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Largely self-contained, rapid introduction to the theory of distributions and its applications to partial differential equations
Contains the computation of fundamental solutions for the Laplacian, heat, wave, Lam\'e, Schrodinger, and Cauchy operators
Each chapter contains exercises and many have solutions at the end of the book
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.
Content Level »Graduate
Keywords »Distributions - Fourier Analysis - Fundamental Solutions - Partial Diffential Equations texbook - Potential Theory
Introduction.- Summary of Topological and Functional Analysis Results.- Weak Derivatives.- The Space D0() of Distributions.- The Fourier Transform.- The Space of Tempered Distributions.- Fundamental Solution.- The Laplace Operator.- The Heat Operator.- The Wave Operator.- The Lame Operator.- Fundamental Solutions for Other Operators.- Hypoelliptic operators.- Sobolev spaces.- Appendix.- References.