Hypoelliptic operator
From Wikipedia, the free encyclopedia
In mathematics, more specifically in the theory of partial differential equations, a partial differential operator P defined on an open subset
is called hypoelliptic if for every distribution u defined on an open subset 
such that Pu is 
(smooth), u must also be .
If this assertion holds with replaced by real analytic, then P is said to be analytically hypoelliptic.
Every elliptic operator is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator
(where k > 0) is hypoelliptic but not elliptic. The wave equation operator
(where 
) is not hypoelliptic.
[edit] References
- Shimakura, Norio (1992). Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I. ISBN 082184556X.
- Egorov, Yu. V.; Schulze, Bert-Wolfgang (1997). Pseudo-differential operators, singularities, applications. Birkhäuser. ISBN 3764354844.
- Vladimirov, V. S. (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0415273560.
This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the GFDL.
