Partial Differential Equations II: Qualitative Studies of Linear Equations

The second volume in a three-volume series builds upon the basic theory of linear PDEs and introduces advanced topics such as pseudodifferential operators, functional analysis, and differential geometrical concepts. It covers spectral theory, scattering of waves, index theory, Brownian motion, and diffusion, and is targeted at graduate students and professional mathematicians. The third edition further expands the material with new theorems and applications.

Format: Hardback
Length: 687 pages
Publication date: 07 December 2023
Publisher: Springer International Publishing AG

This second installment in a trilogy of three volumes expands upon the fundamental theory of linear PDEs presented in volume 1, delving into more advanced topics. The analytical tools introduced herein include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. Moreover, the book endeavors to develop fundamental differential geometrical concepts, with a central focus on curvature. The topics covered in this comprehensive text encompass spectral theory of elliptic differential operators, the theory of wave scattering by obstacles, index theory for Dirac operators, and Brownian motion and diffusion. Designed specifically for graduate students in mathematics and professional mathematicians with a keen interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis, this edition further enhances the material by incorporating new theorems and applications throughout the book. Additionally, it strengthens connections and establishes interrelationships between concepts across chapters.

The third edition of this text takes the content to new heights by incorporating numerous groundbreaking theorems and practical applications throughout. It deepens the connections between various chapters, fostering a more coherent understanding of the subject matter. Furthermore, the book expands its coverage by introducing new sections on rigid body motion, probabilistic results related to random walks, aspects of operator theory related to quantum mechanics, overdetermined systems, and the Euler equation for incompressible fluids. The appendices have also been extensively updated, presenting additional results ranging from weak convergence of measures to the curvature of Kahler manifolds.

Michael E. Taylor, a distinguished Professor of Mathematics at the University of North Carolina, Chapel Hill, NC, has authored this remarkable text. His extensive expertise and profound insights make this book an invaluable resource for scholars and practitioners alike. The first edition of this trilogy received widespread acclaim, with Peter Lax, a renowned expert in the field, praising it as a "treasure trove of information." The second edition continues to uphold the high standards set by its predecessor, making it an essential reference for anyone seeking to delve into the modern theory of partial differential equations and their applications.

Weight: 1324g
Dimension: 162 x 242 x 40 (mm)
ISBN-13: 9783031336997
Edition number: 3rd ed. 2023