Diffusion equations

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Titles in This Series

Diffusion Equations

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     ®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

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