Hypoelliptic operator

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In mathematics, more specifically 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^\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). The heat equation operator

P(u)=u_t - k\Delta u\,

(where k>0) is hypoelliptic but not elliptic. The wave equation operator

P(u)=u_{tt} - c^2\Delta u\,

(where c\ne 0) is not hypoelliptic.

References[edit]

  • 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 0-8218-4790-2. 

This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.