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Preface IX

In Chapter 7, we show that every pseudodifferential operator is an
Hadamard’s finite part integral operator with integrable or nonintegrable
kernel plus possibly a differential operator of the same order as that of the
pseudodifferential operator in case of nonnegative integer order. In addition,
we formulate the necessary and sufficient Tricomi conditions for the inte-
gral operator kernels to define pseudodifferential operators in the domain by
using the asymptotic expansions of the symbols and those of pseudohomoge-
neous kernels. We close Chapter 7 with a presentation of the transformation
formulae and invariance properties under the change of coordinates.
Chapter 8 is devoted to the relation between the classical pseudodifferen-
tial operators and boundary integral operators. For smooth boundaries, all of
our examples in Chapters 1 and 2 of boundary integral operators belong to
the class of classical pseudodifferential operators on compact manifolds hav-
ing symbols of the rational type. If the corresponding class of pseudodifferen-
tial operators in the form of Newton potentials is applied to tensor product
distributions with support on the boundary manifold, then they generate, in a
natural way, boundary integral operators which again are classical pseudodif-
ferential operators on the boundary manifold. Moreover, for these operators
associated with regular elliptic boundary value problems, it turns out that
the corresponding Hadamard’s finite part integral operators are invariant un-
der the change of coordinates, as considered in Chapter 3. This approach also
provides the jump relations of the potentials. We obtain these properties by
using only the Schwartz kernels of the boundary integral operators. However,
these are covered by Boutet de Monvel’s work in the 1960’s on regular elliptic
problems involving the transmission properties.
The last two chapters, 9 and 10, contain concrete examples of bound-
ary integral equations in the framework of pseudodifferential operators on
the boundary manifold. In Chapter 9, we provide explicit calculations of the
symbols corresponding to typical boundary integral operators on closed sur-
3
faces in IR . If the fundamental solution is not available then the boundary
value problem can still be reduced to a coupled system of domain and bound-
ary integral equations. As an illustration we show that these coupled systems
can be considered as some particular Green operators of the Boutet de Mon-
vel algebra. In Chapter 10, the special features of Fourier series expansions
of boundary integral operators on closed curves are exploited.
We conclude the book with a short Appendix on differential operators in
local coordinates with minimal differentiability. Here, we avoid the explicit
use of the normal vector field as employed in Hadamard’s coordinates in
Chapter 3. These local coordinates may also serve for a more detailed analysis
for Lipschitz domains.