Hypoelliptic operator

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

$U\subset {ℝ}^n$

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

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

Every elliptic operator with $C^{\mathrm{\infty }}$ 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 ($P(u)=u_t-k\mathrm{\Delta }u$)

$P={\partial }_t-k{\mathrm{\Delta }}_x$

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

$P=\underset{t}{\overset{2}{\partial }}-c^2{\mathrm{\Delta }}_x$

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

References

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