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VIII Preface

and the potential energies of their solution fields, allowing us to consider
the boundary integral equations in the form of variational problems on the
boundary manifold of the domain.
On the other hand, the Newton potentials as the inverses of the elliptic
partial differential operators are particular pseudodifferential operators on
the domain or in the Euclidean space. The boundary potentials (or Poisson
operators) are just Newton potentials of distributions with support on the
boundary manifold and the boundary integral operators are their traces
there. Therefore, it is rather natural to consider the boundary integral op-
erators as pseudodifferential operators on the boundary manifold. Indeed,
most of the boundary integral operators in applications can be recast as such
pseudodifferential operators provided that the boundary manifold is smooth
enough.
With the application of boundary element methods in mind, where strong
ellipticity is the basic concept for stability, convergence and error analysis of
corresponding discretization methods for the boundary integral equations,
we are most interested in establishing strong ellipticity in terms of G˚arding’s
inequality for the variational formulation as well as strong ellipticity of the
pseudodifferential operators generated by the boundary integral equations.
The combination of both, namely the variational properties of the elliptic
boundary value and transmission problems as well as the strongly elliptic
pseudodifferential operators provides us with an efficient means to analyze a
large class of elliptic boundary value problems.
This book contains 10 chapters and an appendix. For the reader’s benefit,
Figure 0.1 gives a sketch of the topics contained in this book. Chapters 1
through 4 present various examples and background information relevant to
the premises of this book.
In Chapter 5, we discuss the variational formulation of boundary inte-
gral equations and their connection to the variational solution of associated
boundary value or transmission problems. In particular, continuity and co-
erciveness properties of a rather large class of boundary integral equations
are obtained, including those discussed in the first and second chapters. In
Chapter 4, we collect basic properties of Sobolev spaces in the domain and
their traces on the boundary, which are needed for the variational formula-
tions in Chapter 5.
Chapter 6 presents an introduction to the basic theory of classical
pseudodifferential operators. In particular, we present the construction of
n
a parametrix for elliptic pseudodifferential operators in subdomains of IR .
Moreover, we give an iterative procedure to find Levi functions of arbitrary
order for general elliptic systems of partial differential equations. If the fun-
damental solution exists then Levi’s method based on Levi functions allows
its construction via an appropriate integral equation.