Good PDE books for a graduate student?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, but I did not receive a good training in hard analysis.

So I do not mind if the book requires a lot functional analysis, but I would like one that provides more intuition and motivation rather than just hard techniques and estimates.

I know Evan’s is the best, and I read most materials from the second part when I was doing a project on theory of distributions. But I cannot say I enjoy his style very much.

So can someone introduce a gentler book? Thanks so much!

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I really like Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. It has more of a functional analysis flavor than many other books.

Aside from the standard PDE textbooks for beginner (Evans) and for researcher (GT), I have recently run into a great book, Linear and Nonlinear Functional Analysis with Applications, by Philippe G. Ciarlet.

Here I just need to quote the preface of it:

(Begin quote) Why write another textbook on functional analysis and its applications, since there are
already many excellent textbooks around?

Apart from the personal pleasure that such an exercise provides to an author, there are
other reasons: One, which perhaps constitutes the main originality of this text, was to assemble in a single volume the most basic theorems of linear and of nonlinear functional analysis; another reason was to simultaneously illustrate the wide applicability of these theorems by treating an abundance of applications.

Applications to linear and nonlinear partial differential equations treated here include
Korn’s inequality and existence theorems in linear elasticity, obstacle problems, the Babuska–Brezzi inf-sup condition, existence theorems for the Stokes and Navier–Stokes equations of fluid mechanics, existence theorems for the von Karman equations of a nonlinearly elastic plate, and John Ball’s existence theorem in nonlinear elasticity. A variety of other applications deals with selected topics from numerical analysis and optimization theory, such as approximation theory, error estimates for polynomial interpolation, numerical linear algebra, basic algorithms of optimization, Newton’s method, or finite difference methods.

A special effort has been made to enhance the pedagogical appeal of the book. After Chapter 1, which is essentially a review of results from real analysis and the theory of functions that will be used in the text, self-contained and complete proofs of most of the theorems are provided. These include proofs that are not always easy to locate in the literature, or difficult to reconstitute without an extended knowledge of collateral topics; for instance, self-contained proofs are given of the Poincare lemma, of the hypoellipticity of the Laplacian, of the existence theorem for Pfaff systems, or of the fundamental theorem of surface theory.
Numerous figures and problems (almost 400) have also been included. Historical notes and
original references (at least those that I have been able to trace with a reasonable assurance of veracity) have also been included (mostly as footnotes), so as to provide an idea of the genesis of some important results.

It is my belief that this book contains most of the core topics from functional analysis
that any analyst interested in linear and nonlinear applications should have encountered at least once in his or her life. More specifically, linear functional analysis and its applications are the subjects of Chapters 2–6, while nonlinear functional analysis and its applications are the subjects of Chapters 7–9. (End quote)

Here is the Table of Contents, if you will be a researcher in a field relevant to PDE or applied math, you will like this book and appreciate its way of presentation. At least I already fell in love with this book.

You might check out Garabedian’s Partial Differential Equations. One of Evan’s students told me that he thought it was “dated”, and I guess it is a tad on the old side, but it is very readable, not too hard on the analysis, and covers a lot of territory.

As you said Evan’s Partial Differential Equations is a very good book. Gilbarg and Trudinger Elliptic Partial Differential Equations of Second Order is a masterpiece of the subject, but it is a very heavy book and sometimes notation is a nightmare (Schauder’s estimates made me cry 🙁 ). I also would recommend An Introduction to Partial Differential Equations by Renardy and Rogers: as Evan’s book it covers a lot of topics and there you can find chapters dedicated to the functional analysis that is needed.

Books I barely touched but that you could find interesting are Partial Differential Equations: An Introduction by Strauss and Introduction to Partial Differential Equations with Applications by Zachmanoglou and Thoe (the latter is a Dover publication and hence it should be the cheapest)

The book by F. Lin and Q. Han on elliptic PDE’s is a great reference, covering a large variety of topics.