Partial differential equations Definition

Hypoelliptic operator

Builds on Smooth function Elliptic operator

Definition

Let $\Omega \subseteq \R^n$ be an open set and let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \partial^\alpha$ be a linear differential operator on $\Omega$ whose coefficients $a_\alpha$ are smooth functions. A distribution on an open set $U$ is a continuous linear functional on the space of test functions on $U$ . The operator $P$ is hypoelliptic if the following holds for every open set $U \subseteq \Omega$ and every distribution $u$ on $U$ . If $Pu$ is a smooth function on $U$ , then $u$ is a smooth function on $U$ .

Remarks

Equivalently, $\operatorname{sing\,supp} u = \operatorname{sing\,supp} Pu$ for every distribution $u$ , where the singular support is the complement of the largest open set on which the distribution is smooth. Example: every elliptic operator with smooth coefficients is hypoelliptic; this is elliptic regularity. The heat operator $\partial_t - \Delta$ on $\R^{n+1}$ is hypoelliptic but not elliptic. Non-example: the wave operator $\partial_t^2 - \partial_x^2$ on $\R^2$ is not hypoelliptic. For the Heaviside function $H$ , the distribution $u(x, t) = H(x - t)$ satisfies $\partial_t^2 u - \partial_x^2 u = H''(x - t) - H''(x - t) = 0$ , which is smooth, but $u$ is not smooth.

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Tag	`SKS1JPBW2N`
URI	`m2://SKS1JPBW2N`
Kind	definition
Writing	W0 stub
MSC	35H10
Defines	Hypoelliptic operator
Uses	Smooth function, Elliptic operator

Hypoelliptic operator

Definition

Remarks

Reading path

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