This book presents a collection of topics in partial differential equations designed to serve as a bridge to research for Master's students.

Drawing on material originally developed for student projects, the selected topics require only standard prerequisites such as differential calculus, complex analysis, Fourier analysis, the Lebesgue integral, and some functional analysis. They span a broad range of problems in PDE, including Burgers' equation, the wave equation, spectral theory of the Laplacian, and harmonic functions. The topics and their presentation are intended to help readers access the research literature.

Offering a diverse set of themes that highlights the richness of the field and opens the door to research, this book will be of interest to graduate students studying PDEs. With original material closely related to the author's own research, it can also serve as a concise reference for researchers.

Covers an array of diverse topics in PDEs and their applications Includes original material informed by the author's research Contains non-classical examples illustrating the theory

Autorentext

Claude Zuily is an emeritus professor in mathematics at Paris-Saclay University specializing in linear and nonlinear partial differential equations (PDEs) including microlocal analysis. He is the author of several textbooks for mathematics students, both in French and in English.

Inhalt

Chapter 1. The Method of Characteristics.- Chapter 2. Cauchy-Kovalevski Theorem.- Chapter 3. The method of stationary phase.- Chapter 4. Continuity of nonlinear maps on Sobolev and Lebesgue spaces.- Chapter 5. The Burgers Equation.- Chapter 6. Mathematical analysis of surface waves.- Chapter 7. Continuity of the Dirichlet-Neumann Operator.- Chapter 8. Spectral theory of the Laplacian on the sphere and on the torus.- Chapter 9. The hydrogen atom.- Chapter 10. Spectral theory of the Dirichlet problem for the Laplacian.- Chapter 11. Weyl's Law.- Chapter 12. Almgren's Theory, Unique Continuation and Hausdorff Measure of Nodal Sets of Harmonic Functions.- Chapter 13. Nodal domains of eigenfunctions of the Laplacian.