---
tag: A51SQRPM5T
kind: concept
name: Brelot space
msc: [31D05]
sources:
  - url: https://en.wikipedia.org/wiki/Potential_theory
---

## Summary

A Brelot space is a connected, locally connected space with a sheaf of harmonic functions, as in a [[harmonic space|DWN0XEQHHK]].
In it, an increasing sequence of harmonic functions on a domain tends either to a harmonic function or to $+\infty$ everywhere.
Its axioms, due to Marcel Brelot, model elliptic equations such as the Laplace equation.
In a Brelot space a Harnack inequality holds, as for classical [[harmonic functions|0DEB0JMN0W]].
The heat equation gives a harmonic space that is not a Brelot space.

## Explanations

### L1 human

Brelot's axioms capture the behaviour of harmonic functions in Euclidean space without using derivatives.
The key axiom says that an increasing sequence of harmonic functions on a connected open set cannot blow up in one place and stay finite in another.
This reflects the fact that positive harmonic functions cannot be very large at one point and small at a nearby point.
For temperatures that change in time this fails: heat at a later time does not control heat at an earlier time.