Elliptic operator

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http://en.wikipedia.org/wiki/Elliptic_partial_differential_equation

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.

Example: 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

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

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:

$\sum_{k,j} a_{k j}(x) \xi_k \xi_j > 0. \quad$

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 R given by

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

is a uniformly elliptic operator.

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) exhibiting 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.

As an application, suppose a function f satisfies the Cauchy-Riemann equations. Since f then satisfies the Laplacian in particular and since the Laplacian is an elliptic operator, it follows that f is infinitely differentiable.

General Definition

Let D be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its principal symbol σξ(D) with respect to a one-form ξ. (Basically, what we are doing is replacing the highest order covariant derivatives $\nabla$ by vector fields ξ.)

We say D is weakly elliptic if σξ(D) is a linear isomorphism for every ξ.

We say D is (uniformly) strongly elliptic if for some constant c > 0,

$([\sigma_\xi(D)](v),v) \geq c\|v\|^2$

for all $\|\xi\|=1$ and all v. It is important to note that the definition of ellipticity in the previous part of the article is strong ellipticity. Here $(\cdot,\cdot)$ is an inner product. Notice that the ξ are covector fields or one-forms, but the v are elements of the vector bundle upon which D acts.

The quintessential example of a (strongly) elliptic operator is the Laplacian (or its negative, depending upon convention). It is not hard to see that D needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both ξ and its negative. On the other hand, a weakly elliptic first-order operator, such as the Dirac operator can square to become a strongly elliptic operator, such as the Laplacian. The composition of weakly elliptic operators is weakly elliptic.

Weak ellipticity is nevertheless strong enough for the Fredholm Alternative, Schauder estimates, and the Atiyah-Singer Index Theorem. On the other hand, we need strong ellipticity for the Maximum Principle, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.

Published - July 2009