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What are some good PDE books that can used for an independent study with a professor. My background includes:

  1. Linear Algebra at the level of Axler's Linear Algebra Done Right and Friedberg, Insel and Spence's Linear Algebra;
  2. Abstract Algebra at the level of Dummit and Foote's Abstract Algebra;
  3. Complex Analysis at the level of Bak and Newman's Complex Analaysisl
  4. Real Analysis at the level of Rudin's PMA and Pugh's Real Mathematical Analysis.
  5. Multivariable Differential Calculus at the level of Edwards' Advanced Calculus of Several Variables.

I'm also currently revising some of the aforementioned subjects. I also know a bit of measure theory, and I'll be taking a course on it the fall. I don't know functional analysis as of yet. I also don't know a lot about Multivariable Integral/Vector Calculus (theory). I haven't also taken any theory course on ODE's. I suppose I can pick up the basics of Fourier Series, Fourier Transforms etc. during the course of my independent study.

I'm looking for two textbooks that I can keep side by side to introduce myself to the basics of PDE's at the graduate level, as I'll be an incoming graduate student in the fall.

Edit:

  1. The standard suggestions seem to be Walter Strauss Partial Differential Equations (for a first course) and Lawrence Evans' Partial Differential Equations (for a second course). Does it seem like a reasonable for me to keep these two books side by side?

  2. Is the theory of ODE's is an absolute pre-requisite for a course on PDE's taught from any book (at the advanced undergraduate/graduate level)? If not, is there any way I can skirt around it? I don't mind learning measure theory and functional analysis in a self-contained manner from a textbook that covers PDE's

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  • $\begingroup$ You could try reading Evans, which is maybe the most popular graduate level textbook. I think it's also useful to look at Strauss, which is an undergrad level book, just to understand basics clearly. $\endgroup$ – littleO Jul 4 '18 at 6:30
  • $\begingroup$ @littleO Exactly. I edited my question accordingly because these two books seem to be the standard suggestions. What are your thoughts? Is Evans readable prior to having completed measure theory and functional analysis? $\endgroup$ – user82261 Jul 4 '18 at 6:32
  • $\begingroup$ I think you can get by without knowing Theory of ODEs. Strauss + Evans seems like a good combination to me. You might also check out Partial Differential Equations: An Introduction to Theory and Applications by Shearer and Levy, which is for beginning graduate students and is sort of a bridge from Strauss to Evans. "The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis." $\endgroup$ – littleO Jul 4 '18 at 7:10
  • $\begingroup$ @littleO Thanks. I'll check out these books. $\endgroup$ – user82261 Jul 4 '18 at 7:15
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    $\begingroup$ Sandro Salsa, Partial Differential Equations in Action, From Modelling to Theory Third Edition. Or: AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS, YEHUDA PINCHOVER AND JACOB RUBINSTEIN $\endgroup$ – Gustave Jul 5 '18 at 16:27
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Partial Differential Equations by Mikhailov introduces the necessary Lebesgue integration and functional analysis prerequisites in a 60-page preliminary chapter. However, the book presupposes acquaintance with the theory of ordinary differential equations. The preface states the following three prerequisites.

  • A Course of Mathematical Analysis (two volumes) by Nikolsky

  • Foundations of Linear Algebra by Mal'cev

  • Ordinary Differential Equations by Pontryagin

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  • $\begingroup$ Right. My follow-up question would be whether the theory of ODE's is an absolute pre-requisite for any course on PDE's taught from any book (at the advanced undergraduate/graduate level)? If not, is there any way I can skirt around it? I don't mind learning measure theory and functional analysis in a self-contained manner from a textbook that covers PDE's. $\endgroup$ – user82261 Jul 4 '18 at 6:46

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