Английская Википедия:Hypoelliptic operator

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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

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book