Hypoelliptic operator explained

U \subset{ R }^n 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 with

C ^{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\Delta u ) P = \partial _{t} - k\Delta _{x} (where k> 0 ) is hypoelliptic but not elliptic. However, the operator for the wave equation (P(u) = u _{tt} - c ^{2\Delta} u ) P = \partial ^{2} _{t} - c ^{2 \Delta} _{x} (where c\ne 0 ) is not hypoelliptic.

References

• Book: Shimakura , Norio . Partial differential operators of elliptic type: translated by Norio Shimakura . American Mathematical Society, Providence, R.I . 1992 . 0-8218-4556-X .
• Book: Egorov , Yu. V. . Schulze, Bert-Wolfgang . Pseudo-differential operators, singularities, applications . Birkhäuser . 1997 . 3-7643-5484-4.
• Book: Vladimirov , V. S. . Methods of the theory of generalized functions . Taylor & Francis . 2002 . 0-415-27356-0 .
• Book: Folland , G. B. . Fourier Analysis and its applications . AMS . 2009 . 978-0-8218-4790-9.