elliptic regularity

In the theory of partial differential equations, a partial differential operator $P$ defined on an open subset

:$U \subset^n$

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). 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 ($P(u)=u_ - c^2\,\Delta u\,$)
:$P= \partial^2_t - c^2\,\Delta_x\,$
(where $c\ne 0$) is not hypoelliptic.

References

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{{PlanetMath attribution, id=8059, title=Hypoelliptic

Partial differential equations
Differential operators