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Full 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Complete revision of the text to correct errors, remove redundancies, and update outdated material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.
Contents
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.
