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A geometric way of looking at differential equations

In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-principle), we often see the following (all objects smooth):

Give a fibre bundle $\pi:F\to M$ over some manifold $M$, denote by $F^{(r)}$ the associated $r$-jet bundle. A partial differential relation $\mathcal{R}$ is a subset of $F^{(r)}$.

$\mathcal{R}$ is said to be a partial differential equation if it is a submanifold of $F^{(r)}$ with codimension $\geq 1$.

A solution $\Phi$ to the relation $\mathcal{R}$ is a holonomic (in the sense that $\Phi = j^r\phi$ for some section $\phi$ of $F$) section of $F^{(r)}$ that lies in $\mathcal{R}$.

This description is very powerful in the context of the topologically motivated techniques of the h-principle. And for partial differential inequalities where $\mathcal{R}$ is an open submanifold, this allows the convenient setting for the Holonomic Approximation Theorem.

A geometric way of looking at differential operators

Here I recall the famous theorem of Peetre.

Let $E\to M$ and $F\to M$ be two vector bundles. Let $D$ be a linear operator mapping sections of $E$ to sections of $F$. Suppose $D$ is support-non-increasing, then $D$ is (locally) a linear differential operator.

where

A linear differential operator $D:\Gamma E\to\Gamma F$ is a composition $D := i\circ j$ where $j: E\to J^RE$ is the $R$-jet operator, and $i: J^RE \to F$ is a linear map of vector bundles.

Most of the time in applications, $F$ can be taken to be the same bundle as $E$. This way of phrasing things is also convenient for analysis. For example, we can easily define the principal symbol of a linear differential operator in the following way.

Let $D_1$ and $D_2$ be two linear differential operators of order $r$ (that is, $r$ is the smallest natural number such that if $j^r\phi = j^r\psi$, then $D\phi = D\psi$). Let $\pi^r_{r-1}: J^rE \to J^{r-1}E$ the natural projection. We say that $D_1$ and $D_2$ have the same principal part if their corresponding $i_1 - i_2|_{\ker \pi^{r}_{r-1}} \equiv 0$. (In words, their difference is a l.d.o. of lower order. This defines an equivalence relation on linear differential operators of order $r$. Each equivalence class defines a unique linear map of vector bundles $P: \ker \pi^{r}_{r-1} \to F$.

Now, it is known (sec 12.10 in Kolar, Michor, Slovak's Natural operations...) that $\ker \pi^{r}_{r-1}$ is canonically isomorphic to $E\otimes S^r(T^*M)$, where $S^r$ denotes the $r$-fold symmetric tensor product of a space with itself. So we have naturally an interpretation of $P$ as a section of $F\otimes E^* \otimes S^r(TM)$. In the case where $E$ and $F$ are just, say, the trivial $\mathbb{R}$ bundle over $M$, we recover the usual description of the principle symbol of a pseudo-differential operator being (fibre-wise) a homogeneous polynomial of the cotangent space.

Question

Given the above, another way of looking at partial differential equations is perhaps the following.

Let $\pi_X: X\to M$ and $\pi_Y: Y\to M$ be fibre bundles. A system of partial differential operators of order $r$ is defined to be a map $H: J^rX \to Y$ that commutes with projection $\pi_X \circ \pi^r_0 = \pi_Y \circ H$. And a system of partial differential equations is just the statement $H(j^r\phi) = \psi$, where $\phi\in\Gamma X$ and $\psi \in\Gamma Y$. Observe that by considering $H^{-1}(\psi) \subset J^rX$, we clearly have a partial differential relation the sense defined in the first section. If we require that $H$ is a submersion, then $H^{-1}(\psi)$ is also a partial differential equation in the sense defined before.

On the other hand, if $\mathcal{R}$ is an embedded submanifold of $J^rX$, at least locally $\mathcal{R}$ can be written as the pre-image of a point of a submersion; though there may be problems making this a global description. So perhaps my definition is in fact more restrictive than the one given in the first part of this discussion.

My question is: is this last "definition" discussed anywhere in the literature? Perhaps with its pros and cons versus the partial differential relations definition given in the first part of the question? I am especially interested in references taking a more PDE (as opposed to differential geometry) point of view, but any reference at all would be welcome.

