---
tag: EYMQ82Z8XJ
kind: concept
name: Positive harmonic function
msc: [31B05, 31A05]
specializes: [0DEB0JMN0W]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_harmonic_function
same_as:
  wikidata: Q7233271
  wikipedia: Positive harmonic function
---

## Summary

A positive harmonic function is a [[harmonic function|0DEB0JMN0W]] that takes only positive values on its domain.
Positivity is a strong constraint: on a connected open set, the values of such a function at two points control each other, by [[Harnack's inequality|YNH832D50W]].
On the unit disc and on balls, positive harmonic functions are exactly the Poisson integrals of nonzero positive finite Borel measures on the boundary sphere.
They are the starting point of the theory of the Martin boundary.

## Explanations

### L1 human

Think of a steady temperature distribution that is above zero everywhere in a room.
Such a temperature cannot be very large at one point and very small at a nearby point.
For example, on the half-plane $y > 0$ the function $u(x, y) = y$ is harmonic and positive.
On the unit disc, the [[Poisson kernel|TCYSWCTYYM]] $(1 - \abs{z}^2)/\abs{1 - z}^2$ is another example.