---
tag: 9PM8Q2E30Q
kind: concept
name: Potential theory
msc: [31A15, 31B15, 31B05]
sources:
  - url: https://en.wikipedia.org/wiki/Potential_theory
same_as:
  wikidata: Q1154848
  wikipedia: Potential theory
---

## Summary

Potential theory is the study of [[harmonic functions|0DEB0JMN0W]], [[subharmonic functions|Z1MHMGFNFW]] and the potentials generated by distributions of mass or charge.
It began with the gravitational and electrostatic potentials of physics, which satisfy the Laplace equation away from the masses.
Its main topics include the [[Dirichlet problem|8D56GQ4AG8]], [[Green's functions|M36SD56MWQ]], [[capacity|DBQXRTC0VN]], [[harmonic measure|CFD7BPM51W]] and the boundary behaviour of harmonic functions.
In the plane it is tied closely to complex analysis, since the real part of a [[holomorphic function|0NAHXR1FRH]] is harmonic and $\log\abs{f}$ is subharmonic.

## Explanations

### L1 human

The gravitational pull of a planet or the electric field of a charge can be described by a single number at each point, the potential.
Away from the masses and charges, this potential satisfies the Laplace equation, so it is a harmonic function.
Potential theory studies such functions: how they are determined by boundary values, how they behave near the edge of a region, and how charges spread out to minimize energy.
For example, the equilibrium temperature in a plate with fixed edge temperatures is a harmonic function, and finding it is a central problem of the theory.