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 ----