Roadmap to Modern Research in Partial Differential Equations

Question

Let me start off with a description of my background. I am an undergraduate student, with some background in real analysis (Rudin, Principles of Mathematical Analysis), measure theory (Royden, Real Analysis: Parts I and III) and a little functional analysis (first five chapters of Rudin's 'Real and Complex Analysis'). I know no geometry, beyond Loring Tu's 'Introduction to Manifolds'. Additionally, just to give you the entire picture, I have an algebra background from Dummit's and Foote's 'Abstract Algebra', until but not including the section on Galois Theory. I also know a little spectral theory from William Arveson's 'A Short Course in Spectral Theory'.

I recently took a course on ordinary differential equations, where we followed a combination of Arnold's 'Ordinary Differential Equations' and George Simmons's 'Differential Equations with Applications and Historical Notes'. I liked the course, and decided to study partial differential equations from Evans's 'Partial Differential Equations'.

I would like a roadmap that answers the following questions:

1. What related areas of math would I need to know to understand research-level papers in the subject? Here, I understand that there could be many frontiers of the subject, each requiring its own set of prerequisites. I am looking for an answer that explains what exactly the area of research aims at doing and what the prerequisite content to work in that area would be. If certain things are required only in parts as prerequisites, I would very much appreciate it if you could name the theorems or the sections of books I would have to read.

2. What could I read immediately after reading Evans? I've heard that Michael Taylor's three volume series is good, and plan on taking a look at that, but more well-directed (as in, towards a specific sub-area of research) recommendations would be appreciated.

3. If you could estimate the time it would take (on an average) to study the material you suggest, especially if it isn't a book that mentions that, I would appreciate it, since it helps to know whether I am rushing through things (in which case, I often remember nothing later) or am going too slowly (in which case, I will try seeking help from some more advanced students or a professor).

Thanks!

Answer 1

PDE is VERY broad. It will be virtually impossible to answer this question in any generality. I heartily second the recommendation that you read Taylor's three-volume treatise. He really emphasizes the microlocal viewpoint, which is hugely useful in many areas of PDE, and it's a good contrast to Evans. As an aside, Michael Taylor is in my department and that man's knowledge about PDE is truly encyclopedic.

Now, I'll try to address your actual question a bit. If you've REALLY studied Evans, you should be equipped to start dipping your toes into the research literature. Along the way you should also start to get some ideas about what sorts of problems interest you. You could simultaneously begin looking at interesting articles while working your way through new material in Taylor. From there, I would decide what sorts of other "prerequisites" are needed based on the articles you're reading. In my opinion, this is a much more efficient path to studying research-level mathematics than first attempting to gather all of the possible prerequisite knowledge (if the latter is even really possible).

I'll give an example path by way of anecdote. For my MA thesis, I studied a problem related to the Alt-Caffarelli-Friedman functional. I began by working through the relevant portions of Evans (Laplace's equation, Sobolev spaces, second-order elliptic equations and calculus of variations). After that, I jumped right into reading the Alt and Caffarelli paper on the one-phase problem and other relevant research literature (Alt-Caffarelli-Friedman, Littman-Stampacchia-Weinberger, etc). Along the way, I had to take some detours into some particular functional analysis and a healthy dose of geometric measure theory.

To summarize, research in PDE is a huge subject and draws tools from all over mathematics. This will particularly depend on the problem you want to study. Rather than obtaining all of the possible prerequisite knowledge first, it's much better to (1) obtain some basic pde knowledge (Evans + Taylor would do it), (2) begin reading interesting research, (3) identify where your prerequisite knowledge is lacking for the particular problem at hand and then (4) do the necessary prerequisite study (you can often look to the references of the paper you're reading to find good resources for this).

Regarding the timeline: I would say that if you're ready to start studying Evans, then you could be ready to start dipping into research math within a semester or maybe a year (depending upon how quickly you move, how selectively you read, etc).

For the sake of completeness, I want to add a few specific "prerequisites" that have been useful in my corner of the PDE world (assuming the basics as given). Your mileage will vary depending on where you go in PDE.

(1) Harmonic analysis - Littlewood-Paley decomposition, various multilinear estimates and analysis, Fourier analysis, etc. Stein's Harmonic Analysis is great for this material. Additionally, I'd recommend the two-volume Classical and Multilinear Harmonic Analysis by Muscalu and Schlag, as well as Wavelets: Calderón-Zygmund and Multilinear Operators by Coifman and Meyer.

(2) Microlocal analysis - pseudodifferential operators, paradifferential operators and paraproducts, frequency localization, the notion of the wave front set and more. For this material Michael Taylor has some wonderful references: Volumes II and III of his PDE text, Pseudodifferential Operators and Nonlinear PDE, Tools for PDE). Para-differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems by Métivier is a great, standard reference for paradifferential calculus. Hörmander's four volumes are great, of course. Finally, I'd point out: Pseudodifferential Operators and the Nash-Moser Theorem by Alinhac and Gerard, the Coifman-Meyer book I mentioned earlier has some good material on paraproducts and paradifferential operators and Microlocal Analysis for Differential Operators by Grigis and Sjöstrand.

(3) Geometric Measure Theory - theory of rectifiable curves, sets of finite perimeter and so on; the notion of an approximate normal and an approximate tangent space; the notion of blow-up limits; regularity theory of minimal surfaces; and plenty more. Geometric Measure Theory by Federer is a classic reference. Also, Minimal Surfaces and Functions of Bounded Variation by Giusti. Simon's lecture notes are good as is Maggi's Sets of Finite Perimeter and Geometric Variational Problems.