8

I'll start to study PDE's from Evans’ Partial Differential Equations soon. I'd like a few suggestions:

  1. I'm already studying measure theory this semester. Do I need to know functional analysis and multivariable integral/vector calculus (theory) to tackle the first part of the book?

  2. Will a working knowledge of vector calculus suffice? Or should I study the theory on the side as well?

  3. What's a good suggestion for a second book to keep on the side? I'm thinking of keeping a book that focuses more on computational details on the side. I also think Evans' textbook seems a little light on questions. Any suggestion(s) for books that may help?

user82261
  • 1,237
  • It depends how far into the book you want to get into. If you want to look at the first part, that is "Representation Formulas for Solutions", then a knowledge of Real Analysis, Multivariable/Vector Calculus will suffice. If you want to go deeper into the theory, i.e. Sobolev Spaces, then it is essential that you master techniques from Functional Analysis as well. Depending on what background you have, I suggest you leave study of PDEs after you take Functional Analysis. Also, a bit of point-set topology and differential geometry is useful. – Vegeta the Prince of Saiyans Oct 26 '18 at 04:23
  • 1
    Another good book to have on the side is Haim Brezis' "Functional Analysis, Sobolev Spaces, and PDEs". It is meant as a two semester sequence on Functional Analysis, the first part is linear functional analysis and the second part is advanced topics in Functional Analysis, with introduction to PDEs. I like the transition from FA to PDEs made in this book. – Vegeta the Prince of Saiyans Oct 26 '18 at 04:26
  • @LordVader007 I may take a course in functional analysis next semester. Right now, we'll be covering the basics of Hilbert Spaces in the measure theory course. I will also be taking an applied math course next semester that's mostly on functional analysis. My aim till the end of the academic year is to complete the measure theory course and maybe even the functional analysis course. I'd also like to dabble into PDE's and Probability, as I may apply to other graduate programs next academic year. For the time being, I'd like to start off with Evans. – user82261 Oct 26 '18 at 04:36
  • @LordVader007 As time passes, hopefully I will be able to catch up on functional analysis, and progress further with PDE's. Suggestions? – user82261 Oct 26 '18 at 04:36
  • From a purely mathematical point of view, I suggest you first take a course in Functional Analysis, as the subject alone merits its own course, apart from any other treatment done in applied math courses. Only after you have a solid background in FA will you be able to move further in PDEs. In most graduate programs, they won't let you take PDE unless you have both RA and FA. However, this is not meant to discourage you from doing PDEs, you can start building intuition now, which is always nice to have. – Vegeta the Prince of Saiyans Oct 26 '18 at 04:45
  • I speak from my own experience, as I too once sought to undertake an independent study in PDEs. However, I found that my background in FA was lacking, and had to postpone further studies until I took a good rigorous course in FA. – Vegeta the Prince of Saiyans Oct 26 '18 at 04:45
  • @LordVader007 Agreed. But since I may be re-applying to graduate programs in about a year, I think I have to do some work in PDE's and Probability to apply to relevant programs (Stochastic PDE's). What's your suggestion? Could I focus on only certain material on PDE's first, and then pick up functional analysis along the way...starting next semester? Could a balance be struck, so that by the end of this academic year, I'm able to learn both these subjects? Again, suggestions? – user82261 Oct 26 '18 at 04:51
  • @LordVader007 The Professor with whom I met today that in the future, you may even need a bit of Differential Geometry! He said that he recently had to use it. I have to start somewhere. So, what's your recommendation? This is why I'm looking for a right combinations of books to keep with me on the side. I'll only be taking 2 courses next semester; the rest of the time would be spent with 2 independent studies. – user82261 Oct 26 '18 at 04:56
  • I am not in any ways an expert in PDEs (just a mere PhD student), but my suggestion is that if you are applying to programs in that area, it would be also good to attend some kind of colloquia/seminars as well as read papers in (stochastic) PDEs by professors in the department you are applying to. That way, you can mention your progress in the Statement of Purpose (SoP). Keep in mind that the admission committee won't expect that you are a subject matter expert in PDEs, rather they will be looking first at your background/courses taken and test scores, etc... – Vegeta the Prince of Saiyans Oct 26 '18 at 05:12
  • Long story short, you don't need to have expertise (say for example, dynamical systems) to apply to a good program in dynamical systems. On the other hand, (if time permits), you can start learning by yourself PDEs and build a feel for what the subject is like. I suggest then that you work through Evans' book in an almost linear fashion (as much as time allows) and do the problems. Even the classical theory in R^n dimensions isn't that trivial/easy. – Vegeta the Prince of Saiyans Oct 26 '18 at 05:15
  • As the professor said, you do need a bit of differential geometry, as you are taking integrals over n-dimensional smooth manifolds, and will use things like Stokes' theorem over a manifold and partitions of unity. A good reference for differential geometry is Loring Tu's "An Introduction to Smooth Manifolds", it has answers/hints to most exercises. – Vegeta the Prince of Saiyans Oct 26 '18 at 05:17
  • @LordVader007 True. I suppose, as far as background is concerned, I should be studying PDE's and Probability. Also, I suppose I will only be able to study paper (or tack on a thesis/topics courses) only when I study PDE"s and Probability in the first space.

    I have to start somewhere. The question, in part, is motivated from these concerns. I'm trying to keep a good combinations of texts right now, and then move up the ladder (FA etc, non-linear PDE"s etc.). I suppose I will cover linear PDE's right now, and brush up on vector calculus. I'll start reading a FA textbook, and hopefully...

     – user82261 Oct 26 '18 at 05:18
  • pick up the stuff as time goes by. – user82261 Oct 26 '18 at 05:18
  • @LordVader007 On DG, true. But one can only study so much material in a given time. So I'm trying to chart the best possible plan. Will talk to the professor as well. I'll definitely take courses in topology/geometry next year. I want to focus on taking an analysis heavy course load this year. – user82261 Oct 26 '18 at 05:20

0 Answers0