---
tag: DWN0XEQHHK
kind: concept
name: Harmonic space
msc: [31D05, 31C05]
sources:
  - url: https://en.wikipedia.org/wiki/Potential_theory
---

## Summary

A harmonic space is a locally compact space with a sheaf of continuous functions, called harmonic, that obey axioms modelled on classical [[harmonic functions|0DEB0JMN0W]].
The axioms ask that the [[Dirichlet problem|8D56GQ4AG8]] be uniquely solvable on a base of small open sets.
They also ask that suitable increasing limits of harmonic functions be harmonic.
Harmonic spaces are the main setting of [[axiomatic potential theory|YS7RF1PE42]].
Euclidean space with the Laplace equation is a harmonic space, and so is space-time with the heat equation.

## Explanations

### L1 human

In the plane, a harmonic function on a disc is determined by its values on the boundary circle.
A harmonic space is any space where some chosen functions behave in this way on enough small sets.
The chosen functions must also be stable under natural operations, such as gluing local pieces and taking increasing limits.
This is enough to build much of potential theory: superharmonic functions, potentials and the solution of boundary value problems.
The solutions of the heat equation, which describe temperature changing in time, form a harmonic space too.