Hypoelliptic operator

In mathematics, more specifically in the theory of partial differential equations, a partial differential operator $P$ defined on an open subset

:$U\; subset\{mathbb\{R^n$

is called hypoelliptic if for every distribution $u$ defined on an open subset $V\; subset\; U$ such that $Pu$ is $C^infty$ (smooth), $u$ must also be $C^infty$.

If this assertion holds with $C^infty$ 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

:$P(u)=u\_t\; -\; kDelta\; u,$

(where $k>0$) is hypoelliptic but not elliptic. The wave equation operator

:$P(u)=u\_\{tt\}\; -\; c^2Delta\; u,$

(where $c\; e\; 0$) is not hypoelliptic.

References

*cite book
last = Shimakura
first = Norio
title = Partial differential operators of elliptic type: translated by Norio Shimakura
publisher = American Mathematical Society, Providence, R.I
date = 1992
pages =
isbn = 082184556X

*cite book
last = Egorov
first = Yu. V.
coauthors = Schulze, Bert-Wolfgang
title = Pseudo-differential operators, singularities, applications
publisher = Birkhäuser
date = 1997
pages =
isbn = 3764354844

*cite book
last = Vladimirov
first = V. S.
title = Methods of the theory of generalized functions
publisher = Taylor & Francis
date = 2002
pages =
isbn = 0415273560 ----