Partial Differential Equations: Topics in Fourier Analysis explains how to use the Fourier transform and heuristic methods to obtain significant insight into the solutions of standard PDE models. It shows how this powerful approach is valuable in getting plausible answers that can then be justified by modern analysis.
Using Fourier analysis, the text constructs explicit formulas for solving PDEs governed by canonical operators related to the Laplacian on the Euclidean space. After presenting background material, it focuses on:
- Second-order equations governed by the Laplacian on Rn
- The Hermite operator and corresponding equation
- The sub-Laplacian on the Heisenberg group
Designed for a one-semester course, this text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Through its coverage of fundamental examples of PDEs, the book prepares students for studying more advanced topics such as pseudo-differential operators. It also helps them appreciate PDEs as beautiful structures in analysis, rather than a bunch of isolated ad-hoc techniques.
Table of Contents
The Multi-Index Notation
The Gamma Function
The Heat Kernel
The Free Propagator
The Newtonian Potential
The Bessel Potential
Global Hypoellipticity in the Schwartz Space
The Poisson Kernel
The Bessel-Poisson Kernel
The Heat Kernel of the Hermite Operator
The Green Function of the Hermite Operator
Global Regularity of the Hermite Operator
The Heisenberg Group
The Sub-Laplacian and Twisted Laplacians
Convolutions on the Heisenberg Group
Wigner Transforms and Weyl Transforms
Spectral Analysis of Twisted Laplacians
Heat Kernels Related to the Heisenberg Group
Green Functions Related to the Heisenberg Group
M.W. Wong is a professor in and former chair of the Department of Mathematics and Statistics at York University in Toronto, Canada. From 2005 to 2009, he was president of the International Society for Analysis, its Applications and Computations (ISAAC).
"… nicely presented material … includes theorem with bibliographical references, exercises, and historical notes. … a useful complement to the existing literature for those who study partial differential equations."
—Zentralblatt MATH 1286