Page 430 -

414 8. Pseudodifferential and Boundary Integral Operators

transmission conditions under change of local coordinates of Γ lead to the re-
sult of Theorem 8.4.6 which implies that all these boundary integral operators
are invariant under the change of coordinates. In particular, the Hadamard
finite part integral operators in this class transform into Hadamard’s finite
part integral operators without producing local differential operator contri-
butions.
We also collect in this chapter the relevant mapping properties and jump
relations for the boundary potentials (or Poisson operators) as well as for
the volume potentials, which are covered by Boutet de Monvel’s work for
even more general pseudodifferential operators satisfying the transmission
conditions including those of operators with symbols of rational type.
The last section is devoted to the concept of strong ellipticity and Fred-
holm properties of boundary pseudodifferential operators.
The presentation of this chapter is partially based on the book by
Chazarain and Piriou [39], the PHD dissertation by Kieser [156] and
H¨ormander’s book [131, Vol. III] and covers only a small part of Boutet
de Monvel’s analysis (see [19, 20, 21], Grubb [110] and Schulze [273]).

8.1 Pseudodifferential Operators on Boundary
Manifolds

In Section 3.3 we introduced the parametric representation of the boundary
Γ = ∂Ω (cf.(3.3.6),(A.0.1)). This defines Γ as an (n − 1)–dimensional pa-
rameterized surface. We may also consider Γ as a manifold immersed into
n
IR in the sense of differential geometry and associate Γ with an atlas A
which is a family of local charts {(O r ,U r ,χ r ) | r ∈ I}. Each of the local
charts is a triplet with U r ⊂ IR n−1 an open subset of the parametric space

IR n−1 ; and where the representation x = T r ( )= χ r (−1) ( )for ∈ U r

defines a parameterized patch O r := T r (U r ) of the surface Γ, or, reversely,
U r = χ r (O r ). The mappings T r and χ r are both bijective and bicontinuous,
(−1)
hence, T r = χ r is a homeomorphism, see Fig. 8.1.1. For an atlas we require
Γ = ∪ O r . Moreover, if O rt := O r ∩ O t = ∅ then the mapping
r∈I
Φ rt := χ t ◦ T r = χ t ◦ χ (−1) : χ r (O rt ) → χ t (O rt ) (8.1.1)
r
is supposed to be a sufficiently smooth diffeomorphism. (For details see also
Section 3.3.) Note that any union of two atlases is again an atlas on Γ.
Definition 8.1.1. Let A : D(Γ) →E(Γ) be a continuous linear operator.
m
Then A is said to be in the class L (Γ) of pseudodifferential operators if for
every chart (O r ,U r ,χ r ) the associated local operator
∗
:= χ r∗ Aχ : D(U r ) →E(U r ) (8.1.2)
A χ r r
m
belongs to L (U r ). Here,