 | Summaries and Customer Reviews are supplied by Amazon.com | 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. |
Average Customer Rating:   Very good PDE book Used this for a two year sequence PDE class, our teacher is fantastic and she was a student from Evans'. Love it. Clearly not for graduate students. Still very good! Evans is not really worried about graduate students. However this book is essential for anyone that wants to have real (first) contact with nonlinear PDE. I'm sorry for the beginners but it has to be this way. If you are looking for some engineering do not buy it. Mathematician? What are you waiting for? A Fine Treatise on the Subject This is a superb exposition of a difficult, yet enriching subject. This book is intended only as a beginning text (in a relative sense) and is by no means an attempt to give an exhaustive view of many topics discussed therein.
The first few chapters discuss classical solution techniques to frequently encountered PDEs such as the heat and Laplace equation. Methods of solution are discussed including Fourier transform methods and other classical methods to obtain strong solutions and/or representation formulas. The author, from this point, focuses on weak solution techniques for second order PDEs and systems in addition to conservation laws and other nonlinear PDEs. There is also a self-contained chapter on Sobolev spaces that proves to be fairly useful.
There is a necessary mathematical maturity needed to fully benefit from this text. The reader should be relatively comfortable with standard topics from classical analysis. It would help if the reader has seen Lebesgue spaces and is familiar with basic functional analysis and operator theory although many of these topics are reviewed in the apendices.
While this book is dense and difficult at times, it has a prominent place on my bookshelf. Solid opening, weak ending. 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. The best book in PDE If you want to learn PDE you have to study this book... as simple as that. |  |
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