Elliptic operator

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In mathematics, an elliptic operator is one of the major types of differential operator. 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

$P u = 0 \quad$

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.

1 Regularity properties
2 Second order operators
3 See also
4 References
5 External links

[edit] Regularity properties

Let P be an elliptic operator with infinitely differentiable coefficients. If f = P u is infinitely differentiable, then so is u. This is a very special case of a regularity theorem for a differential equation.

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.

[edit] Second order operators

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

$P\phi = \sum_{k,j} a_{k j} D_k D_j \phi + \sum_\ell b_\ell D_{\ell}\phi +c \phi$

where $D_k = \frac{1}{\sqrt{-1}} \partial_{x_k}$ . Such an operator is called elliptic if for every x the matrix of coefficients of the highest order terms

$\begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \cdots & a_{1 n}(x) \\ a_{2 1}(x) & a_{2 2}(x) & \cdots & a_{2 n}(x) \\ \vdots & \vdots & \vdots & \vdots \\ a_{n 1}(x) & a_{n 2}(x) & \cdots & a_{n n}(x) \end{bmatrix}$