Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext)

The theory of distributions constitutes an essential tool in the study of partial differential equations. This textbook would offer, in a concise, largely self-contained form, a rapid introduction to the theory of distributions and its applications to partial differential equations, including computing fundamental solutions for the most basic differential operators: the Laplace, heat, wave, Lam\'e and Schrodinger operators.​ "The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity." -- Prové de l'editor Front Matter....Pages i-xxi Weak Derivatives....Pages 1-16 The Space $$\mathcal{D}^{\prime}(\Omega )$$ of Distributions....Pages 17-87 The Schwartz Space and the Fourier Transform....Pages 89-108 The Space of Tempered Distributions....Pages 109-187 The Concept of a Fundamental Solution....Pages 189-200 Hypoelliptic Operators....Pages 201-216 The Laplacian and Related Operators....Pages 217-275 The Heat Operator and Related Versions....Pages 277-288 The Wave Operator....Pages 289-308 The Lamé and Stokes Operators....Pages 309-339 More on Fundamental Solutions for Systems....Pages 341-374 Solutions to Selected Exercises....Pages 375-414 Appendix....Pages 415-447 Back Matter....Pages 449-460