---
tag: X4V2S56YQQ
kind: concept
name: Superharmonic function
msc: [31B05, 31A05]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Summary

A superharmonic function is a lower semicontinuous function whose value at each point is at least its average over spheres around that point.
A function $u$ is superharmonic exactly when $-u$ is [[subharmonic|Z1MHMGFNFW]], so the two theories are mirror images.
The potentials of positive charges, such as the Newtonian kernel $\abs{x}^{2-n}$ in $\R^n$ for $n \ge 3$, are the basic examples.
Superharmonic functions form the upper class of the Perron method and are the main objects of classical [[potential theory|9PM8Q2E30Q]].

## Explanations

### L1 human

A [[harmonic function|0DEB0JMN0W]] equals its average over every small sphere around a point.
A superharmonic function sits at or above that average, like a tent pole pushing a membrane up from below.
It cannot have a strict local minimum inside its domain.
The gravitational potential $1/\abs{x}$ of a point mass in space is superharmonic, with the value $+\infty$ at the mass.
At the origin the average of $1/\abs{x}$ over the sphere of radius $r$ is $1/r$, which is below the value $+\infty$.