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Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice one to the audience consisting of graduate students who had taken the standard first-year analysis courses but who had little background in PDE. The later one makes fairly free use of the techniques of real and complex analysis.

Could any expert here suggest a PDE book in the level between these two ones, or a complementary reading for Folland's book?

I don't know if this is a appropriate question here --- I have no idea if the question is too localized, I learned from the internet that these two books are very classic for PDE though. Any suggestions for a good update of the question will be really appreciated.

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10 Answers 10

Fritz John's book "Partial Differential Equations" is one of the more elementary but still substantial PDE books I have run across.

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It also has the virtue of being a thin volume. –  Steve Huntsman Aug 8 '11 at 1:41

L. Craig Evan's textbook, also called "Partial Differential Equations", is also a pretty standard text. It is aimed at the graduate level, but assumes very little analysis at the outset. Furthermore, all the analysis that you do need to know is contained in a very extensive appendix that contains such things as basic multi-variable calculus identities, metric space inequalities, measure theory, and functional analysis. I've used this book for the past two years, and I enjoy it a lot as well.

I might also be slightly biased because the author is in my department. =P

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I'd second that recommendation; Evans is a great book, and a good preparation for Folland. –  user59001 Apr 29 at 11:51

I like Jeffrey Rauch's book Partial Differential Equations. Fritz John's book is also very good.

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I think that the book of Fritz John is still the most appropriate beginner graduate level book, and it does lie, in difficulty, between the books of W. Strauss and G. Folland. The book of Strauss is the main text for undergraduate PDEs in my department, whereas the book of F. John was the main textbook at Courant Institute, when I was a graduate student there.

The great thing about John's book is that it contains a significant number non-trivial results, it covers many areas of PDEs and it is still neither too advanced nor too big.

However, to the best of my knowledge, the most popular, in graduate departments, book on PDEs is the one of L.C. Evans (now in 2nd edition). Its problem (if that's a problem) is that it is too big, but this is because it covers extensively too many topics.

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Evans's one is indeed very big... –  Jack Dec 21 '13 at 20:37

I want to add that when we put up Evans and Renardy & Rogers, one could also take a solid look at McOwen's text, Partial Differential Equations: Methods and Applications. I've referred to it several times with confidence. Great for various examples and concise information.

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'Partial Differential Equations' by Paul Garabedian is an excellent text 'between' Strauss and Folland. The book rewards repeated reading, and contains a wealth of material and insight. It doesn't use the analytical machinery of modern PDE (e.g. does not use Sobolev spaces). It includes topics like the Perron construction, the properties of the Neumann function (as opposed to the Green's function) for a domain, and the Hamilton-Jacobi PDE. My advisor gave me his copy many years ago, saying 'this will be a good friend'.

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While Evans (which I think fits your requirement) is probably the 'standard' text, I slightly prefer Renardy and Rogers' book, which covers roughly the same material.

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In addition to the excellent books mentioned in the previous answers, Qing Han wrote a new book called A basic course in partial differential equations, which emphasizes a priori estimates from the very beginning.

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I think that an excellent book for linear PDE's is by F. Treves, and is called "Basic Linear Partial differential equations". It is excellent because it uses the language of distributions from the very first page, and also, explains details by giving both the "big picture" based on symmetry arguments rather than just beating estimates to death (being able to estimate is crucial, but it's important to know what's at the heart of those estimates). There are several exercises, which even explore connections with several complex variables from time to time.

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For classical PDE theory, I would agree with several other posters that Evans is a great pre-Folland text. I would also suggest a less conventional text for more modern aspects of PDE theory: An Introduction to Pseudo-Differential Operators by Man Wah Wong. It's a short and very easy to read text (perhaps even easier than Evans) that can be used as a supplement or as a stand-alone text.

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