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8. Pseudodifferential and Boundary Integral
Operators

This chapter concerns the relation between the boundary integral operators
and classical pseudodifferential operators. A large class of boundary integral
operators including those presented in the previous chapters belong to the
special class of classical pseudodifferential operators on compact manifolds.
We are particularly interested in strongly elliptic systems of pseudodifferen-
tial operators providing G˚arding’s inequality, see Theorem 8.1.4. The partic-
ular class of operators in the domain having symbols of rational type enjoys
many special properties such as their relation to Newton potentials, which
n
define genuine pseudodifferential operators in IR and which satisfy in par-
ticular the transmission conditions covered in the work of Boutet de Monvel
for a more general class of pseudodifferential operators. The traces of their
(k)
composition with tensor product distributions involving δ (i.e. the trace
Γ
of Poisson operators by Boutet de Monvel [19, 20, 21] and Grubb [110]),
generate, in a natural way, boundary integral operators as pseudodifferential
operators on the boundary manifold.
In fact, it should be mentioned that Boutet de Monvel found a complete
calculus of Green operators (see Chapter 9) where all compositions of these
operators belong to the same class. For the special class of pseudodifferential
operators with symbols of rational type, corresponding results are presented
in the Theorems 8.5.5 and 8.5.8.
To obtain these results, we present a detailed analysis of the boundary
potentials (or Poisson operators) in the vicinity of the boundary which is
based on properties of the pseudohomogeneous expansions of the Schwartz
kernel (see the extension properties Theorems 8.3.2 and 8.3.8). It should
be noted that in contrast to the requirements of C –extensions as for the
∞
transmission conditions, here we only need finitely many conditions and cor-
responding finite regularity of Γ. Our conditions can be obtained directly
from the Schwartz kernels which is costumarily employed in practical appli-
cations. Once the resulting boundary operator given by the original Schwartz
kernel reduced to Γ satisfies the Tricomi conditions it automatically becomes
a pseudodifferential operator on the boundary.
The relations between various conditions such as the extension condi-
tions, Tricomi conditions, parity conditions as well as transmission condi-
tions are summarized in Table 8.3.1. Moreover, the invariance properties and