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I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts:

An elliptic partial differential equation is given as being a second-order partial differential equation of the form $$Au_{xx} + 2Bu_{xy} + Cu_{yy}+Du_{x} + Eu_{y} + F = 0$$ that satisfies the condition $B^{2}-AC < 0$. The classification seems to be connected with conic sections.

And then there's the definition of an elliptic operator which is defined as a linear differential operator $L$ of order $m$ on a domain $\Omega$ in $\mathbb{R}^{d}$ given by $$Lu = \sum_{|\alpha| \leq m}a_{\alpha}(x)\partial^{\alpha}u$$ (where $\alpha$ is a multi-index) is called elliptic if for every $x$ in $\Omega$ and every non-zero $\zeta$ in $\mathbb{R}^{d}$ $$\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha} \neq 0$$

I just have a couple of questions about these concepts? Firstly, why are PDE's classified in this way where it relates to conic sections?(elliptic, parabolic,hyperbolic) Secondly, what is the connection between elliptic partial differential equations and elliptic operators? I thought that an elliptic operator would be an elliptic PDE in operator form, in the sense that say $x-y=0$ was an elliptic PDE then $f(x,y) = x-y$ would be an elliptic operator. But it seems that there is no connection between elliptic operators and elliptic PDE's?

Thanks for any help.

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    $\begingroup$ For the elliptic operator $L$, take $d = m = 2$, and suppose for simplicity that the $a_\alpha$ are all constant. Do you see a relation then? $\endgroup$ – Daniel Fischer Jul 6 '14 at 12:37
  • $\begingroup$ @DanielFischer Having difficulty seeing what you are hinting at? Given the conditions you gave are you implying a relation between $B^{2}-AC < 0$ and $\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha}$? $\endgroup$ – user116403 Jul 6 '14 at 13:01
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    $\begingroup$ Yes. If you look at the quadratic form $Ax^2 + 2Bxy + Cy^2$, the condition $B^2-AC < 0$ means it is positive (or negative, if $A < 0$) definite, so the quadrics $Ax^2 + 2Bxy + Cy^2 = k$ are ellipses. In the general case, the condition $\sum_{\lvert\alpha\rvert = m} a_\alpha(x)\zeta^\alpha \neq 0$ for $\zeta\neq 0$ means for $m = 2$ that the level sets of the bilinear form are ellipsoids. For general $m$, it's a generalisation. An elliptic PDE is one of the form $Lu = 0$, where $L$ is an elliptic differential operator. $\endgroup$ – Daniel Fischer Jul 6 '14 at 15:32
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    $\begingroup$ Yes, the ellipticity is equivalent to the (positive or negative) definiteness of the associated quadratic form. And for a second order PDE in $\mathbb{R}^2$, there is no doubt, parabolicity is equivalent to the quadratic form having rank $1$ (so it's semidefinite), and hyperbolicity to it being indefinite. The level curves of the quadratic form then are ellipses, parabolas, or hyperbolas, respectively. $\endgroup$ – Daniel Fischer Jul 6 '14 at 18:58
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    $\begingroup$ The image of a connected set under a continuous map is connected. The only connected subsets of $\mathbb{R}$ are the intervals (possibly empty or degenerate, $[a,a]$). If an interval contains positive as well as negative numbers, it must also contain $0$. For the case where the domain is an interval, you know the special case as the intermediate value theorem. $\endgroup$ – Daniel Fischer Jul 6 '14 at 19:31
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Why are PDE's classified in this way where it relates to conic sections?

They are not. The parallel with conic sections is an artifact of second-order PDE in two dimensions. It is not a classification of PDE in general, as one quickly discovers when encountering higher order equations and higher dimensions. The properties recognized in two dimensions can be usefully identified in other settings (e.g., hyperbolic equations preserve singularities of initial data, while elliptic/parabolic smoothen them out...) but they do not form a "classification".

I'll quote from the beginning of PDE textbook by Lawrence C. Evans:

Many texts describe PDE as if functions of the two variables $(x,y)$ or $(x,t)$ were all that matter. [...] I also find it unsatisfactory to "classify" partial differential equations: this is possible in two variables, but creates the false impression that there is some kind of general and useful classification scheme...

Your other question:

what is the connection between elliptic partial differential equations and elliptic operators?

An operator is something that takes a function and produces another function. An equation is what you get by equating the output of an operator to a known function. That it, $Lu=g$ where $u$ is unknown function, $L$ is a differential operator, and $g$ is a known function, sometimes called the source term.

The notion of ellipticity of an equation depends only on $L$, not on $g$. So, an equation is elliptic if $L$ satisfies the definition of an elliptic operator. There are numerous inequivalent definitions of what ellipticity means, most of which have nothing to do with conic sections. Ellipticity is just a word, like "regularity". Its meaning is to be obtained from context.

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  • $\begingroup$ Thanks for your response. Very helpful. $\endgroup$ – user116403 Jul 7 '14 at 20:01

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