Elliptic operator

In mathematics, an elliptic operator is one of the major types of differential operator P. It can be defined on spaces of complex-valued functions, or some more general function-like objects. What is distinctive is that the coefficients of the highest-order derivatives satisfy a positivity condition.

An important example of an elliptic operator is the Laplacian. Equations of the form

$Pu=0\quad$

are called elliptic partial differential equations if P is an elliptic operator. The usual partial differential equations involving time, such as the heat equation and the Schrödinger equation, also contain elliptic operators involving the spatial variables, as well as time derivatives. Elliptic operators are typical of potential theory. Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous). More simply, steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations.

Properties

Let $P$ be an elliptic operator with constant coefficients.

• If $Pu=f$ and $f$ is infinitely differentiable, then so is $u$. Any differential operator (with constant coefficients) enjoying this property is called a hypoelliptic operator; thus, every elliptic operator is hypoelliptic. The property also means that every fundamental solution of an elliptic operator is infinitely differentiable off 0.

Second order operators

For expository purposes, we consider initially second order linear partial differential operators of the form

$P\phi =\sum _{k,j}a_{kj}D_{k}D_{j}\phi +\sum _{\ell }b_{\ell }D_{\ell }\phi +c\phi$

where $D_{k}=-i\partial _{x_{k}}$. Such an operator is called elliptic if for every x the matrix of coefficients of the highest order terms

${\begin{bmatrix}a_{11}(x)&a_{12}(x)&\cdots &a_{1n}(x)\\a_{21}(x)&a_{22}(x)&\cdots &a_{2n}(x)\\\vdots &\vdots &\vdots &\vdots \\a_{n1}(x)&a_{n2}(x)&\cdots &a_{nn}(x)\end{bmatrix}}$

is a positive-definite real symmetric matrix. In particular, for every non-zero vector

${\vec {\xi }}=(\xi _{1},\xi _{2},\ldots ,\xi _{n})$

the following ellipticity condition holds:

$\sum _{k,j}a_{kj}(x)\xi _{k}\xi _{j}>0.\quad$

In many applications, this condition is not strong enough, and instead a uniform ellipticity condition must be used:

$\sum _{k,j}a_{k,j}(x)\xi _{k}\xi _{j}>C|\xi |^{2}\,$

where C is a positive constant.

Example. The negative of the Laplacian in Rⁿ given by

$-\Delta =\sum _{\ell =1}^{n}D_{\ell }^{2}$

is a uniformly elliptic operator.

See also

• Elliptic complex
• Hyperbolic partial differential equation
• Ultrahyperbolic wave equation
• Hypoelliptic operator
• Parabolic partial differential equation
• Semi-elliptic operator
• Weyl's lemma

References

• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2

• D. Gilbarg and Neil Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. ISBN 3-540-41160-7

External links

• Linear Elliptic Equations at EqWorld: The World of Mathematical Equations.

• Nonlinear Elliptic Equations at EqWorld: The World of Mathematical Equations.