PDEs as a tool in other domains in mathematics

Question

According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural sciences, such as Physics, Chemistry, Biology, Continuum Mechanics, and so on.

What I am interested in, here, is examples where PDEs were used to establish a result in an other mathematical field. Let me provide a few.

1. Topology. The Atiyah-Singer index theorem.
2. Geometry. Perelman's proof of Poincaré conjecture, following Hamilton's program.
3. Real algebraic geometry. Lax's proof of Weyl-like inequalities for hyperbolic polynomial.

Only one example per answer. Please avoid examples in the reverse sense, where an other mathematical field tells something about PDEs (examples: Feynman-Kac formula from probability, multi-solitons from Riemann surfaces). This could be the matter of an other MO question.

Answer 1

Maybe ... Cauchy-Riemann equations ... they may have been used a time or two over the years ...

Answer 2

The Hodge theorem (each de Rham cohomology class on a compact Riemann manifold has a unique harmonic representative) has a wide range of applications in complex algebraic geometry, much deeper than showing the finite-dimensionality of the cohomology. One of my favorite results that depend on the Hodge theorem is the Kodaira embedding theorem, which characterizes those compact complex manifolds that can be embedded holomorphically into projective space. See Griffiths-Harris. That a compact manifold has finite-dimensional cohomology groups can be shown in a more elementary way. I am sure this is somewhere in Bott-Tu's book.

Answer 3

The diffeomorphism group of a closed surface of negative Euler characteristic has contractible components. This is theorem by Earle and Eells (Journal of Differential Geometry 3, 1969). The crucial ingredient for their proof is the solvability of the Beltrami differential equation. Later, Gramain found a purely topological, rather elementary proof of that result but - at least for me - the proof using PDEs is much easier to understand.

Answer 4

The work of Uhlenbeck, Taubes, Donaldson, and others on Yang-Mills connections is a gorgeous application of nonlinear elliptic PDE theory.

Answer 5

The Nash isometric embedding theorem

Answer 6

PDEs are massively used in the theory of harmonic maps.

My personal favourite is a nice theorem by Lemaire and Sacks-Uhlenbeck.

Theorem. Suppose $M$ is a compact Riemann surface, possibly with boundary, $N \subset \mathbb R^n$ is compact. If $\pi_2(N) = 0$, then any map $u_0: M \to N$ is homotopic to a smooth harmonic map.

The key ingredient of the proof relies on existence and uniqueness of global weak "energy" solutions $u:\ M\times[0,\infty])\to N$ to a nonlinear Cauchy problem for the $L^2$-gradient flow $$\begin{cases} u_t-\triangle_M u=A(u)(\nabla u,\nabla u)_M & \mbox{in }M\times[0,\infty),\\\ u=u_0 & \mbox{at }t=0\mbox{ and on }\partial M\times[0,\infty)\end{cases}$$ which converge to a smooth harmonic map $u_{\infty}:\ M\to N$ as $t\to\infty$.

Answer 7

Another not-quite-yet connection which I learned from Lax's Hyperbolic PDE book: one can, technically speaking, extract the Riemann hypothesis from the scattering rates of certain "automorphic waves". (This is where my knowledge ends; those interested can look at Chapter 9 of the the book.)

Answer 8

How about Hodge theory. I.e. that each DeRham cohomology class of a smooth compact manifold has a harmonic representative (one has to of course choose a Riemannian metric to make sense of harmonic). This for instance allows one to show that the Betti numbers of a compact manifold are all finite and is the usual way to show this (the only way?).

Answer 9

Riemann's existence theorem which states that every compact Riemann surface has a non-constant meromorphic function (and hence is an algebraic curve). Standard proofs use harmonic functions, i.e., solutions of the Laplace equation.

Answer 10

Some other probability PDE techniques:

1) Percolation: The Aizenman Barsky proof of exponential decay in subcritical percolation hinged on establishing a number of differential inequalities.

2) Conformal Invariance and SLE: Many conformal invariance proofs reduce to showing that the discrete stochastic process in question satisfies a Riemann Hilbert boundary value problem along with defining a flow on the state space which is divergence and curl free. This makes it clear how Cardy's formula arises as the hypergeometric function which solves the appropriate differential equation.

Answer 11

The work of Meeks and Yau using minimal surfaces is a beautiful application of nonlinear elliptic PDE's to low-dimensional topology.

