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?
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.
Useful is a relative word.
What type of problems can be attacked using h-principle? What type of results can be obtained?
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.
You asked about the $h$-principle, but I'll say something about convex integration instead.
Here is a survey by DeLellis and Szekelyhidi about instances of the "h-Principle" where convex integration is used to construct low regularity solutions to many equations of fluid mechanics:
http://arxiv.org/abs/1111.2700
These analytic results, however, are different in flavor to what you usually call the $h$-principle in topology and geometry. In topology a nontrivial instance of the $h$-principle might say something like "you can invert the sphere $S^2 \subseteq {\mathbb R}^3$ through a regular family of immersions"; what makes it non-trivial is that there could have been a topological obstruction to doing so (for instance, you can't invert $S^1 \subseteq {\mathbb R}^2$ because the inclusion map $i$ and $-i$ have different degrees). In these analytic results, you're not exactly interested in homotopies.
You can use the method of convex integration (at least, basically the same kind of convexity argument -- Gromov himself prefers not to call this convex integration) to construct wild solutions to PDE. For example, there are bounded solutions to incompressible Euler which are in the energy space and can have any prescribed energy density $\frac{1}{2} |v|^2(x,t)$ (in particular, they can be compactly supported in space and time). This is a bit shocking because sufficiently regular solutions to Euler conserve energy. The fact that weak solutions need not conserve energy is tied to ideas regarding the theory of turbulence, which is the main motivation for all these studies.
If you read Springer or Gromov you may not immediately recognize the similiarities between the analysis convex integration and the topology/geometry version (for example, sometimes Baire category arguments are used in analysis to simplify the technical arguments, sometimes at the expense of some regularity in the solution). But the arguments closely parallel Nash's proof that short maps can be approximated by $C^1$ isometric embeddings, which is where the story of convex integration begins. More recent developments regarding isometric embeddings can be found in the references to the survey linked above. One main challenge regarding both Euler and the isometric embedding problem is to find the degree of regularity at which there is a transition from flexibility to rigidity.
Preceding the developments in fluid mechanics, convex integration was also used by Kirchheim, Muller and Sverak to exhibit elliptic systems coming from Euler Lagrange equations with solutions that are Lipschitz but nowhere $C^1$ -- this flexibility result contrasts the result of Evans that minimizers of the same kinds of functionals are smooth off a closed set of measure 0. There are also many related investigations in the calculus of variations tied to the stability of differential inclusions $\nabla u \in K$, especially regarding how they arise in the mathematical theory of materials. For example, James and Ball presented the idea that if $u : \Omega \subseteq {\mathbb R}^3 \to {\mathbb R}^3$ is the configuration of a crystal, its deformation gradient $\nabla u$ minimizes free energy $\int_\Omega W(\nabla u) dx$ by taking values pointwise in the set $K$ of critical points of $W$. Muller's book "Variational Models for Microstructures and Phase Transitions" has more on this topic (for example regarding how you can explain microstructures as patterns which are "trying to minimize" such a functional), but I think this is a bit more distant from the original question. The relevance is only that convex integration can be used to produce wild solutions to $\nabla u \in K$; but here $K$ might even be a finite set, and $u$ is only Lipschitz, so it's fairly different from the topological setting.
Dear Pawel,
The H-principle is a technique by which you solve differential equations and inequalities by first defining a space of pre-solutions and then showing that every pre-solution can be deformed to a solution. The existence of pre-solutions is usually a topological problem and to show that the differential equation/inequality satisfies de h-principle is another matter (Gromov's book introduces something like four general techniques to do this).
To define pre-solutions (I'm just using this term to explain things) one needs the language of jet spaces. For example, suppose you want to look at the laplacian in the plane. Consider the space of $2$ jets where each point is classically denoted by $(x,y,z,q,p,r,s,t)$. The $2$-jet of a function $u$ is $$(x,y) \mapsto (x,y,u(x,y),\partial_xu(x,y),\partial_yu(x,y),\partial^2_{xx}u(x,y), \partial^2_{xy}u(x,y), \partial^2_{yy}u(x,y)) $$
Harmonic functions are functions whose $2$-jets lie inside the submanifold $r + t = 0$ inside the space of $2$-jets. In this case, the pre-solutions I've been talking about would be given by maps of the form $$ (x,y) \mapsto (x,y,u(x,y),q(x,y),p(x,y),r(x,y),s(x,y),t(x,y)) $$ such that $r(x,y) + t(x,y) = 0$. What is missing is that $q(x,y)$ is not the partial derivative of $u$ with respect to $x$ and so on: the map (more precisely, this section of the $2$-jet bundle) is not necessarily "holonomic" in Gromov's terminology.
