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 $C^{\mathrm{\infty }}$ (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

• 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 978-0-8218-4790-9.

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