TITLE: Partial differential equations in "pure mathematics"
QUESTION [44 upvotes]: 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?

REPLY [3 votes]: 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.