Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 [Hardcover]

Lawrence C. Evans (Author)

Book Description

ISBN-10: 0821807722 | ISBN-13: 978-0821807729 | Publication Date: June 1, 1998

This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: 1) representation formulas for solutions, 2) theory for linear partial differential equations, and 3) theory for nonlinear partial differential equations.

Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and much more.

The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he emphasizes the modern interplay between functional analytic insights and calculus-type estimates within the context of Sobolev spaces. Treatment of all topics is complete and self-contained. The book's wide scope and clear exposition make it a suitable text for a graduate course in PDEs.

Editorial Reviews

Review

"...Highly recommend the book for students as well as for lectures in PDE." -- Monatshefte fur Mathematik

"...entirely possible that the ...text could eventually become the benchmark. ...extremely well written..." -- SIAM Review

"For a student wishing to specialise in the theory of PDEs it provides a very solid foundation." -- The Mathematical Gazette

"Highly recommend the book for students as well as for lectures in PDE." -- Monatshefte fr Mathematik

"It is a standard treatment with good notation. It is extremely well written, with a very attractive format." -- SIAM Review

"This excellent textbook [is recommended] as the first textbook for anyone who wants to learn the theory of [PDE]" -- European Mathematical Society Newsletter

"Well written; proofs are given in full detail and pictures are inserted when needed; moreover, the exposition is perfectly self-contained, the development within each part and section being rigorous and complete." -- Mathematical Reviews

For a student wishing to specialise in the theory of PDEs it provides a very solid foundation. -- The Mathematical Gazette

Product Details

• Hardcover: 662 pages
• Publisher: Amer Mathematical Society (June 1, 1998)
• Language: English
• ISBN-10: 0821807722
• ISBN-13: 978-0821807729
• Product Dimensions: 10.2 x 7.3 x 1.5 inches
• Shipping Weight: 2.8 pounds
• Average Customer Review: 4.5 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #66,594 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

51 of 53 people found the following review helpful:

4.0 out of 5 stars Solid opening, weak ending., March 3, 2007

By 

T. Butler "mathman" (Fort Collins, CO United States) - See all my reviews
(REAL NAME)   

This review is from: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 (Hardcover)

If you are just getting started in learning PDEs and want to see all the classical problems/solutions (Poisson, Laplace, Heat, and Wave Equations), then this book might be a little advanced for you, but it is solid in this content if you have a solid background in analysis (probably best to have at least one high-level analysis class that covers all the multivariable calculus material as you will find that your ability to identify and use Green's Theorems will make life much easier as you get started). This is considered "Part I" of the book.

Once you have covered all the nice problems that don't exist in practice, you are ready to move onto general linear PDE theory in Part II of the book. I would recommend you complete a course in measure theory before you start in on chapter 5, which covers Sobolev spaces. I would then recommend that you complete a course in functional analysis before starting chapter 6 or 7 (chapters 5-7 are Part II of this book). This is not necessary as you will have access to a fairly complete appendix of functional analysis results in this book, but once you understand functional analysis and measure theory, then you will be able to grasp the idea of an elliptic (or in chapter 7, parabolic or hyperbolic) operator acting on a function space (the function space being a Sobolev space) more easily and these ideas won't seem so abstract. Overall, the second part of this book is great if you have a lot of the prerequisites I just suggested because many of the proofs can easily be made to be three to five times longer as many steps that link ideas in functional analysis are skipped. The proofs on higher regularity will be hard to understand your first time through, so I wouldn't worry about it too much. Read through the chapters and then read through the regularity stuff again. If you just want to get the basic ideas you can skip either the parabolic or hyperbolic section in chapter 7 because the techniques in solving either type of problem are fairly similar.

Once you are done with the linear PDE theory and are ready to start chapter 8, I recommend putting the book down and getting a different one. Evans gets fairly abstract in the nonlinear part of the book (Part III). I would recommend getting "Navier-Stokes Equations: Theory and Numerical Analysis" by Temam as it is a great source for nonlinear PDE theory and has more results and better proofs than Evans on this subject. I just feel like the Evans book is a great book to learn from for your first two semesters of PDEs at a graduate level, but after that it is time to change texts.

35 of 38 people found the following review helpful:

3.0 out of 5 stars A Decent Introduction to PDE, June 27, 2005

By 

White Rabbit - See all my reviews

This review is from: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 (Hardcover)

I've seen a lot of positive reviews of this text, and I feel the need to explain some cons of this book. Before that, I will say this is probably the best introduction to PDE theory out there. This is NOT a book for people looking for a dissertation on undergraduate methods of solution (separation of variables, fourier series, etc.). If that is what you are looking for, go to Haberman or perhaps Strauss.

Ok, so here are the problems I see with this text. First, there is no mention of distributions in this book. Evans addresses this in the intro., saying it's not necessary. I find that hard to swallow, given that fundamental solutions play a big part in the text. Despite this, Evans devotes parts of the book to going into very esoteric subjects like mean value theorems for the heat equation. The other glaring gap in this text is the absence of Schauder estimates; a corner-stone for linear elliptic theory. On a note of personal preference, I would have like to have seen more of the book dedicated to a functional analytic foundation; the appendicies that are present are simply not enough.

Overall, the book gives a decent introduction; but is far from being self-contained and is not enough of a foundation for people wishing to pursue research in PDE. Evans does acknowledge this in his introduction, but I think its something that is frequently overlooked in reviews of this text.

22 of 23 people found the following review helpful:

5.0 out of 5 stars PDE making sense, February 10, 2004

By A Customer

This review is from: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 (Hardcover)

This is a textbook for a first-year graduate course in PDE (for mathematics students). You should take courses in analysis (on the level of Rudin) and measure theory before you expect to understand everything in this book.

This is by far the best book on PDE. The text is extremely clear, and most of the rather technical proofs are prefaced with "heuristic" calculations to help the reader understand what is going on. The chapter on the calculus of variations is the best exposition I have found of the subject, and Evans completely dispenses with the awful "delta" notation which never made any sense.

The text doesn't make much use of the Fourier transform and doesn't even mention distributions, and this gives his book a definite nonlinear flavor (which is a good thing). This should become the standard introduction to PDE on the graduate level.