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Preface

This book is devoted to the mathematical foundation of boundary integral
equations. The combination of finite element analysis on the boundary with
these equations has led to very efficient computational tools, the boundary
element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.)
[267]). Although we do not deal with the boundary element discretizations
in this book, the material presented here gives the mathematical foundation
of these methods. In order to avoid over generalization we have confined
ourselves to the treatment of elliptic boundary value problems.
The central idea of eliminating the field equations in the domain and re-
ducing boundary value problems to equivalent equations only on the bound-
ary requires the knowledge of corresponding fundamental solutions, and this
idea has a long history dating back to the work of Green [107] and Gauss
[95, 96]. Today the resulting boundary integral equations still serve as a
major tool for the analysis and construction of solutions to boundary value
problems.
As is well known, the reduction to equivalent boundary integral equations
is by no means unique, and there are primarily two procedures for this re-
duction, the ‘direct’ and ‘indirect’ approaches. The direct procedure is based
on Green’s representation formula for solutions of the boundary value prob-
lem, whereas the indirect approach rests on an appropriate layer ansatz. In
our presentation we concentrate on the direct approach although the corre-
sponding analysis and basic properties of the boundary integral operators
remain the same for the indirect approaches. Roughly speaking, one obtains
two kinds of boundary integral equations with both procedures, those of the
first kind and those of the second kind.
The basic mathematical properties that guarantee existence of solutions
to the boundary integral equations and also stability and convergence analy-
sis of corresponding numerical procedures hinge on G˚arding inequalities for
the boundary integral operators on appropriate function spaces. In addition,
contraction properties allow the application of Carl Neumann’s classical suc-
cessive iteration procedure to a class of boundary integral equations of the
second kind. It turns out that these basic features are intimately related
to the variational forms of the underlying elliptic boundary value problems