---
tag: AQ5RN6FFT0
kind: concept
name: Superharmonic potential
msc: [31B05, 31B15]
specializes: [X4V2S56YQQ]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Summary

A potential, in the sense of classical potential theory, is a nonnegative [[superharmonic function|X4V2S56YQQ]] whose greatest harmonic minorant is $0$.
It is the part of a superharmonic function that comes from charge, with no harmonic background left over.
By the [[Riesz decomposition theorem|CBZ20SS8RQ]], on a domain with a Green function the potentials are exactly the Green potentials of positive measures.
The Newtonian kernel $\abs{x}^{2-n}$ on $\R^n$ for $n \ge 3$ is the basic example.

## Explanations

### L1 human

A positive superharmonic function can be split into a harmonic piece and a piece produced by charges.
A potential is a function made only of the charge piece.
Its harmonic piece is zero, which roughly means it fades away at the edge of the domain.
The potential $1/\abs{x}$ of a unit point charge in space is a potential in this sense, while a positive constant is not.