In the theory of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s, a partial

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

P

defined on an

open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...

U \subset^n

is called hypoelliptic if for every

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

u

defined on an open subset

V \subset U

such that

Pu

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 In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

with

C^\infty

coefficients is hypoelliptic. In particular, the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the

heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...

(

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 The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...

(

P(u)=u_ - c^2\,\Delta u\,

) :

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

(where

c\ne 0

) is not hypoelliptic.

References

* * * * {{PlanetMath attribution, id=8059, title=Hypoelliptic Partial differential equations Differential operators