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# Hypoelliptic operator

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

$U subset\left\{mathbb\left\{R\right\}\right\}^n$

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 is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator

$P\left(u\right)=u_t - kDelta u,$

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

$P\left(u\right)=u_\left\{tt\right\} - c^2Delta u,$

(where $cne 0$) is not hypoelliptic.

## References

• Shimakura, Norio Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I.
• Egorov, Yu. V.; Schulze, Bert-Wolfgang Pseudo-differential operators, singularities, applications. Birkhäuser.
• Vladimirov, V. S. Methods of the theory of generalized functions. Taylor & Francis.

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