Partial differential equations: topics in fourier analysis

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. Provides explicit formulas for the solutions of PDEs important in physics ; Solves the equations using methods based on Fourier analysis; Presents the equations in order of complexity, from the Laplacian to the Hermite operator to Laplacians on the Heisenberg group; Covers the necessary background, including the gamma function, convolutions, and distribution theory; Incorporates historical notes on significant mathematicians and physicists, showing students how mathematical contributions are the culmination of many individual efforts. Includes exercises at the end of each chapter.

"A Chapman & Hall book."

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.