Hypoelliptic operator
# Hypoelliptic operator

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 with $C^{\infty }$ coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the heat equation ($P(u)=u_{t}-k\,\Delta u\,$)

    $P=\partial _{t}-k\,\Delta _{x}\,$

(where $k>0$) is hypoelliptic but not elliptic. However, the operator for the wave equation ($P(u)=u_{tt}-c^{2}\,\Delta u\,$)

    $P=\partial _{t}^{2}-c^{2}\,\Delta _{x}\,$

(where $c\neq 0$) is not hypoelliptic.

