Hypoelliptic operator

In the theory of partial differential equations, a partial differential operator ${\displaystyle P}$ defined on an open subset

${\displaystyle U\subset {\mathbb {R} }^{n}}$

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

If this assertion holds with ${\displaystyle C^{\infty }}$ replaced by real-analytic, then ${\displaystyle P}$ is said to be analytically hypoelliptic.

Every elliptic operator with ${\displaystyle C^{\infty }}$ 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 (${\displaystyle P(u)=u_{t}-k\,\Delta u\,}$)

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

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

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

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