NationMaster - Encyclopedia: Elliptic operator

In mathematics, an elliptic operator is one of the major types of differential operator P. It can also 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. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

An important example of an elliptic operator is the Laplacian. Equations of the form In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$P u = 0 quad$

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 spacial variables, as well 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. The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... Potential theory may be defined as the study of harmonic functions. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

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

Second order operators

For expository purposes, we consider initially second order linear partial differential operators of the form In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

$Pphi = sum_{k,j} a_{k j} D_k D_j phi + sum_ell b_ell D_{ell}phi +c phi$

where $D_k = frac{1}{i} partial_{x_k}$ . Such an operator is called elliptic iff for every x the matrix of coefficients of the highest order terms IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

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

is a positive-definite real symmetric matrix. In particular, for every non-zero vector In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

$vec{&# 0; = (&# 0;1, &# 0;2, ldots , &# 0;n)$

the following inequality holds:

$sum_{k,j} a_{k j}(x) &# 0;k &# 0;j > 0. quad$

Example. The negative of the Laplacian in Rⁿ given by In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

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

is an elliptic operator.

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.

Categories: Differential operators | Partial differential equations

Results from FactBites:

Elliptic curves (746 words)
	On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian D. Gonzalez-Aviles.
	Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
	On an elliptic analogue of Zagier's conjecture, Jörg Wildeshaus.

Elliptic operator - Wikipedia, the free encyclopedia (245 words)
	In mathematics, an elliptic operator is one of the major types of differential operator P.
	An important example of an elliptic operator is the Laplacian.
	Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous).

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Second order operators GA_googleFillSlot("encyclopedia_square");

See also

References

External links

Second order operators