Jet bundles and partial differential operators

Question

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.

Answer 1

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.