Partial Differential Equations - Topics in Fourier Analysis

Description

Partial Differential Equations: Topics in Fourier Analysis, Second Edition 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; and 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. New to the Second Edition: Three brand new chapters covering several topics in analysis not explored in the first edition Complete revision of the text to correct errors, remove redundancies, and update outdated material Expanded references and bibliography New and revised exercises. Cover Half Title Title Page Copyright Page Contents Preface 1. The Multi-Index Notation 2. The Gamma Function 3. Convolutions 4. Fourier Transforms 5. Tempered Distributions 6. The Heat Kernel 7. The Free Propagator 8. The Newtonian Potential 9. The Bessel Potential 10. Global Hypoellipticity in the Schwartz Space 11. The Poisson Kernel 12. The Bessel–Poisson Kernel 13. Wave Kernels 14. The Heat Kernel of the Hermite Operator 15. The Green Function of the Hermite Operator 16. Global Regularity of the Hermite Operator 17. The Heisenberg Group 18. The Sub-Laplacian and the Twisted Laplacians 19. Convolutions on the Heisenberg Group 20. Wigner Transforms and Weyl Transforms 21. Spectral Analysis of Twisted Laplacians 22. Heat Kernels Related to the Heisenberg Group 23. Green Functions Related to the Heisenberg Group 24. Theta Functions and the Riemann Zeta-Function 25. The Twisted Bi-Laplacian 26. Complex Powers of the Twisted Bi-Laplacian Bibliography Index