---
tag: NA4BTZ6WRB
kind: concept
name: Minimal harmonic function
msc: [31C35, 31B05]
specializes: [EYMQ82Z8XJ]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_boundary
---

## Summary

A minimal harmonic function is a [[positive harmonic function|EYMQ82Z8XJ]] that has no positive harmonic minorants other than its own constant multiples.
Minimal functions play the role of extreme points in the convex cone of positive harmonic functions on a domain.
By Martin's theory, every positive harmonic function is an integral of minimal ones.
The minimal ones correspond to the points of the minimal Martin boundary.

## Explanations

### L1 human

Positive harmonic functions can be added and scaled by positive numbers.
A minimal one cannot be split into two genuinely different positive harmonic pieces.
On the unit disc, the Poisson kernel with its pole at one boundary point is minimal.
It is the response to a unit mass at a single point of the circle.
The constant function $1$ on the disc is not minimal, since it is the average of the Poisson kernels over the circle.