Partial Differential Equations: The Center of the Universe

I didn’t always intend to do PDE. My first research-level attraction was to Functional Analysis. Something about working in infinite-dimensional spaces seemed to have an air of heaven. Edward Nelson taught me that the strength of Functional Analysis was that it shed light on other parts of analysis, such as Fourier Analysis and Distribution Theory.

I got interested in Noncommutative Harmonic Analysis, which involves Lie Group Representation Theory. Part of the power of this theory arises from the ability to understand representations of the group via work on the representations of its Lie algebra. But there is a catch, for infinite-dimensional representations. To exponentiate the Lie algebra representations, you need lots of analytic vectors to be available. Lars Garding had an intriguing approach to some important results on analytic vectors; it used an analysis of the Heat Equation on the Lie group. Hmmm!

This intrusion of PDE into the analytical heart of many interesting mathematical subjects began to catch my attention. There it was at a crucial juncture in justifying the Hodge decomposition in the theory of harmonic forms. It arose in the foundations of the theory of Riemann surfaces, even in modern treatments, via Serre duality. I got hold of a copy of Bers, John, and Schechter’s book on PDE and was hooked.

Over the course of time I have paid more attention to PDE in classical physics, as a tool for understanding waves, fluid flows and such, the task for which the subject was created. PDE also plays an important role in twentieth century physics, with Einstein’s equation, Schrodinger’s equation, and Dirac’s equation, amongst others. Work in these areas often draws on topics mentioned above, such as the use of noncommutative harmonic analysis in study of spectra of Schrodinger operators arising from gauge fields. So I have kept up my older interests, and also maintained an abiding fondness for books on PDE (see below).

 

Partial Differential Equations, Vols. 1-3, appear in Springer’s Applied Math Sciences series, Vols. 115-117. A second edition has come out in 2011. This edition contains a number of additions and corrections, including seven new sections, given as follows.

Volume 1: Chapter 4, Section 7; Chapter 6, Section 8.
Volume 2: Chapter 7, Section 15; Chapter 10, Section 11.
Volume 3: Chapter 13, Appendix A; Chapter 17, Sections 6 and 7.

Click on each volume to see what it has
Volume 1┬áPartial Differential Equations I — Basic Theory
Volume 2┬áPartial Differential Equations II — Qualitative Studies of Linear Equations
Volume 3┬áPartial Differential Equations III — Nonlinear Equations