Strongly Elliptic Systems and Boundary Integral Equations

by William McLean

• ISBN13: 9780521663755
• ISBN10: 052166375X
• Format: Paperback
• Copyright: 2000-01-28
• Publisher: Cambridge University Press

Summary

Partial differential equations provide mathematical models of many important problems in the physical sciences and engineering. This book treats one class of such equations, concentrating on methods involving the use of surface potentials. William McLean provides the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains. Included are chapters on three specific examples: the Laplace equation, the Helmholtz equation and the equations of linear elasticity. The book affords an ideal background for studying the modern research literature on boundary element methods.

Table of Contents

Preface	xi
Introduction	1	(16)
Exercises	15	(2)
Abstract Linear Equations	17	(40)
The Kernel and Image	18	(2)
Duality	20	(7)
Compactness	27	(5)
Fredholm Operators	32	(6)
Hilbert Spaces	38	(4)
Coercivity	42	(3)
Elementary Spectral Theory	45	(7)
Exercises	52	(5)
Sobolev Spaces	57	(56)
Convolution	58	(3)
Differentiation	61	(3)
Schwartz Distributions	64	(5)
Fourier Transforms	69	(4)
Sobolev Spaces -- First Definition	73	(2)
Sobolev Spaces -- Second Definition	75	(4)
Equivalence of the Norms	79	(4)
Localisation and Changes of Coordinates	83	(2)
Density and Imbedding Theorems	85	(4)
Lipschitz Domains	89	(7)
Sobolev Spaces on the Boundary	96	(4)
The Trace Operator	100	(6)
Vector-Valued Functions	106	(1)
Exercises	107	(6)
Strongly Elliptic Systems	113	(45)
The First and Second Green Identities	113	(5)
Strongly Elliptic Operators	118	(10)
Boundary Value Problems	128	(5)
Regularity of Solutions	133	(8)
The Transmission Property	141	(4)
Estimates for the Steklov-Poincare Operator	145	(11)
Exercises	156	(2)
Homogeneous Distributions	158	(33)
Finite-Part Integrals	159	(7)
Extension from Rn \ {0} to Rn	166	(3)
Fourier Transforms	169	(5)
Change of Variables	174	(7)
Finite-Part Integrals on Surfaces	181	(6)
Exercises	187	(4)
Surface Potentials	191	(26)
Parametrices	192	(5)
Fundamental Solutions	197	(3)
The Third Green Identity	200	(2)
Jump Relations and Mapping Properties	202	(9)
Duality Relations	211	(4)
Exercises	215	(2)
Boundary Integral Equations	217	(29)
Operators on the Boundary	217	(2)
Integral Representations	219	(7)
The Dirichlet Problem	226	(3)
The Neumann Problem	229	(2)
Mixed Boundary Conditions	231	(3)
Exterior Problems	234	(5)
Regularity Theory	239	(2)
Exercises	241	(5)
The Laplace Equation	246	(30)
Fundamental Solutions	247	(3)
Spherical Harmonics	250	(8)
Behaviour at Infinity	258	(2)
Solvability for the Dirichlet Problem	260	(6)
Sovability for the Neumann Problem	266	(2)
Exercises	268	(8)
The Helmholtz Equation	276	(20)
Separation of Variables	277	(3)
The Sommerfeld Radiation Condition	280	(6)
Uniqueness and Existence of Solutions	286	(3)
A Boundary Integral Identity	289	(4)
Exercises	293	(3)
Linear Elasticity	296	(13)
Korn's Inequality	297	(2)
Fundamental Solutions	299	(2)
Uniqueness Results	301	(4)
Exercises	305	(4)
Appendix A. Extension Operators for Sobolev Spaces	309	(8)
Exercises	315	(2)
Appendix B. Interpolation Spaces	317	(17)
The K-Method	318	(3)
The J-Method	321	(8)
Interpolation of Sobolev Spaces	329	(4)
Exercises	333	(1)
Appendix C. Further Properties of Spherical Harmonics	334	(7)
Exercises	338	(3)
References	341	(6)
Index	347	(6)
Index of Notation	353