Hypoelliptic operator

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

: $P= \partial^2_t - c^2\,\Delta_x\,$

(where $c\ne 0$ ) is not hypoelliptic.

References

{{cite book

| last = Shimakura

| first = Norio

| title = Partial differential operators of elliptic type: translated by Norio Shimakura

| publisher = American Mathematical Society, Providence, R.I

| date = 1992

| pages =

| isbn = 0-8218-4556-X

}}

{{cite book

| last = Egorov

| first = Yu. V.

|author2=Schulze, Bert-Wolfgang

| title = Pseudo-differential operators, singularities, applications

| publisher = Birkhäuser

| date = 1997

| pages =

| isbn = 3-7643-5484-4

}}

{{cite book

| last = Vladimirov

| first = V. S.

| title = Methods of the theory of generalized functions

| publisher = Taylor & Francis

| date = 2002

| pages =

| isbn = 0-415-27356-0

}}

{{cite book

| last = Folland

| first = G. B.

| title = Fourier Analysis and its applications

| publisher = AMS

| date = 2009

| pages =

| isbn = 978-0-8218-4790-9

}}

Category:Partial differential equations

Category:Differential operators