Partial Differential Equations

by Lawrence C. Evans

This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail. … Evans' book is evidence of his mastering of the field and the clarity of presentation. ―Luis Caffarelli, University of Texas It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations … Every graduate student in analysis should read it. ―David Jerison, MIT I usePartial Differential Equationsto prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's … I am very happy with the preparation it provides my students. ―Carlos Kenig, University of Chicago Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge … An outstanding reference for many aspects of the field. ―Rafe Mazzeo, Stanford University

Genres: Mathematics, Textbooks, Reference, Science, Technical

749 pages, Hardcover

First published June 1, 1998

Reviews

[Name · Rating 4 out of 5]

The bible of PDE's. If you're not familiar with his notation he introduces it in the beginning of the text. This work is readable and understandable which is often rare for mathematics at this level. Evans is thorough in his covering of nearly the breadth of PDE's up to the point of publication. Enough for a graduate level sequence in the subject, and perhaps then some. A sufficient mastery of multivariate calculus and ordinary differential equations is necessary to tackle this work. Some previous exposure to partial differential equations would be beneficial.

[Tomáš Ševček · Rating 5 out of 5]

This book provides an excellent treatment of PDEs for (pure/general) mathematicians. If you are interested in PDEs, you should definitely consider reading this book (albeit this book requires 'active reading' with pen and paper as parts of the proofs tend to be left as an exercise for the reader). The first part introduces various methods to find classical solutions of certain linear and nonlinear equations (separation of variables, characteristics, transform methods, similar solutions, Green's functions, power series, etc.). You should be familiar with vector calculus, parametric improper integrals, the Fourier series and ODEs (among others) to fully appreciate this part of the book. The second part introduces Sobolev spaces and analyzes the properties of weak solutions of second-order linear elliptic, parabolic and hyperbolic equations. The third part introduces techniques to tackle nonlinear equations (the calculus of variations, fixed-point theorems, nonlinear semigroup theory, etc.). For these parts, functional analysis is necessary. Not much measure theory is generally required except for the theorems for changing the order of a limit and the Lebesgue integral (the appendix contains the bare minimum on functional analysis and the measure theory).
The book is a great choice if you want to learn how to prove the (non)existence and uniqueness of (classical and weak) solutions, analyze their regularity, derive certain bounds and so on. However, if you are more of an applied mathematician looking to learn how to actually derive mathematical models of real-world phenomena with PDEs or if you want to study specific parts of (applied) PDEs like reaction-diffusion systems, traveling wave solutions and their stability, steady-state solutions and their stability, dispersive waves and similar concepts, then you will probably be better off with different PDE textbooks, such as those by Logan, Debnath or Haberman or even with an applied textbook (like Murray or Kot for biology).

[HikBook]

5.6 Sobolev inequalities.
If a function u belongs to the Sobolev space W^{1,p}, does u automatically belong to a certain oher spaces?
Yes, but, this depends on p and n.

The estimate |u|_L^q<=C|Du|_L^p implies that q=np/(n-p) (with 1<=p
Application of Schaefer's Theorem in PDE
9.2 Fixed point method.
Example 2. (A cuasilinear elliptic PDE). Application of Schaefer's Theorem for solving the boundary-value problem -Laplacian(u)+b Du+au=0 in U and u=0 on the boundary of U.
Here, U is bounded with a "nice" boundary. The function b:R^n->R is smooth, Lipschitz continuous and |b(xi)|<=C(|xi|+1) (growth condition).

[Owen Melia · Rating 4 out of 5]

I learned this year that the Evans textbook is classic for learning the theory of PDEs. I can see why. It's self-contained but assumes a background in vector calculus. The book is well-written and cleverly organized. Before getting into general theory, it goes over four important example PDEs to give a flavor of the results and techniques. It doesn't give a lot of physical intuition for the PDEs, so if you're into that type of stuff it's BYOPL (bring your own physics lecturer).

[Matthew Zhang · Rating 4 out of 5]

Useful if a little dry.