Gromov's incredible idea is that for lots and lots of differential equations and inequalities in geometry any pre-solution can be deformed to a solution. Alas, the method does not seem to be too useful in mathematical physics, although I think some people in elasticity have used it to construct weak solutions to some variational problems.
I've always wanted to use this method, but no luck yet ...
One place for h-principles, and where PDEs come up is jet bundles. (There seems to be a MathOverflow question about PDEs and jet bundles, here). For example, consider the 1-jet space of maps $\mathbb R^n\rightarrow \mathbb R$. That's the space of differentiable maps from $\mathbb R^n$ to $\mathbb R$, together with possible first derivatives of maps, and this forms a bundle over $\mathbb R^n$ with fiber $\mathbb R\times\mathbb R^n$ (the second copy of $\mathbb R^n$ is the tangent space). So a given map $f: \mathbb R^n\rightarrow\mathbb R$ and its derivative $Df$ together form a section of a jet bundle. But a general section is any map $f$ together with a family of homomorphisms $T_x\mathbb R^n\rightarrow T_{f(x)}\mathbb R$ covering $f$. The sections that are actually of the form $(f, Df)$ are called holonomic.
To make an $h$-principle in this setup, you can first ask a question of all jet maps, and then ask if holonomic solutions exist. The advantage of this is that bundle homomorphisms have Gauss maps to classifying spaces, so you can use algebraic topology and homotopy theory to answer existence questions. For example, given a symplectic manifold ($M, \omega$) of dimension $2n$ there is the space $\Lambda_n$ of $n$-planes in $TM$ that are Lagrangian. So to ask about Lagrangian embeddings, you would ask does an $n$-manifold $L$ admit an embedding into $M$ such that $TL\subset\Lambda_n$?'' What a successful $h$-principle does is to say, essentially,if you can solve the differential condition $TL\subset\Lambda_n$ then you can get to an actual map whose derivative does this.''
Unfortunately, $h$-principles work best for open conditions. The condition $f^*\omega=0$ is a closed condition, that is, the space of Lagrangian planes is closed in $TM$. An open condition can be made instead: given $M$, find it an almost complex structure $J$ such that $\omega(v, Jv)>0$. Then consider the space of all totally real $n$-planes of $TM$: all planes that do not contain any complex line. That includes all Lagrangian planes, and $h$-principles for this do exist. For example, for a closed surface $S$ embedded in $\mathbb C^2$, we get that $TS$ is isomorphic to the normal bundle of $S$, and (with some trouble) all the bundle theory works out to a requirement that $\chi(S)=0$ if $S$ is orientable, and $\chi(S)=0$ mod 4 if not. Examples of all such surfaces embedded as Lagrangian submanifolds of $\mathbb C^2$ exist /except/ for the Klein bottle: it is known that no Lagrangian Klein bottle exists in $\mathbb C^2$, but I believe the proof of existence of a totally real Klein bottle came from an $h$-principle.
So, a PDE, being an equation, is usually a closed condition. A partial differential relation may be open, such as ``the derivative map never drops below maximum rank'' (thus making a nonsingular immersion). That is where $h$-principles are at their most powerful: studying singularities of smooth maps between manifolds.
Sorry this is only the start of an answer, but I also recommend Eliashberg's and Gromov's book. In particular, Gromov has a theorem from 1967 that given a fiber bundle $E\rightarrow M$ ($M$ a manifold, not closed) with an action of Diff $M$ on the total space (such as a jet bundle), and a relation $R$ that is open and Diff $M$ invariant, then an $h$-principle holds for $R$. And examples of singularities of maps and how to fix them follow from this.
David Spring's book Convex Integration Theory: Solutions to the $h$-principle in geometry and topology has a chapter that specifically deals with non-linear PDEs (Chapter 9). The upshot is that convex integration, a particular method of proving the $h$-principle developed by Gromov from early ideas of Nash, can in fact be used to prove some existence theorems for systems of non-linear PDEs. However, these systems have to satisfy some conditions that, I believe, exclude linear and even quasilinear systems, but allow some fully non-linear ones. I'm afraid I don't understand these conditions well enough to reproduce them here. Please get a hold of the book for more precise statements of these results.
