PDEs as a tool in other domains in mathematics

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 PoincarĂ© 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.

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"According to MathSciNet, PDEs is an important topic of its own." I wish I can use that for the Intellectual Merit section of my next NSF proposal... – Willie Wong Oct 11 2010 at 11:41
@Dennis: big-list questions should always be community wiki (edit the question, tick the small check box to the bottom right of the text-field). – Willie Wong Oct 11 2010 at 11:42
+1. I wonder if any PDE has been used in Number Theory, even in the Analytic branch. – To be cont'd Oct 11 2010 at 11:42
@Willie. OK. I'll correct it if I can. But please, write Denis with one N (I'm French, nobody's perfect). – Denis Serre Oct 11 2010 at 11:57
"I'm French, nobody's perfect" are four of the funniest words I've read in some time. – Pete L. Clark Oct 11 2010 at 15:29

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.

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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.

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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.

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

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Two comments (don't take me too seriously, this is all in good fun): (a) See Gerald Edgar's answer (b) When we loop all the way back to Analysis as an application of PDEs to a different field, something should be said about how fragmented knowledge in mathematics has become. – Willie Wong Oct 12 2010 at 10:40

The Nash isometric embedding theorem

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

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

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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.

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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.

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

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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?).

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 A compact manifold is homeomorphic to a finite CW complex (take a Morse function, or a triangulation, etc.) and thus has finite Betti numbers. – Tom Church Mar 19 2012 at 18:07

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

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 Always a good thing to have an expert in the field a question is rooted in present to answer it,Deane. Me,I'm just trying to find the time to finish Evans' text on the subject......... – Andrew L Oct 12 2010 at 7:35

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.

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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 $t\to\infty$.

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There's also a similar result for the Yang-Mills heat flow. – Willie Wong Oct 11 2010 at 17:39

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.

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Isn't that a little unfair? The Calabi conjecture is Aubin and Yau, but the other two are Schoen and Yau. – Deane Yang Oct 11 2010 at 16:14
That's why I said "involve" and not "due to". =) And large parts of Yamabe is also due to Aubin, no? But the "same mathematician" comment is more meant to illustrate how connected geometry and PDE actually are (through geometric analysis, which the winds of fortune seems to prefer to classify nowadays as geometry, and not PDEs), that the division into "different fields" is somewhat illusory. – Willie Wong Oct 11 2010 at 17:29
(In other words, that was suppose to be a subtle hint that the answer I gave above is not a very good answer at all, since I was able to cheat by thinking to myself: geometry and PDEs, hum, what did Yau do?) Aldo, I deliberately left the name out because, if you knew who worked on all three, more likely you also knew who his co-authors/competitors were. If you didn't, well, after looking it up it will be clear those were not one-man efforts. Maybe I should've written "the same mathematicians"? – Willie Wong Oct 11 2010 at 17:34
It's not a big deal, but I like to try to give explicit attribution as much as possible, because most people will not dig into these things carefully but they should be given some sense of who were the main people involved. – Deane Yang Oct 11 2010 at 19:02
I would say Yau was not directly involved with the solution of the Yamabe problem. He is involved only in the sense that the Positive Mass Theorem is used by Schoen in the final solution. One must also mention Trudinger's contribution to the Yamabe problem, who pointed out the flaw in the original argument by Yamabe and repaired the proof for some cases. – timur Oct 12 2010 at 4:53