Answer 12

Some other results in Geometry that do not require reaching very far to see its connection to PDEs: the resolution of the Yamabe Problem, the proof of the Calabi Conjecture (now the Calabi-Yau theorem), and the proof of Positive Energy Theorem.

(I violate the 1 example per answer rule, since these three are all from geometry, and involve the same mathematician.)

Edit: As Deane pointed out below, I should be more precise about the attribution. A well known contributor to the solution of those three problems above is S.T. Yau. Others who have worked on those problems include Rick Schoen, who collaborated with Yau on the proof of the Positive Energy theorem and (hence) the Yamabe problem, and Thierry Aubin who also contributed much to the understanding of the Yamabe Problem, as well as making significant progress toward the Calabi conjecture.

Edit 2: And of course, as Timur pointed out below, I inadvertantly left out Neil Trudinger as one of the main contributors to the Yamabe problem. (One of the reasons I didn't want to be too precise on references in the beginning was to avoid mistakes like this.) Also please note that this is a Community Wiki article, so please feel free to just edit it to fix any insufficiencies you see.

Answer 13

Great! Since this has somehow bubbled to the top, I have yet-another occasion for a not-entirely-selfish rant! :)

Although @WillieWong's answer was a few years ago, it does point (if a bit obscurely) in a significant direction, for example. Immediately, one should note that there is no known (to me, and ... I care) proof of RH that immediately/simply uses PDE ideas on modular curves.

But those ideas, from Haas, Hejhal, Faddeev, Colin-de-Verdiere, and Lax-Phillips, do solidly establish a connection of spectral properties of (certain "customized" extensions of) Laplacians on modular curves and related canonical objects.

A more obviously legit example of interaction of number-theoretic "physical objects" and "PDE" was Milnor's example of two tori with the same spectrum for the Laplacian. But, yes, that wasn't really about PDE.

A problem with reporting modern relevance of PDE to number theory is that there are "complicating" additions, ... :) ... often under-the-radar amounting to things stronger than Lindelof Hypothesis, if not actually the Riemann Hypothesis.

Rather than recap things better documented elsewhere (many peoples' arXiv preprints, my own vignettes and talks various places, ...) please forgive my returning to the homily that "PDE" are merely assertions of relations, as Newton intuited for the planets, and others have observed/inferred for many more things.

(Any hysterically provincial remarks about turf, or "specialties-as-ignorance-of-other" are obviously toxic... despite their dangerous prevalence and popularity...)

Thus, srsly, people, "PDE" means "a kind of condition on functions..." ... If people weren't so caught up in ... oop, sorry, the kind of people do get caught up in... :) ... it'd be obvious that "infinitesimal" conditions would be natural...

Thus, an explanatory but not really useful answer is, that we seem to see that This is not related to That because the respective proprietores have no vested interest in letting on that anyone else could ... perform their guild's function.

(Yes, it is informative to review Europe's late-renaissance guild-culture...)

And, as in many rants, I wish to reassure everyone by my disclaimer, "wait, what was the question, ... again...? " :)

(But, yes, this is a serious-and-important issue, in many ways, so, yeah, just some kidding-at-the-end.)

Answer 14

Graph theory, e.g. http://arxiv.org/abs/math/0009120

Answer 15

The elliptic regularity theorem can be used to establish the classical result that holomorphic (and harmonic) functions are $C^\infty$.

Answer 16

Reilly used PDEs to give a very elegant proof that spheres are the only embedded hypersurfaces of constant mean curvature in $\mathbb{R}^n$.

Let $\Sigma$ be such a hypersurface, bounding a region $\Omega$. He showed that any solutions to the PDE $\Delta u = -1$ in $\Omega$ with $u=0$ on $\Sigma$ must be a second order polynomials with leading term proportional to $|x|^2$. One sees that level sets of this function are spheres by completing the square.

Answer 17

The $\bar{\partial}$ and $\bar{\partial}$neumann problems which are of importance in integrable systems and complex analysis.

Answer 18

Dynamical systems. Roughly speaking, a dynamical system $\dot{x} = a(x)$ is stable if and only if the 1st order linear partial differential equation $\mathcal{L}_a v + \ell = 0$ has a positive solution $v$. Here $v$ is called a Lyapunov function for the system, $\mathcal{L}$ is the Lie derivative, and $\ell > 0$ has to be chosen so that the equation has a solution but is otherwise arbitrary.