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X Preface

Classical model problems ψdOs in Ω ⊂ R n

(Chap. 1 and Chap. 2) (Chap. 6)

Generalized multilayer Classical ψdOs and IOs
potentials on Γ = ∂Ω in Ω ⊂ R n

(Chap. 3) (Chap. 7)

Sobolev spaces and trace BIEs and ψdOs on Γ

theorems (Chap. 4) (Chap. 8 and Chap. 9)

Variational formulations of Fourier representation of
BVPs and BIEs BIOs and ψdOs on Γ ⊂ R 2
(Chap. 5) (Chap. 10)
Abbreviations:
Ω ⊂ IR n – A given domain with compact boundary Γ
BVPs – Boundary value problems
BIEs – Boundary integral equations
ψdOs – Pseudodifferential operators
IOs – Integral operators
BIOs – Boundary integral operators

Fig. 0.1. A schematic sketch of the topics and their relations

Our original plan was to finish this book project about 10 years ago.
However, many new ideas and developments in boundary integral equation
methods appeared during these years which we have attempted to incorpo-
rate. Nevertheless, we regret to say that the present book is by no means
complete. For instance, we only slightly touch on the boundary integral op-
erator methods involving Lipschitz boundaries which have recently become
more important in engineering applications. We do hope that we have made
a small step forward to bridge the gap between the theory of boundary inte-
gral equation methods and their applications. We further hope that this book
will lead to better understanding of the underlying mathematical structure of
these methods and will serve as a mathematical foundation of the boundary
element methods.