---
tag: B4TTNHSTRX
kind: concept
name: Elliptic partial differential equation
msc: [35J15, 35J30, 47F10]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation
same_as:
  wikidata: Q3607962
  wikipedia: Elliptic partial differential equation
---

## Summary

An elliptic partial differential equation is one whose [[principal symbol|F31T650DKY]] vanishes for no nonzero covector.
For second-order equations this means that the matrix of second-order coefficients is definite.
Elliptic equations model equilibrium states, such as steady temperatures, electrostatic potentials and membranes at rest.
Their solutions are as smooth as the data allow, and boundary value problems are the natural problems for them.
See the [[definition|3BYHP9E9VF]].

## Explanations

### L1 human

The model elliptic equation is the [[Laplace equation|S8RG6N1PRT]] $\partial_x^2 u + \partial_y^2 u = 0$.
Its solutions, the [[harmonic functions|0DEB0JMN0W]], have the property that the value at a point is the average of the values on any circle around it.
So an elliptic equation averages things out: there are no preferred directions, no time, and no travelling waves.
A soap film stretched over a wire frame is a picture of a solution: its shape inside is determined by the frame.