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

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

is called hypoelliptic if for every distribution u defined on an open subset $${ {\displaystyle V\subset U}$$ such that $${ {\displaystyle Pu}$$ 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). The heat equation operator

$${{\displaystyle P(u)=u_{t}-k\,\Delta u\,}$$

(where k > 0 {\displaystyle k>0} k>0) is hypoelliptic but not elliptic. The wave equation operator

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

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

Shimakura, Norio (1992). Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I. ISBN 0-8218-4556-X.
Egorov, Yu. V.; Schulze, Bert-Wolfgang (1997). Pseudo-differential operators, singularities, applications. Birkhäuser. ISBN 3-7643-5484-4.
Vladimirov, V. S. (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0.
Folland, G. B. (2009). Fourier Analysis and its applications. AMS. ISBN 0-8218-4790-2.

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