Английская Википедия:Hypoelliptic operator
In the theory of partial differential equations, a partial differential operator <math>P</math> defined on an open subset
- <math>U \subset{\mathbb{R}}^n</math>
is called hypoelliptic if for every distribution <math>u</math> defined on an open subset <math>V \subset U</math> such that <math>Pu</math> is <math>C^\infty</math> (smooth), <math>u</math> must also be <math>C^\infty</math>.
If this assertion holds with <math>C^\infty</math> replaced by real-analytic, then <math>P</math> is said to be analytically hypoelliptic.
Every elliptic operator with <math>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>P(u)=u_t - k\,\Delta u\,</math>)
- <math>P= \partial_t - k\,\Delta_x\,</math>
(where <math>k>0</math>) is hypoelliptic but not elliptic. However, the operator for the wave equation (<math>P(u)=u_{tt} - c^2\,\Delta u\,</math>)
- <math> P= \partial^2_t - c^2\,\Delta_x\,</math>
(where <math>c\ne 0</math>) is not hypoelliptic.
References
