---
tag: 3KP89PT5P3
kind: statement
role: theorem
name: Minimum principle for harmonic functions
msc: [31B05, 35B50]
uses: [0DEB0JMN0W, T62HHSRHP5]
about: [0DEB0JMN0W]
sources:
  - url: https://en.wikipedia.org/wiki/Maximum_principle
---

## Statement

Let $U \subseteq \R^n$ be a domain, that is, a connected open set, and let $u \colon U \to \R$ be [[harmonic|0DEB0JMN0W]].

1. (Strong minimum principle.) If there is $a \in U$ with $u(a) = \inf_U u$, then $u$ is constant on $U$.
2. (Weak minimum principle.) If $U$ is bounded and $u$ extends continuously to the closure $\overline{U}$, then $\min_{\overline{U}} u = \min_{\partial U} u$.

## Remarks

Proof: $-u$ is harmonic, and the [[maximum principle|T62HHSRHP5]] for $-u$ is exactly this statement for $u$.
Consequence: if $U$ is bounded, $u$ is continuous on $\overline{U}$ and $u \ge 0$ on $\partial U$, then $u \ge 0$ on $U$; if moreover $u$ is not constant, then $u > 0$ on $U$.
Example: on the unit disc, $u(x, y) = 1 + x$ is $\ge 0$ on the boundary circle, with value $0$ only at $(-1, 0)$, and it is positive inside.
Boundedness is needed in part 2: on the half-plane $\set{x > 0}$, $u = -x$ is $0$ on the boundary and negative inside.
For subharmonic functions only the maximum principle holds, and for superharmonic functions only the minimum principle.

## Explanations

### L1 human

A harmonic function cannot have a dip inside its domain.
Its smallest values on a bounded region are reached on the edge.
For a steady temperature distribution, the coldest point is on the boundary.

### L2 human

Together with the maximum principle, it says that the values of a harmonic function inside a bounded domain lie between its smallest and largest boundary values.
The positivity form, nonnegative boundary values give nonnegative solutions, is the comparison principle used everywhere in potential theory, for example to show that Green's functions are positive.