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主 编 杨劲根
副主编 楼红卫 李振钱 郝群
编写人员（按汉语拼音为序）
陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林
1. 序言
2. 非数学专业的数学教材
3.数学分析和泛函分析
4.单复变函数
5.多复变函数
6.代数
7.数论
8.代数几何
9.拓扑与微分几何
10.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
书名： Hyperbolic Partial Differential Equations
作者： Peter D. Lax
出版商： American Mathematical Society, Providence, Rhode Island
页数： 217
适用范围：数学专业研究生
预备知识：泛函分析，常微分方程
习题数量：无
推荐强度： 10
书评： 本书是柯朗研究所系列讲义丛书之一。其作者是美国著名数学家 Peter D. Lax 教授。他在双曲型方程尤其是拟线性双曲型守恒律方程组及其计算，孤立子理论， 拟微分算子理论等诸多方面作出了开创性及里程碑式的工作。 本书恰是作者对双曲型方程理论知识方面的讲解。 全书包含十章及五个附录，介绍了线性双曲型方程及方程组的一些基本概念，能量估计， 解的存在性以及解的其他性质，还介绍了线性双曲型方程及方程组的差分格式及其差分格式的稳定性， 以及散射理论和拟线性双曲型守恒律方程组一些基本理论。 附录中最后一章由美国著名数学家 Morawetz 教授执笔。对于想从事双曲型方程研究的读者来说， 这是一本很好的入门书。（张永前）
书名： Partial Differential Equations, An Introduction
作者： Walter A. Strauss
出版商： John Wiley ＆ Son Inc.
页数： 425
适用范围：大学数学系本科高年级学生或低年级研究生
预备知识：微积分，线性代数，常微分方程
习题数量：较大
习题难度： 较难
推荐强度： 9
书评： Walter A. Strauss 是美国 Brown 大学数学系教授，著名的偏微分方程专家。本书自出版以来， 被美国多所著名大学作为本科生的偏微分方程课程的教科书。 全书包含十四章及一个附录，介绍了几类重要的偏微分方程的来源和基本性质以及基本的研究方法， 并在最后两章中介绍了一些来自物理学的非线性偏微分方程方面的进一步课题。 书中习题类型甚广，而且还配有部分习题的答案供读者参考。 本书适合作为偏微分方程的教材及参考书。（张永前）
国外评论摘选
1 ） This 1992 title by Strauss (professor at MIT) has become a standard for teaching PDE theory to junior and senior applied maths and engineering students in many American universities. Last year, being an informal teaching assistant for the class, I found many of the students struggling with the concepts and exercises in the book. Admittedly the style of writing here is very dense and if the reader does not have a very strong background in the topic, chances are high he or she will face a grand level of frustration with the exposition and the subject as a whole. One would need perseverance and dedication working numerous hours with this text before things start to settle in. After about the second or third chapter onward, those who were still taking the class had an easier time understanding the material and doing the excercises.
Contentwise, after a brief and important introductory chapter (which should not be skipped by any reader!) the book first focuses on the properties and methods of solutions of the one-dimensional linear PDEs of hyperbolic and parabolic types. Then after two separate chapters, one on the trio of Dirichlet, Neumann, and Robin conditions and the other on the Fourier series, the author embarks upon the discussion of elliptic PDEs via the methods of harmonic analysis and Green's functions. Subsequently there is a brief introduction to the numerical techniques for finding approximate solutions to the three types of PDEs, mostly centered on the finite differences methods.
The beginning of roughly the second half of the text is devoted to the higher-dimensional wave equations and boundary conditions in plane and space, utilizing the machinery of Bessel and Legendre functions, and ending up with a section on angular momentum in quantum mechanics. In the following, Dr. Strauss brings up the discussion of the general eigenvalue problems, and then proceeds with a treatment of the advanced subject of weak solutions and distribution theory. (This topic is normally skipped in an undergraduate course.) The last two chapters are a pure delight to read, dealing with the PDEs from physics as well as a survey of the nonlinear phenomena (shocks, solitons, bifurcation theory). A few appendixes at the end, summarize the analysis background needed for the course and must be consulted before and during the first reading.
All in all this is a very splendid source for all the applied maths and engineering students, that can be used in conjuction with other references to help break through the conceptual barriers. In fact, I recommended the book by Stanley Farlow to our students and many found the presentation there very modular and accessible. For example, some of the Strauss' homework problems, such as solving the Poisson equation on an annulus, were subjects of a single chapter in Farlow. In any event, I am very much hoping to see a new and more student-friendly edition of the Strauss' text be prepared and issued in the near future.
