Lawrence C. Evans’ seminal work, “Partial Differential Equations,” has been a cornerstone in the field of PDEs since its first edition. Now, with the release of the second edition, Evans has further solidified his book’s position as an indispensable resource for graduate students and researchers alike. This updated version not only presents the fundamental concepts and techniques in the theoretical study of PDEs but also delves into modern approaches, particularly in the realm of nonlinear equations.
The second edition, published on March 22, 2022, is a substantial expansion of its predecessor. It introduces a new chapter dedicated to nonlinear wave equations, offering fresh insights and the latest research in this dynamic area. Additionally, the book includes over 80 new exercises, providing ample opportunities for readers to test their understanding and apply the concepts learned. Several new sections have also been added, and the bibliography has been significantly expanded, reflecting the most recent developments in the field.
A Deeper Look into the Content
One of the key highlights of this edition is the new chapter on nonlinear wave equations. This addition significantly broadens the scope of the book, covering critical cases of PDEs that are of great interest to researchers. The chapter offers a thorough treatment of the subject, presenting both theoretical foundations and practical applications. For example, the book discusses the nonlinear wave equation:
$$
u_{tt} – \Delta u = f(u)
$$
where $u(x, t)$ is the unknown function, $\Delta$ is the Laplacian operator, and $f(u)$ is a nonlinear function of $u$. This equation has numerous applications in physics, such as in the study of vibrating strings or membranes.
Another notable feature is the expanded exercise set. With over 80 new problems, the book caters to a wide range of difficulty levels and topics. This makes it an excellent choice for both classroom use and self-study. The exercises are carefully crafted to reinforce the concepts presented and to encourage readers to think critically about the material. For instance, one exercise might ask the reader to prove the uniqueness of the solution to the following initial value problem:
$$
\begin{cases}
u_t + u_x = 0, & x \in \mathbb{R}, t > 0 \\
u(x, 0) = g(x), & x \in \mathbb{R}
\end{cases}
$$
where $g(x)$ is a given function.
The book’s comprehensive approach is another strength. It covers both classic and modern aspects of PDEs, making it suitable for a graduate-level course as well as a reference for researchers. The text strikes a balance between depth and accessibility, providing detailed explanations without sacrificing clarity. For example, the book presents a thorough treatment of the heat equation:
$$u_t – \Delta u = f(x, t)$$
where $u(x, t)$ represents the temperature at position $x$ and time $t$, and $f(x, t)$ is a given heat source. The book discusses various methods for solving this equation, such as the method of separation of variables and the use of Green’s functions.
Praise from Experts
“Partial Differential Equations” has garnered praise from some of the most respected names in the field. Luis Caffarelli, a renowned mathematician and professor at the University of Texas at Austin, commends Evans’ work for its “insightful combination of technical detail and clear presentation.” Similarly, Carlos Kenig, a distinguished professor at the University of Chicago, recognizes the book’s value as both a pedagogical tool and a reference material.
These endorsements underscore the book’s significance in the mathematical community. Its depth, clarity, and comprehensive scope make it a preferred choice among educators and students alike.
Why Every Aspiring Mathematician Should Read This Book
For anyone pursuing advanced studies in analysis, “Partial Differential Equations” is a must-read. The book provides a solid foundation in the theory of PDEs while also introducing readers to cutting-edge topics. Its detailed exposition prepares students not just for academic success but also for future research endeavors in the field of mathematics.
Moreover, the book’s clear and engaging writing style makes it accessible to a wide audience. Whether you are a graduate student, a researcher, or simply someone with a keen interest in PDEs, this book will undoubtedly enhance your understanding and appreciation of the subject.
A Classic in Mathematical Literature
As part of the prestigious Graduate Studies in Mathematics series, “Partial Differential Equations” is more than just a textbook; it is a comprehensive guide that reflects the ever-evolving landscape of PDEs. Its status as a classic in mathematical literature is well-deserved, and its continued relevance in educational and professional settings is a testament to its quality and impact.
In conclusion, the second edition of Lawrence C. Evans’ “Partial Differential Equations” is an essential addition to any mathematician’s library. Its comprehensive coverage, clear explanations, and extensive exercise set make it an invaluable resource for graduate students and researchers alike. If you are serious about pursuing a deeper understanding of PDEs, this book is an investment that will pay dividends for years to come.