Differential operators | Partial differential equations

Hypoelliptic operator

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). In addition, the operator for the heat equation (where ) is hypoelliptic but not elliptic. However, the operator for the wave equation (where ) is not hypoelliptic. (Wikipedia).

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Related pages

Differential operator | Elliptic operator | Partial differential equation | Distribution (mathematics) | Heat equation | Wave equation | Mathematical analysis