Talk:Elliptic operator

Heat Equation
Reading this article one gets the ideia that the Heat equation is also an Eliptic pde (the page elliptic partial differential equation redirects here), when it is in fact a Parabolic equation. Not knowing if the mention to the Heat equation in this page is correct (that is contains an Elliptic operator), could someone clarify it please? Jlsilva 11:26, 27 September 2007 (UTC)

Discussion
There's nothing here about the properties of elliptic operators; it only covers the definitions. The page needs to review the basic properties of these operators, and say why these types of operators are so widespread and so useful. It also needs more links to specific examples of elliptic operators. Ewjw 09:07, 19 December 2005 (UTC)

Why Elliptic?
I agree with the comment above; unless I knew what an elliptic operator was I wouldn't gain anything from the current version of the page. In particular, why is it called "elliptic"? What is elliptic about it? &mdash;BenFrantzDale 22:11, 17 February 2006 (UTC)


 * Like ellipses, in this case. The criterion comes from the 'symbol'; this is easy to talk about in case you have a homogeneous polynomial P in the partial derivatives as operator, with constant coefficients. For example for the Laplacian you get a sum of squares quadratic form. The important thing is whether P is definite for real values, or indefinite. So, roughly speaking,


 * P(x,y) = ax2 + by2


 * gives rise to an elliptic operator, substituting partial derivative wrt x for x and so on, if and only if


 * P(x,y) = 0


 * is the equation of an ellipse. It is hyperbolic if a and b have opposite signs (like a wave equation). Charles Matthews 22:57, 17 February 2006 (UTC)

$$b_l$$ is not defined. What are these? Myrkkyhammas 21:53, 8 May 2007 (UTC)

Positive definite?
I'm fairly new to this area of maths and came to this page after reading some of the first chapter of "Riemmanian Holonomy Groups and Calibrated Geometry" by Joyce. His definition of elliptic is that $$a^{ij}\xi_{i}\xi_{j} \not= 0$$, so by continuity either positive definite or negative definite, not the claimed "iff positive definite" of this page. That seems to make sense since if P(u)=f is an expression of a positive definite equation -P(u)=-f is negative definite but just as valid.

Is this viable reasoning? All it needs is some mention of this choice in sign on the main page I think but it's a point which does need clarification. AlphaNumeric 20:06, 15 May 2007 (UTC)
 * In all the sources I ever saw (and there have been a few), the quadratic form of an elliptic operator is always positive definite. Of course, you can define things as in the book by Joyce you mention above, but I don't think that's common. Oleg Alexandrov (talk) 01:53, 16 May 2007 (UTC)

I dont understand this article
This material is new for me. I cant understand anything from this article (and i do have some small background in analysis). just for example: "
 * $$ P\phi = \sum_{k,j} a_{k j} D_k D_j \phi  + \sum_\ell b_\ell D_{\ell}\phi  +c \phi $$

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

this all passage is unclear to me, what is "i"? what is the meaning of ":$$ D_k D_j \phi $$"?

amit


 * I have changed i into $$\sqrt{-1}$$ to address one of your concerns. I'm not sure what the difficulty is with $$D_k D_j\phi$$.  This is just an iterated partial derivative. Silly rabbit 13:16, 7 July 2007 (UTC)


 * thanks silly rabbit,
 * - i think Leibniz's notation (: $$\frac{d}{dx}.$$) is clearer (more poplar) then Euler's notation, which i was not aware of.
 * - i actually needed to understand what is elliptic operator, when i enter this article. I'm sorry to say i still don't.
 * maybe I lack the proper background, but still, i could not figure out what this elliptic operator is all about.
 * alltough it was addresses above, i still not sure what "elliptic" about this operator. more important, what is special about it? how it is different ::from other operators types? in what way it is significant?
 * as i said before perhaps my background is not sufficient, but in this case i think more links should be added.
 * Amit man 19:53, 7 July 2007 (UTC)

Split
I propose that it be split into two subarticles. The first, Elliptic operator, will focus on linear elliptic operators. There is a school of mathematics where "elliptic operator" means precisely that, and the article shall make clear from the outset that it will deal exclusively with the linear case. The other article will be Elliptic partial differential equation, which will deal with potentially nonlinear elliptic equations, and the typical analytic behavior one expects from solutions of such equations. I think the two areas of study are sufficiently distinct from one another that this should be done for the sake of clarity. The elliptic operators of Atiyah and Singer do not closely resemble those of Monge and Ampère. Silly rabbit (talk) 12:32, 11 March 2008 (UTC)
 * I agree. Why hasnt this been split yet? 128.83.68.15 (talk) 18:42, 30 October 2008 (UTC)
 * Add my voice to the amens. I made the same observsation, here. 41.58.15.94 (talk) 22:04, 30 December 2012 (UTC)

Definition?
This article appears to lack a definition of "elliptic operator." —Preceding unsigned comment added by 129.97.134.34 (talk) 05:14, 8 July 2008 (UTC)


 * That's not actually true, if you look further down the page "Such an operator is called elliptic if for every x the matrix of coefficients of the highest order terms..." However, this is not really a very satisfying definition since it is not the most general linear elliptic operator, nor does it allow for the possibility of nonlinear ones.  The article needs to be completely rewritten, and it is on my to-do list, but at a regrettably low priority.   siℓℓy rabbit  (  talk  ) 05:19, 8 July 2008 (UTC)

Definition
I don't understand why there is an $$x$$ in the $$\sum{ a_{\alpha}(x) \xi^{\alpha}}$$. The symbol comes from taking Fourier Transforms and should not appear here. — Preceding unsigned comment added by Beowulf333 (talk • contribs) 14:40, 7 November 2011 (UTC)

Assessment comment
Substituted at 02:02, 5 May 2016 (UTC)