In mathematics, more specifically in the theory of partial differential equations, a partial differential operator
P
defined on an open subset
U
⊂
R
n
is called hypoelliptic if for every distribution
u
defined on an open subset
V
⊂
U
such that
P
u
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
t
−
k
Δ
u
(where
k
>
0
) is hypoelliptic but not elliptic. The wave equation operator
P
(
u
)
=
u
t
t
−
c
2
Δ
u
(where
c
≠
0
) is not hypoelliptic.
