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?

  • $\begingroup$ 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. $\endgroup$ – LordVader007 Oct 26 '18 at 4:23
  • $\begingroup$ 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. $\endgroup$ – LordVader007 Oct 26 '18 at 4:26
  • $\begingroup$ @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. $\endgroup$ – user82261 Oct 26 '18 at 4:36
  • $\begingroup$ @LordVader007 As time passes, hopefully I will be able to catch up on functional analysis, and progress further with PDE's. Suggestions? $\endgroup$ – user82261 Oct 26 '18 at 4:36
  • $\begingroup$ 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. $\endgroup$ – LordVader007 Oct 26 '18 at 4:45

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