What is "hypoelliptic operator"

n. (context calculus English) A differential operator that preserves mathematical smoothness.

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

U ⊂ R

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

If this assertion holds with C replaced by real analytic, then P is said to be analytically hypoelliptic.

Every elliptic operator with C coefficients 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 − kΔu 

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

P(u) = u − cΔu 

(where c ≠ 0) is not hypoelliptic.