Hypoelliptic operator

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

[math]\displaystyle{ U \subset{\mathbb{R}}^n }[/math]

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

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

Every elliptic operator with [math]\displaystyle{ C^\infty }[/math] 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 ([math]\displaystyle{ P(u)=u_t - k\,\Delta u\, }[/math])

[math]\displaystyle{ P= \partial_t - k\,\Delta_x\, }[/math]

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

[math]\displaystyle{ P= \partial^2_t - c^2\,\Delta_x\, }[/math]

(where [math]\displaystyle{ c\ne 0 }[/math]) 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.