What is the geometric/algebraic representation of a (partial) differential relation?

Question

My first idea is a kind of equation that may be an inequality too(?) such as:

$$ F(x,u(x),\partial u(x), \partial^2u(x),...)\leq 0 $$

In Gromov's book he generalizes PDEs into partial differential relations. But what exactly is the meaning of these?

In this summary of it he sets them up this way:

Let $p:X\to V $ be a smooth fibration, and let $X^{(r)}$ be the space or $r$-jets of smooth sections of $p$. A section $\phi$ of the bundle $X^{(r)} \to V$ is called holonomic if it is the $r$-jet of a section of $p$. A differential relation $ \mathcal{R} $ of order $r$ imposed on sections of $p$ is a subset $\mathcal{R} \subset X^{(r)}$. A section $f$ of $p$ is a solution of $\mathcal{R}$ if the $r$-jet of $f$ takes values in $\mathcal{R}$.

I can't really do much with this, this definition feels really abstract to me.

In these lecture papers the author defines PDRs the following way:

Let $M$ be a smooth $n$-manifold. Let $E$ be a smooth fibration over $M$, $E \to M$. The space of $k$-jets of the bundle $E$, $J_kE$, is the sections of a bundle $E^{(k)}$ over $M$ whose fiber $J_kE\vert_x$ at a point $x \in M$ is the space of smooth sections of $E$ in a neighborhood of $x$ modulo the equivalence relation that $f \sim g$ if they agree to order k in a neighborhood of $x$ (i.e., if the first $k$ derivatives of $f-g$ vanish when restricted to some arbitrarily small $\mathbb{R}^n \cong U \subset M$ containing $x$). Note that there is a canonical map $j^{(k)}: \Gamma(E)\to J_kE$ from sections of $E$ to $k$-jets of sections of $E$.

Defintion 2.1. A differential relation $\mathcal{R}$ of order $k$ is a subspace of $E^{(k)}$. The space of (holonomic) solutions Sol$_{\mathcal{R}}$ of $\mathcal{R}$ is the image $\Gamma(E)$ in $\Gamma(\mathcal{R})$, i.e., the sections of $\mathcal{R}$ which are $k$-jets of actual section of $E$.

These are quite similar definitions, still i find myself wondering what exactly i am reading.

What is the geometric(visual)/algebraic representation of this? How could i explain this in simpler terms?

Answer 1

I found the answer in BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 3, July 2012, Pages 447–453 - SELECTED MATHEMATICAL REVIEWS

...partial differential relations ... are, for the most part, under-determined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. [These] alluded to above are usually either equations or inequalities.

Classical PDEs of order $n$ look like this: $$ F(x,u(x),\partial u(x), \partial^2u(x),...,\partial^nu(x))= 0 $$

And as their domain is in a $n-1$ dimensional hypersurface, their solution is too.

A PDR on the other hand is way more under-determined and covers the entire $n$-hypervolume for

$$ F(x,u(x),\partial u(x), \partial^2u(x),...\partial^nu(x))\leq 0 $$

In the example it can take up any negative point on the $x$-axis probably. And this has way more solutions too. This is what the author meant i think by "dense in spaces of solutions".