, d\mu(y)$. It is the potential of the measure $\mu$ with respect to the Newtonian kernel. In $\R^3$ it is the gravitational potential of Newton's theory, up to the gravitational constant and a sign. It is harmonic off the support of $\mu$. In the sense of distributions it solves Poisson's equation $\Delta U^\mu = -(n-2)\sigma_n \mu$, where $\sigma_n$ is the area of the unit sphere.