Partial Differential Equations

A unified Hilbert Space Approach

Rainer Picard, Des McGhee

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110250268
• eBook ISBN: 9783110250275
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev lattice structure, a simple extension of the well-established notion of a chain (or scale) of Hilbert spaces.

The focus on a Hilbert space setting (rather than on an apparently more general Banach space) is not a severe constraint, but rather a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations.

In contrast to other texts on partial differential equations, which consider either specific equation types or apply a collection of tools for solving a variety of equations, this book takes a more global point of view by focusing on the issues involved in determining the appropriate functional analytic setting in which a solution theory can be naturally developed. Applications to many areas of mathematical physics are also presented.

The book aims to be largely self-contained. Full proofs to all but the most straightforward results are provided, keeping to a minimum references to other literature for essential material. It is therefore highly suitable as a resource for graduate courses and also for researchers, who will find new results for particular evolutionary systems from mathematical physics.

Table of contents

• Preface
• Contents
• Nomenclature
• 1 Elements of Hilbert Space Theory
• 2 Sobolev Lattices
• 3 Linear Partial Differential Equations with Constant Coefficients
• 4 Linear Evolution Equations
• 5 Some Evolution Equations of Mathematical Physics
• 6 A "Royal Road" to Initial Boundary Value Problems
• Conclusion
• Bibliography
• Index