2 ） I used this book in a tough applied math course, and the quality of this book did not help matters much. There are a couple of good things about this book. The material chosen is appropriate and reasonably comprehensive for an intro PDE text. In other words, the table of contents is a nice read. The notation is very clean and concise throughout, as is the typesetting. The bibliography was also useful, pointing me to some great supplementary texts. Now for the bad parts. An intro PDE book should explain clearly the basic concepts behind PDEs, including how certain famous equations (wave, heat, Laplace , etc.) arise in physical modeling. It should explain in detail the various computational techniques for finding analytical solutions to these equations. It should explain relevant elementary theorems needed for these computational techniques. This book attempts to do all of these things, but does so poorly. The basic problem is that the book's explanations and examples are too terse and incomplete for an introductory text. Analytically solving a PDE is a relatively difficult task, involving several computational steps and techniques. Examples of these techniques should be worked in detail, but in this book, they frequently omit steps or fail to explain where or how a particular technique is being applied. Theorems are often not stated, or if they are, proofs are either omitted or partially sketched. This makes the book difficult for beginners, but it is not a terrible reference if you have already been exposed to the material.
My advice: given the price of this book and its mediocre quality, you would do better by looking elsewhere for an intro PDE text.
书名： Partial Differential Equations
作者： Lawrence C. Evans
出版商： American Mathematical Society, Providence, Rhode Island
页数： xviii+662
适用范围：数学专业研究生
预备知识：微积分，线性代数，常微分方程
习题数量：较大
习题难度：适中
推荐强度： 10
书评： Lawrence C. Evans 是著名的偏微分方程专家，美国加州大学伯克利分校的教授。本书 被美国多所著名大学采用作为研究生偏微分方程课程的教科书或者参考书。 作者在这本教材中介绍了偏微分方程的许多重要课题，重点介绍了偏微分方程各种现代处理方法。 内容主要分为三部分。第一部分介绍了一些偏微分方程解的表示， 其中包含了一阶非线性偏微分方程的特征线方法， Hamilton-Jacobi 方程和双曲守恒律方程组解的表示方法 以及一些特殊非线性方程的行波解等。 第二部分主要介绍了处理二阶线性椭圆型方程及抛物型方程的现代方法。 第三部分介绍处理非线性方程的变分和非变分方法以及处理 Hamilton-Jacobi 方程和双曲守恒律方程组 的一些现代方法。每章之后都给出了相关内容的出处的说明以及与正文内容紧密配合的习题。 本书极适合作为研究生的偏微分方程教科书。（张永前）
国外评论摘选
1) 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.
2) I have taught a one-year course in PDE based on Evans' book and found it extremely cogent and stimulating both for myself and for the students. The treatment is up-to-date, with a definite nonlinear flavor. Beyond that, the exercises are very good, and the treatment is sufficiently detailed to make class preparation fairly fast. It does demand mathematical dexterity and maturity of the students right from the start, though.
3 ） 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.
书名： Partial Differential Equations
作者： Fritz John
出版商： Springer-Verlag, New York-Berlin
页数： x+249
适用范围：大学数学系本科
预备知识：微积分，线性代数，常微分方程
习题数量：中等
习题难度：适中
推荐强度： 10
书评： 本书是系列丛书《 Applied Mathematical Sciences 》的第一卷。其作者是已故著名数学家﹑美国纽约大学 Courant 研究所 Fritz John 教授。其内容包括：一阶偏微分方程， Cauchy-Kovalevskaya 定理 , Holmgren 定理 , Lewy 的著名反例，波动方程及 Poisson 方程﹑热传导方程和相应高阶方程及对称双曲组的性质及 相应定解问题的基本求解方法。既介绍了特征线方法等经典方法， 也介绍了差分方法及 Hilbert 空间理论等近代方法。本书已被四次再版， 作者在每个新版中加入了一些新的内容，如在第四版中加入了关于 Cauchy-Kovalevskaya 定理解的存在区域大小的讨论等。 1991 年 Springer-Verlag 出版社又重印了其第四版， 可见其影响。我国科学出版社也在 1986 年翻译出版了其第四版。该书由朱汝金教授翻译。 书后习题与正文内容紧密配合，有助于对所介绍方法的理解。 适合偏微分方程的初学者作为入门及参考书。（张永前）