Maximum principle for potentials

Remarks

Proof idea: first assume that $U^\mu$ restricted to $S$ is continuous. The continuity principle then makes $U^\mu$ continuous on $\R^n$. On the open set $\R^n \setminus S$ the potential is harmonic, it is at most $M$ on the boundary, which lies in $S$, and it tends to $0$ at infinity. The maximum principle for harmonic functions gives $U^\mu \le M$ there. The general case follows by restricting $\mu$ to compact subsets of $S$ on which $U^\mu$ is continuous, which exist by Lusin's theorem, and letting these subsets increase.

Example: let $\mu$ be the uniform measure of total mass $m$ on the unit sphere in $\R^3$. By the shell theorem, $U^\mu(x) = m$ for $\abs{x} \le 1$ and $U^\mu(x) = m / \abs{x}$ for $\abs{x} \ge 1$. So $U^\mu \le m$ on the support, and indeed $U^\mu \le m$ everywhere.

The positivity of $\mu$ is needed. Let $\sigma$ be the uniform probability measure on the unit sphere of $\R^3$ and let $\mu = -\sigma$. Then $U^\mu = -1$ on the support, but $U^\mu(x) = -1/\abs{x} > -1$ for $\abs{x} > 1$. For logarithmic potentials in the plane the same statement holds: if $\int \log(1/\abs{z - w}) \, d\mu(w) \le M$ on the support of $\mu$, then it is at most $M$ on all of $\C$.

Going deeper

For a student

The principle is due to Maria and to Frostman. It is the key step that turns an inequality on the support of a measure into an inequality everywhere. It is used in the proof of Frostman's theorem to show that the equilibrium potential never exceeds its value on the conductor. For Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha < n$ and $\alpha > 2$, only a weaker version holds, with $M$ replaced by a constant multiple of $M$.