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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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    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? – Daniel Fischer Jul 06 '14 at 12:37
  • @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 $\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha}$? –  Jul 06 '14 at 13:01
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    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. – Daniel Fischer Jul 06 '14 at 15:32
  • @DanielFischer Thanks for your response. If we represent the quadratic form in matrix form as $q(z) = z^{T}Mz$ where $z$ is column vector and $M$ is symmetric square matrix, then positive definiteness means that $q(z)$ is positive for all $z \neq 0$, how is this equivalent with $B^{2}-AC < 0$ as you stated? Also, if you were defining a general case (as you stated), would you not consider $\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha} > 0$ for all $\zeta \neq 0$ rather than $\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha} \neq 0$ for $\zeta \neq 0$ as stated? –  Jul 06 '14 at 18:04
  • The matrix in question is $$M = \begin{pmatrix} A & B\ B&C\end{pmatrix}.$$ $B^2 - AC < 0$ means $\det M > 0$. There is a criterion for positive definiteness - by Hurwitz or Sylvester, I forgot who - a symmetric matrix is positive definite if and only if all top-left square minors have positive determinant. For negative definiteness, you need a factor $(-1)^d$ for the $d\times d$ minor. So here $B^2 - AC < 0$, or $\det M > 0$ says the matrix is positive or negative definite, depending on the sign of $A$. If $\sum_{\lvert\alpha\rvert=m} a_\alpha(x)\zeta^\alpha\neq 0$ for all $\zeta\neq 0$, then – Daniel Fischer Jul 06 '14 at 18:11
  • the value has the same sign for all $\zeta\neq 0$ (since the form is continuous, and $\mathbb{R}^n\setminus {0}$ is connected [for $n > 1$], so if there were values of opposite sign, there would also be zero values), whether that is positive or negative is unimportant. – Daniel Fischer Jul 06 '14 at 18:13
  • @DanielFischer So could one say that ellipticity of a PDE is equivalent to the positive or negative definiteness of the quadratic form? If so what would parabolic and hyperbolic be equivalent to? Are you noting the continuity simply from observing that it is the sum of linear equations(which are obviously continuous)? –  Jul 06 '14 at 18:33
  • It's not a quadratic form anymore if the degree of the operator is greater than $2$. A parabolic operator/equation would correspond to a singular form (take the heat equation as the paradigmatic example, one coordinate is missing in the highest order), I'm not sure whether semidefiniteness is required for parabolic operators/equations, PDEs are not my cup of tea; and hyperbolic operators/equations have positive as well as negative eigenvalues (wave equation as the paradigmatic example), not sure whether the form is required to be non-degenerate there. – Daniel Fischer Jul 06 '14 at 18:40
  • @DanielFischer I haven't been studying it for very long but it seems quite dense. I know the generalization isn't a quadratic form anymore but I'm asking if for the case of $q(x,y) = Ax^{2}+2Bxy + Cy^{2}$ you could say that the ellipticity of the second order pde is equivalent to saying that the quadratic form associated with the pde is either positive or negative definite? –  Jul 06 '14 at 18:49
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    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. – Daniel Fischer Jul 06 '14 at 18:58
  • @DanielFischer Kwl thanks. Could I ask one more question, how is connectedness on $\mathbb{R}^{n}\setminus {0}$ used to show that $\sum_{|\alpha|=m}a_{\alpha}(x)\zeta^{\alpha} = 0$ at some point if it changes sign? –  Jul 06 '14 at 19:27
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    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. – Daniel Fischer Jul 06 '14 at 19:31
  • @DanielFischer Yes I understand, thanks for the discussion, very helpful. –  Jul 06 '14 at 19:33

1 Answers1

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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.