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One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are PDE's arising "naturally" in "pure" mathematics like Geometry, Topology, etc.? Of course, I can always write an arbitrary surface as one or more PDE's, but they don't seem to drive much research in the subject, afaik. I understand that PDE's originated in Physics, but hasn't the subject grown away from physical models into more abstract examples?

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If you haven't seen PDEs in geometry then you haven't done enough (differential) geometry! –  Zhen Lin Jul 19 '11 at 7:22
I think this should be CW. –  Marek Jul 19 '11 at 11:00
@Marek: It would be more effective if you flagged the question with that request, rather than leaving it as a comment, since Moderators are the only ones who can CW-ify a question. Cheers. –  Willie Wong Jul 19 '11 at 11:45
@Willie: ah, I forgot about that. Thanks. –  Marek Jul 19 '11 at 11:55
Related thread: mathoverflow.net/questions/74183/… –  timur Jul 24 '12 at 17:30
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8 Answers

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I believe that one of the examples can be Ricci flow which are parabolic PDEs describing the deformation of metric for Riemann manifold. The new result is e.g. the solution of Thurston's gometrization conjecture which Perelman did using Ricci flow.

In general as you have mentioned, many PDEs appears from the applied science: physics, finance etc. though I am not sure that PDEs related to Markov processes appeared only after considerations of applied problems.

Finally, there is a theory of Harmonic functions which was based on the Laplace PDE $\Delta u = 0$. These functions are generalizations of linear functions on the real line for the multidimensional case and they inherit many of nice properties of linear 1-dim functions. Although this equation appeared from the physical problem, it has been also important for the needs of the pure mathematics.

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Some more examples from geometry and topology:

The Atiyah-Singer Index Theorem connects analysis of elliptic differential operators, with the topology of smooth compact manifolds. There are many approaches to proving the Index Theorem, but there is one in particular which does so via studying the heat equation on the compact manifold.

On non-compact Riemannian manifolds, an uniform geometric structure at infinity has deep connections to the spectrum of the associated (conformal) Laplacian, this leads to deep connections with scattering properties of wave equations evolving on such manifolds.

The topology of four dimensional manifolds is connected to their geometries manifesting in Yang-Mills theory, which gives rise to very interesting non-linear elliptic partial differential equations. There's quite a nice book about this whole field.

The study of over-determined systems of partial differential equations (such as this one) can trace a not-insignificant-part of its roots to E. Cartan's program to understand differential geometry using the moving frames method.

On the other direction, PDEs can give rise to interesting problems in other fields. For example, the fundamental solution of a linear constant coefficient hyperbolic partial differential equation has deep connections to topology and geometry of algebraic varieties. And the so-called strong Huygen's principle for the wave equation is closely connected to the geometry of homogeneous spaces.

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Don't forget the minimal surface equation, one of many PDE's arising in differential geometry as alluded to by Zhen Lin. In his Comprehensive Introduction to Differential Geometry, Spivak includes a chapter on PDE's under the title And Now A Brief Message From Our Sponsor - a nice way of highlighting the many interactions between the two subjects.

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Calabi conjecture stated that on compact Kähler manifold there exists a unique Kähler metric with prescribed Ricci form. This problem can be reformulated in terms of a (non-linear) PDE and has been settled by Shing-Tung Yau who indeed constructed a solution to this PDE (the uniqueness was settled prior to that by Calabi himself).

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The natural linear PDEs that arise in physical situations also arise in "pure math" contexts, for similar reasons, namely, calculus/analysis with symmetry constraints. In my own direct experience:

The spectral theory of automorphic forms, in the simplest/smallest examples exemplified in Iwaniec' "Spectral methods...", and certainly on rank-one spaces, characterizes automorphic forms/functions as being essentially eigenfunctions for a suitable (invariant) Laplace-Beltrami operator. Regularity properties of elliptic operators are used almost without mention. Fundamental solutions, or Green's functions, play the expected geometric-analysis roles in basic estimates on automorphic forms.

Selberg's-and-others' "trace formula(s)" can be viewed as studies of the trace of the resolvent of this Laplace-Beltrami operator on suitable subspaces of $L^2$.

Heat-kernel ideas have been used by many people to prove essential results (Mueller on trace-class of discrete spectrum, Lang-Jorgensen projects, etc.)

Scattering-theory and wave-equation ideas were used by Fadeev and his collaborators, by Lax-Phillips, and others, to study Eisenstein series and the continuous spectrum.

For that matter, the classical linear PDE on Euclidean spaces, Laplace's, heat, wave equation, or one products of circles, or on spheres, seems to me to be primordial analysis of Euclidean geometry, whether or not one is interested in physics or mechanics. After all, I suspect that the aspects of all these things we put in these categories are substantially manifestations of human perceptions and psychology rather than something innately called "physics-or-mechanics", as opposed to "pure mathematics".

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One has the classical Gauss-Bonnet Theorem:

If $M$ is a compact surface with Riemannian metric $g$, and if $K_g$ is the sectional curvature of $g$, then $\int_M K_g dvol = 2\pi\chi(M)$.

One can view this as saying that a necessary condition in order for a function $K:M\rightarrow\mathbb{R}$ to be the sectional curvature of some metric on $M$ is that it integrate to $2\pi\chi(M)$. A natural question arises: Is it sufficient?

Well, since $K$ is built out of the second partial derivatives of $g$, this amounts to proving the existence or nonexistence of a solution to a PDE.

The answer, as it turns out, is yes: The necessary condition is also sufficient. I believe this result is due to Kazdan and Warner but I'm not sure. (I'll look it up tomorrow morning, if I remember to).

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Another example is the de Rham complex on a manifold, or even simply on $\mathbb{R}^n$. One way to look at it is as a natural set of partial differential equations, in various dimensions; for instance finding closed 1-forms is equivalent to solving the differential equation $\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y} = 0$.

To quote Bott & Tu from "Differential Forms in Algebraic Topology", p. 15: "The de Rham complex may be viewed as a God-given set of differential equations".

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The $\overline{\partial}$-equation can be considered a PDE on a complex maifold. The equation is $\overline{\partial}u = f$ where $u$ and $f$ are complex differential forms, and $f$ is $\overline{\partial}$-closed (i.e. $\overline{\partial}f = 0$). This equation is directly related to the Dolbeault cohomology of the manifold.

In Hörmander's An Introduction to Complex Analysis in Several Variables, he proves Cartan's Theorem B which states that, for a Stein manifold, the $\overline{\partial}$-equation always has a solution. The method he uses has very little to do with complex analysis (at least at first glance), but rather uses standard techniques from PDE's.

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The method does not use complex analysis but the result has a lot of applications in complex analysis (although there are at least a couple of alternatives to this PDE-based approach in several complex variables, as far as I understand). –  timur Dec 21 '13 at 3:05
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