Hypoelliptic operator

In mathematics, more specifically in the theory of partial differential equations, a partial differential operator defined on an open subset

is called hypoelliptic if for every distribution defined on an open subset such that is (smooth), must also be .

If this assertion holds with replaced by real analytic, then is said to be analytically hypoelliptic.

Every elliptic operator with coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator

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

(where ) is not hypoelliptic.


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