Note: For the reference-request part of this question, I would also appreciate pointers to whom I can ask/e-mail on these matters.

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Peetre's theorem can only be stated this way for compact manifolds. For non-compact manifolds you have to allow differential operators of (globally) infinite order (locally they still have finite order). For example, you can glue a sequence of differential operators with increasing orders into one differential operator using a partition of unity. –  Dmitri Pavlov Sep 28 '11 at 12:51
    
Could you say why you're interested in such a formal setup for PDE's? –  Deane Yang Sep 28 '11 at 14:18
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Assuming he has time, Robert Bryant is the best person to answer your question. You might want to look at the book of Bryant, Chern, Gardner, Goldschmidt, and Griffiths. Although the focus on the book is using exterior differential systems (which are equivalent to systems of PDE's as you define above), Goldschmidt approaches the subject from the jet bundle point of view. –  Deane Yang Sep 28 '11 at 14:25
    
@Deane: well, it is a bit embarrassing. In what I am working on at the moment this will be formal set-up that will end up being ¨tossed away¨; so at the moment I am just trying to cover my behind by making sure I am not saying anything irresponsible. :-) (Though this may become genuinely the focus later.) As to why I need this formal set-up: I want to get a good notion of a ¨quasilinear¨ partial differential equation independent of the choice of connection, and to define its linearisation and its linearised principal symbol. –  Willie Wong Sep 28 '11 at 15:23
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For first and second order systems one can do what I want without referring to all this abstract nonsense. But as far as I can work out this point of view is necessary if I want to deal with higher order systems of equations. –  Willie Wong Sep 28 '11 at 15:24

1 Answer 1

up vote 6 down vote accepted

If $\mathcal{R}\subset J^rX$ is closed, then there's a smooth function $f:J^rX\to\mathbb R$ with $\mathcal{R}=f^{-1}(0)$. So you can construct a differential operator $H:J^kX\to M\times \mathbb{R}$ by $H(\theta):=(\pi_X(\pi^r_0(\theta)),f(\theta))$ and the equation $\mathcal{R}$ will be given by $H(j^r\phi)=0$.

So there is no big difference between the two definitions. If you are only interested in the "space" of solutions of the differential equation, then i'd say that the set $\mathcal{R}$ is enough, or put differently, you could choose the differential operator which suits you best to represent the equation.

Edit In response to Willies comment: Here's a counterexample to what you are asking for: recall that there's no submersion from $\mathbb{RP}^2$ to something, which has $\mathbb{RP}^1\subset \mathbb{RP}^2$ as a fiber. So take $M=\mathbb{R}$, $X=M\times \mathbb{RP}^2$ and $\mathcal{R}=M\times \mathbb{RP}^1\subset X$. Then there's no fiber bundle $Y\to M$ allowing a submersion $H:X\to Y$ with $\mathcal{R}=H^{-1}(\psi)$ for any $\psi\in \Gamma(Y)$. This is probably a silly example since the PDE is of order zero, but I'm sure one can come up with examples in higher order.

Anyway: if what you are interested in is an intrinsic notion of overdeterminedness of a PDE you might want to take a look at Bryant and Griffiths Characteristic Cohomology of Differential systems. Roughly the codimension of the characteristic variety serves as such a measure. And the characteristic variety can be defined completely without referring to an operator describing the equation. As Deane says, much of this can be found in the book Exterior differential systems. There are also the books by Vinogradov, Krasil'shchik and Lychagin.

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Of course. I wasn't clear about it in my question (sorry!), but of particular interest would also be the issue of when $H$ can be taken to be a submersion. In some sense the submersion requirement would be a good one, since for "dimension counting" purposes (whether an equation is over/under/just-okay-determined) it is usually a good idea for the equation to be defined by $\mathcal{R}$ a submanifold, not just an arbitrary closed set. –  Willie Wong Sep 28 '11 at 12:37
    
I'm accepting this answer because, as pointed out by Michael, the issue is mostly topological. Deane's comments also helped greatly. In particular, Goldschmidt's 1967 JDG paper "Integrability Criteria..." mentioned the same issue, namely that the topological condition from the point of view of a submersion maybe too restrictive (see the paragraph before Defn. 7.2). –  Willie Wong Sep 29 '11 at 13:26

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