Elliptic operator

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.

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.

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

Second order operators

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

:$$Pphi = 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

:$$egin{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}

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^"n" given by

:$$ - Delta = sum_{ell=1}^n D_ell^2

is a uniformly elliptic operator.

See also

* Hopf maximum principle
* Elliptic complex
* Hyperbolic partial differential equation
* Ultrahyperbolic wave equation
* Parabolic partial differential equation
* Semi-elliptic operator
* Weyl's lemma

References

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External links

* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc3.pdf Linear Elliptic Equations] at EqWorld: The World of Mathematical Equations.

* [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc3.pdf Nonlinear Elliptic Equations] at EqWorld: The World of Mathematical Equations.