This book presents a self-contained exposition of the theory of initial-boundary value problems for diffusion equations. Intended as a graduate textbook, the book is of interest to mathematicians as well as theoretical physicists. Because it uses as little knowledge of functional analysis as possible, the book is accessible to those with a background in multivariable calculus, elementary Lebesgue integral theory, and basic parts of the theory of integral equations. Itô treats diffusion equations with variable coefficients associated with boundary conditions and the corresponding elliptic differential equations. The fundamental solution of the initial-boundary value problem and Green's function for the elliptic boundary value problem are constructed, and the existence of solutions of these problems is proved. In addition, the book discusses several important properties of the solutions.
Readership: Graduate students of pure and applied mathematics and of theoretical physics.
Author(s): Seizo Ito
Series: Translations of Mathematical Monographs, Vol. 114
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+X+225+B
Cover
Titles in This Series
Diffusion Equations
Copyright
®1992 by the American Mathematical Society.
ISBN 0-8218-4570-5
QA377.I7813 1992 515' . 3 5 3-dc20
LCCN 92-24069
Contents
Preface to the English Edition
Preface to the Japanese Edition
Introduction
§0. Physical background for diffusion equations
§1. Preparation for the mathematical investigation of diffusion equations; outline of the contents of this book
§2. Preliminary notions and notation
§3. Diffusion equations and the definition of fundamental solutions
CHAPTER 1 Fundamental Solutions of Diffusion Equations in Euclidean Spaces
§4. Preliminaries for fundamental solutions
§5. Construction of the fundamental solution (in the case of Euclidean space)
CHAPTER 2 Diffusion Equations in a Bounded Domain
§6. Preparatory investigation of boundary conditions
§7. Construction of the fundamental solution (in the case of a bounded domain)
§8. Uniqueness of the fundamental solution and the nonnegativity of the fundamental solution
§9. Existence and uniqueness of the solution of inhomogeneous initial-boundary value problems
§10. Positivity of the fundamental solution and the strong maximum principle for diffusion equations
§11. Dependence of solutions on the coefficients in the equation, on the boundary condition, and on the domain where the equation is considered
CHAPTER 3 Diffusion Equations in Unbounded Domains
§12. Construction of a fundamental solution
§13. Properties of the fundamental solution, existence of solutions of inhomogeneous initial-boundary value problems
§14. Fundamental solution in the temporally homogeneous case
§15. Eigenfunction expansion associated with the elliptic operator (A, Bo) in a bounded domain
§16. Remarks on the case of a domain with piecewise smooth boundary; examples of eigenfunction expansion
§17. Some counterexamples concerning the uniqueness of solutions and related problems
CHAPTER 4 Elliptic Boundary Value Problems
§18. Green's function for elliptic boundary value problems
§19. Existence of solutions of elliptic boundary value problems. I
§20. Invariant measure for the fundamental solution
§21. Existence of solutions of elliptic boundary value problems. II. The Neumann function
§22. Properties of A-harmonic functions
1. Maximum principle
2. Harnack theorems
3. Removable isolated singularity.
§23. Weak solutions and genuine solutions
CHAPTER 5 Some Related Topics in Vector Analysis
§24. Solenoidal and potential components of a vector field
§25. Helmholtz decomposition, incompressible flow given boundary data
Supplementary Notes and References
Subject Index
Titles in This Series
Back Cover
