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I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me.

What is known about applications of methods from homological algebra to the analysis of solutions of PDE on domains in $\mathbb{R}^n$?

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I suggest you to read Sheaves on Manifolds by M. Kashivara, P. Schapira – Norbert Oct 16 '12 at 14:54
Thanks for the reference, but I'm especially looking for applications to PDE on open, bounded domains in real Euclidean space, not on manifolds. – anon Oct 16 '12 at 14:59
Isn't such a domain a manifold? – Norbert Oct 16 '12 at 15:04
But not really interesting as a manifold, is it? – anon Oct 16 '12 at 15:34
This is probably somewhat irrelevant, but there is an algebrization (or rather algebro-geometrization) of the idea of a PDE on a manifold. It is called a D-module. Then of course you can play all sorts of homological games with it.This shows up in representation theory, algebraic geometry. – DBS Jul 9 '13 at 6:39
up vote 2 down vote accepted

Homological techniques are very often seen in the literatures of PDE treated from a differential geometrical point of view. An extensive overview can be found in: Homological methods in equations of mathematical physics. Also I recommend one of my favorite book here: The Geometry of Physics: An Introduction by Theodore Frankel. Some simple google-fu gives me a recent book also: Cohomological Analysis of Partial Differential Equations and Secondary Calculus.

I am working on computational physics, and the methodologies arised from de Rham cohomology have been used extensively in construction of the finite element spaces for equations in electromagnetism: Finite element exterior calculus, homological techniques, and applications. Mostly people in my field are interested in solving the Hodge Laplacian acting on a $k$-form(magnetic flux or electric field).

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You might be interested in reading Chang's "Infinite Dimensional Morse Theory" or Jost's "Riemannian Geometry and Geometric Analysis". Moreover, I guess it would be useful to give you an example: there is a result due to Bahri and Coron that states as follows: if $\Omega \subset \mathbb{R}^N$, $N\ge 3$ is a smooth bounded domain, you can find a solution of

$$\begin{cases}-\Delta u = u^{\frac{N+2}{N-2}}&\text{in }\Omega\\ u > 0 & \text{in }\Omega\\ u = 0&\text{on }\partial \Omega,\end{cases}$$

provided that the domain $\Omega$ is "topologically nontrivial", that is if the homology group $H_q(\Omega, \mathbb{Z}_2) \ne 0$ for some $q > 0$. Of course, there are a lot of more recent works in the study of pdes which involves techniques from algebraic topology.

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