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I have been helping a student identify whether PDEs are hyperbolic, parabolic or elliptical, and we can do the calculation, but I don't know what that does for us. What is that knowledge good for? What is the point?

It seems those terms "elliptic", "parabolic" and "hyperbolic" are used in a lot of contexts, and I am wondering what those contexts have in common. Why are those names used?

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    $\begingroup$ Because different techniques are available for different types of PDE. $\endgroup$ Oct 13, 2021 at 17:05
  • $\begingroup$ In addition to what @ArcticChar points out, the idea of a well- or correctly-posed problem—including considerations of function spaces, appropriate boundary/initial conditions and so forth—will be different. $\endgroup$ Oct 14, 2021 at 0:54
  • $\begingroup$ Related: math.stackexchange.com/q/4294133/169085 $\endgroup$
    – Alp Uzman
    May 7, 2022 at 10:02

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I'll take a stab at some of this. I use ellipticity in examples in some places, but this could be replaced with the other two for the most part.

First, I argue that words like elliptic, parabolic, and hyperbolic are used in common discourse by analysts to describe equations or phenomena via implicit analogy, and that analogy is how we think about PDE most of the time.

The truth is that we do not understand PDE very well. We barely understand the simplest examples: there are people out there right now trying to understand harmonic functions better, and harmonic functions have been studied for centuries. If you give me an arbitrary PDE, the odds of me deducing something interesting about it directly are extremely small. This isn't like with ODE, where you interpret it as a dynamical system and maybe it's complicated but at least you know more or less what to ask and to look at. With PDE, it's not clear what the right questions are to ask. Maybe you stare at it and notice that it has some trivial solution, like $0$: is this meaningful information or a random accident? Maybe it has symmetries or invariances... but how do you tell? We know from experience that maybe it makes sense to assign boundary conditions... but where, and on what, and how?

What we have going for us is experience. I have spent many (too many?) hours of my life thinking about harmonic functions, solutions to $\Delta u = 0$. I understand very little about them, but I can at least ask questions about them that turn out to be interesting. I also have some mental picture of what solutions are doing, and some ways to support this mental picture with "sophisticated PDE machinery" like the maximum principle. So when you give me some weird PDE, it's reasonable for me to ask: is this like the Laplace equation, where I can maybe ask some of my favorite questions and bring my (inadequate) arsenal of tools to bear on it?

For example, look at this PDE: $\sum_{ij}\partial_{ij}u \partial_i u \partial_j u = \nabla u \cdot D^2u \nabla u = 0$. I don't really know whether this has enough resemblance to the Laplace equation for my knowledge of harmonic functions to be relevant, but I can try and check. Does it have a maximum principle? (yes) Can I solve the Dirichlet problem? (yes) Can I prove some regularity properties of solutions? (some, not many) Does it have some analogue to Dirichlet's principle? (no) Does it have a mean value property? (not that I'm aware of) And so on. After I study this for long enough, I might reasonably decide that, yes, the analogy to the Laplace equation here is a powerful one, and the word I use to capture that idea is "elliptic."

[I don't feel like you should be able to classify this equation as elliptic easily, for the record; I have even seen prominent mathematicians classify it as parabolic (and not entirely ironically).]

The moral here is, if someone tells you that an equation is elliptic, what they are saying is that there are important similarities between its structure and the behavior of its solutions and the Laplace equation. They may say something very precise as well (it's elliptic in the sense of Hörmander, it's elliptic in the sense of Agmon-Douglis-Nirenberg, it's a second order fully nonlinear uniformly elliptic equation, etc.), but those are best thought of as criteria that some PDE experts felt are sufficient for a good analogy to the Laplace equation.

Three corollaries of this:

  • There is no single particular thing "ellipticity" tells you, when used in this broad sense. "Elliptic equations are ones with a maximum principle!" Except for those elliptic systems that don't have one ... while various first-order equations do, though no one would ever call them elliptic. "Elliptic equations solve elliptic boundary value problems!" Phrasing this in a way which is not tautological is more difficult than you might think: what are the "elliptic" boundary value problems for $\Delta^2 u = 0$? We might say that most of the time elliptic equations have well-posed boundary value problems with data on the entire boundary of a domain, while parabolic and hyperbolic equations might not ... but that's a pretty weak statement. Etc.
  • It's not very useful to think of these descriptors (or others like dispersive, transport, and so on) as a grand classification of PDE. You see this with things like the infinity Laplace equation above, and you see this when your class of hyperbolic or parabolic equations in your classification starts including things which defy good analogies to simpler model equations.
  • The better your understanding of the model equation, the more relevance the descriptors will have to you. Saying an equation is like the Laplace equation is pointless unless you know a decent amount about harmonic functions and have a good mental picture of how you expect them to behave. This is also relevant to the OP: "I have been helping a student identify whether PDEs are hyperbolic, parabolic or elliptical." Yes, this will not be easy to motivate unless the student knows something about some equation of that type. On the bright side, students might only deal with more limited collections of equations, for which the analogies are stronger (and where maybe we can make reasonable generalizations, like "Second-order elliptic equations of the kind we saw in class have well-posed Dirichlet problems on nice domains").

Finally, as far as the names: if you try classifying second-order constant coefficient differential equations of two variables, you notice that the characteristic surfaces (which essentially determine which model equation you are most like in the strongest possible sense, linear isomorphism of the underlying space) are conic sections, and these are the possible conic sections (well, the nondegenerate ones, at any rate). I imagine in other places where analogies to conic sections or quadratic forms are appropriate, you'll see similar terminology. The connection between these actual conics and solutions to the equations is not very meaningful in general.

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