According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs)".
I'd like to know if h-principle and theory from M. Gromov's "Partial Differential Relations" is a useful tool in the field of nonlinear PDE's.
What type of problems can be attacked using h-principle?
What type of results can be obtained?
Best Answer
Useful is a relative word.
This is easier to answer. Here's my favourite example of an h-principle due to Gromov and Lees, independently. (I can recommend Lees's paper for brevity (when weighed against Gromov's book). Sadly the only place I know where it's available is in Duke Maths Journal and hence not free).
Definition: An immersion $f: L\to\mathbf{C}^n$ of a closed $n$-manifold is called Lagrangian if $f^*\omega=0$ where $\omega$ is the standard symplectic 2-form $\sum_{i=1}^ndx_i\wedge dy_i$.
The condition that an immersion be Lagrangian is a (very flexible) nonlinear PDE. Flexibility here means roughly that there are many solutions (for example, if you take any compactly supported function $H$ on $\mathbf{C}^n$ you can use it to construct a Hamiltonian vector field $X_H$ on $\mathbf{C}^n$ satisfying $\omega(X_H,V)=dH(V)$ for any $V$ and pushing $f$ around using the flow of such a vector field will give you more Lagrangians). More pertinently, flexibility means that Lagrangian immersions satisfy an h-principle...
One might ask "How does the space of Lagrangian immersions sit inside the space of all smooth immersions?", but that would be the wrong question because a Lagrangian immersion has slightly more data than just the underlying smooth immersion. Namely, a Lagrangian immersion gives you a "Lagrangian Gauss map" which sends a point to the Lagrangian tangent space considered as a linear subspace in $\mathbf{C}^n$.
One might then ask "How does the space of Lagrangian immersions sit inside the space of all smooth immersions equipped with an abstract Lagrangian Gauss map?". By abstract Lagrangian Gauss map, I mean that there is a map of bundles $F:TL\to T\mathbf{C}^n$ which lives over $f$ and which sends tangent spaces to Lagrangian tangent spaces. Note that $F$ doesn't have to be $df$!
Now the h-principle tells you the answer to the second question: the space of Lagrangian immersions is a deformation retract of the space of smooth immersions with an abstract Lagrangian Gauss map!
Moreover, any given immersion can be approximated by a Lagrangian immersion which is arbitrarily close to it (in the sense that it lives in an arbitrarily small neighbourhood, however the tangent spaces will vary wildly).
Moremoreover, the Lagrangian Gauss map of the resulting immersion will be homotopic to the given abstract Gauss map.
Not every immersion can be given such a Gauss map, but it's a topological condition to check: it's equivalent to triviality of the complexified tangent bundle.
The same immersion can have different (non-homotopic) Lagrangian Gauss maps.
Here is a simple example. Note that you don't have to start with an immersion because you can always approximate something by an immersion. So start with the map sending $S^1$ to the origin into $\mathbf{C}$. (Oriented) Lagrangian subspaces of $\mathbf{C}$ are just (oriented) lines through the origin.The oriented Lagrangian Grassmannian is therefore $S^1$ and any map $S^1\to S^1$ will do as the Lagrangian Gauss map. Suppose you take the trivial map $S^1\to S^1$ sending everything to a point. A nearby Lagrangian immersion whose Gauss map is homotopic to this is the figure 8 immersion. If instead you take the degree 1 map $S^1\to S^1$ then a nearby Lagrangian immersion would be the inclusion of a small circle centred at 0. If you've met it before, the degree of this Gauss map is (half) the minimal Maslov number. In higher dimensions there are various cohomological/homotopic Maslov invariants because the space of Lagrangian subspaces is more complicated (it's the homogeneous space $U(n)/O(n)$), but the most important is the analogue of this one.
So the h-principle gives you a huge pool of solutions to your nonlinear PDE and tells you something about what they can look like.
This is made all the more useful by the fact that a generic Lagrangian immersion has at worst double points and one can surger these double points to obtain embedded Lagrangian submanifolds, which are and have long been a beautiful and mysterious class of objects. What is most mysterious about them is that they don't satisfy an h-principle, so we don't know how to construct/classify them. Indeed there are examples due to Luttinger of smoothly embedded tori in $\mathbf{C}^2$ which are not isotopic to Lagrangian tori.