---
tag: AS8A9F5SKE
kind: concept
name: Biharmonic equation
msc: [31B30, 35J30]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Biharmonic_equation
same_as:
  wikidata: Q859808
  wikipedia: Biharmonic equation
---

## Summary

The biharmonic equation is the fourth-order equation $\Delta^2 u = 0$, where $\Delta^2 = \Delta \Delta$ is the [[Laplacian|0E0B6RA8K0]] applied twice.
Its solutions are called biharmonic functions.
It models the small bending of a thin elastic plate and, in two dimensions, the stream function of slow viscous flow.
It is the basic example of a higher-order elliptic equation, for which tools such as the maximum principle no longer hold.

## Explanations

### L1 human

A [[harmonic function|0DEB0JMN0W]] has zero Laplacian.
A biharmonic function only needs its Laplacian to be harmonic.
So every harmonic function is biharmonic, but there are many more biharmonic functions.
For example, $u(x, y) = x^2 + y^2$ has Laplacian $4$, which is constant and so harmonic; hence $u$ is biharmonic, although it is not harmonic.