A Short Introduction to Partial Differential Equations

This book provides a short introduction to partial differential equations (PDEs). It is primarily addressed to graduate students and researchers, who are new to PDEs. The book offers a user-friendly approach to the analysis of PDEs, by combining elementary techniques and fundamental modern methods.

The author focuses the analysis on four prototypes of PDEs, and presents two approaches for each of them. The first approach consists of the method of analytical and classical solutions, and the second approach consists of the method of weak (variational) solutions.

In connection with the approach of weak solutions, the book also provides an introduction to distributions, Fourier transform and Sobolev spaces. The book ends with an appendix chapter, which complements the previous chapters with proofs, examples and remarks.

This book can be used for an intense one-semester, or normal two-semester, PDE course. The reader isexpected to have knowledge of linear algebra and of differential equations, a good background in real and complex calculus and a modest background in analysis and topology. The book has many examples, which help to better understand the concepts, highlight the key ideas and emphasize the sharpness of results, as well as a section of problems at the end of each chapter.

A short introduction to some of the most important techniques for PDEs A smooth transition from analytical formula methods to modern techniques Provides the readers key concepts, results & techniques for progressing with their research

Auteur

The author is a professor of Mathematics at University of Ottawa, Canada (since 2002). He holds a PhD in Mathematics from University Henri Poincare, Nancy, France, 1997. Before joining University of Ottawa in 2002, he had held postdoctoral positions at INRIA France and PIMS, UBC, Vancouver, Canada. The author's main research area are Shape Optimization and Partial Differential Equations, as they are integral part of his research. The author has taught PDEs at University of Ottawa for many years, and this book represents this teaching experience.

Contenu

1 Notations and review.- 2 Partial differential equations.- 3 First order PDEs. Classical and weak solutions.- 4 Second-order linear elliptic PDEs. maximum principle and classical solutions - 5 Distributions.- 6 Sobolev spaces.- 7 Second order linear elliptic PDEs. Weak solutions.- 8 Second order parabolic and hyperbolic PDEs.- 9 Annex.