D-modules and Algebraic Solutions of PDEs - MathOverflow

D-modules and Algebraic Solutions of PDEs

2011-10-09T17:51:27Z

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by S. C. Coutinho" the justification on the importance of D-modules; they provide an algebraic tool towards the solution of differential equations. This is the story I always hear! Do someone have a reference or more information about D-modules and algebraically solution of PDEs ?.

Answer by B. Bischof for D-modules and Algebraic Solutions of PDEs

2011-10-10T04:57:18Z

Perhaps you are looking for something deeper, but right there at the beginning of Hotta, Takeuchi, and Tanisaki's book on D-mods in the introduction is the connection to Linear PDEs.

I quote:

Therefore, systems of linear partial differential equations can be identified with the D-modules having some finite presentations like (0.0.3), and the purpose of the theory of linear PDEs is to study the solution space HomD(M, O). Since the space HomD(M, O) does not depend on the concrete descriptions (0.0.2) and (0.0.3) of M (it depends only on the D-linear isomorphism class of M), we can study these analytical problems through left D-modules admitting finite presentations. In the language of categories, the theory of linear PDEs is nothing but the investigation of the contravariant functor HomD(•, O) from the category M(D) of D-modules admitting finite presentations to the category M(C) of C-modules.

Answer by Paul Siegel for D-modules and Algebraic Solutions of PDEs

2011-10-10T06:59:23Z

My guess is that this is a reference to Bernshtein's proof using the theory of D-modules that every constant coefficent partial differential operator $D$ has a fundamental solution, i.e. there is a distributional solution to the PDE $Du = \delta$ where $\delta$ is the Dirac distribution. This was an important theorem in analysis due to Malgrange and Ehrenpreis in the 1950's, and I think it came as a bit of a surprise that it can be done purely algebraically - all of the analytic input is encapsulated by a few basic facts about distributions. Everyone knows that the fundamental theorem of algebra is proved in analysis class; maybe this means the fundamental theorem of analysis will one day be proved in algebra class!

Here is a link to Bernshtein's paper: http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-mod-dif-FAN.pdf. This was written in the 70's, and I think since then the argument has been cleaned up and made even more algebraic.