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3. Representation Formulae, Local Coordinates
and Direct Boundary Integral Equations

In order to generalize the direct approach for the reduction of more general
boundary value problems to boundary integral equations than those pre-
sented in Chapters 1 and 2 we consider here the 2m–th order positive elliptic
systems with real C –coefficients. We begin by collecting all the necessary
∞
machinery. This includes the basic definitions and properties of classical func-
tion spaces and distributions, the Fourier transform and the definition of
Hadamard’s finite part integrals which, in fact, represent the natural regu-
larization of homogeneous distributions and of the hypersingular boundary
integral operators. For the definition of boundary integral operators one needs
the appropriate representation of the boundary manifold Γ involving local
coordinates. Moreover, the calculus of vector fields on Γ requires some basic
knowledge in classical differential geometry. For this purpose, a short excur-
sion into differential geometry is included. Once the fundamental solution
is available, the representation of solutions to the boundary value problems
is based on general Green’s formulae which are formulated in terms of dis-
tributions and multilayer potentials on Γ. The latter leads us to the direct
boundary integral equations of the first and second kind for interior and
exterior boundary value problems as well as for transmission problems. As
expected, the hypersingular integral operators are given by direct values in
terms of Hadamard’s finite part integrals.
The results obtained in this chapter will serve as examples of the class of
pseudodifferential operators to be considered in Chapters 6–10.

3.1 Classical Function Spaces and Distributions

For rigorous definitions of classical as well as generalized function spaces we
first collect some standard results and notations.
Multi–Index Notation
n
Let IN 0 be the set of all non–negative integers and let IN be the set of
0
all ordered n–tuples α =(α 1 , ··· ,α n ) of non–negative integers α i ∈ IN 0 .
n
Such an n–tuple α is called a multi–index. For all α ∈ IN , we denote by
0
n
|α| = α 1 +α 2 +···+α n the order of the multi-index α.If α, β ∈ IN , we define
0
α + β =(α 1 + β 1 , ··· ,α n + β n ). The notation α ≤ β means that α i ≤ β i for
1 ≤ i ≤ n.Weset