In the theory of partial differential equations, a partial differential operator
defined on an open subset is called hypoelliptic if for every distribution
defined on an open subset
If this assertion holds with
replaced by real-analytic, then
Every elliptic operator with
In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic).
In addition, the operator for the heat equation (
) is hypoelliptic but not elliptic.
However, the operator for the wave equation (
This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.