---
tag: N5JN15Z5J6
kind: definition
name: Polyharmonic function
msc: [31B30, 35J30]
defines: 17R5V5ZVSR
uses: [0E0B6RA8K0, 02BH3MCBQB]
sources:
  - url: https://en.wikipedia.org/wiki/Biharmonic_equation
---

## Definition

Let $n \ge 1$, let $m \ge 1$ be an integer, and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]], and let $\Delta^m$ be $\Delta$ applied $m$ times.
A function $u \colon \Omega \to \R$ is *polyharmonic of order $m$* if it is of class $C^{2m}$ and
$$\Delta^m u = 0 \quad \text{on } \Omega.$$

## Remarks

For $m = 1$ this is the definition of a [[harmonic function|0DEB0JMN0W]], and for $m = 2$ that of a solution of the [[biharmonic equation|AS8A9F5SKE]].
A function that is polyharmonic of order $m$ is polyharmonic of every order $m' \ge m$.
Example: $u(x) = \abs{x}^4$ on $\R^n$ is polyharmonic of order $3$: $\Delta \abs{x}^4 = 4(n+2)\abs{x}^2$, $\Delta^2 \abs{x}^4 = 8n(n+2)$ and $\Delta^3 \abs{x}^4 = 0$.
Non-example: the same $u$ is not polyharmonic of order $2$, since $\Delta^2 u = 8n(n+2) \ne 0$.
Non-example: $u(x) = e^{x_1}$ is polyharmonic of no order, since $\Delta^m u = e^{x_1} \ne 0$ for all $m$.