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150 3. Representation Formulae

By taking the traces on both sides of (3.9.18) we arrive with (3.8.6) or
(3.8.10) at
ϕ ϕ ϕ

γu = N = γF −Q Ω PN = γF + C Ω N
λ λ λ
where

F(x):= E(x, y) f(y)dy .
Ω
Now, applying the boundary operator M, we find

ϕ −1 ϕ
Mγu = = MγF + MC Ω M .
λ λ
ϕ

If f = 0, the right–hand side again is a projector for = Mγu,which
λ
are generalized Cauchy data and boundary data to any solution u of the
homogeneous equation Pu =0 in Ω. We call C Ω , defined by

ϕ −1 ϕ R

C Ω := MC Ω M = C Ω (N 1 ϕ + N 2 λ) , (3.9.19)

λ λ S
the modified Calder´on projector associated with P, Ω and M, which satisfies
2
C = C Ω . (3.9.20)

Ω
In terms of the finite–part integrals defined in (3.8.26), we have a relation
corresponding to (3.8.28) for the modified Calder´on projector as well:
1
−1
MDPM χ = − C Ω χ + χ on Γ. (3.9.21)

2
For the boundary value problem (3.9.1) and (3.9.3), ϕ is given. Consequently,
for the representation of u in (3.9.18) the missing boundary datum λ is to be
determined by the boundary integral equations
ϕ ϕ

= C Ω + MγF on Γ. (3.9.22)

λ λ
As for the Laplacian, (3.9.22) is an overdetermined system for λ in the sense
that only one of the two sets of equations is sufficient to determine λ.
0
−1 −1
I
Applying RM = and SM = to (3.9.22) and employing
0 I
(3.9.21) leads to the following two choices of boundary integral equations in
terms of finite part integral operators:
Integral equations of the ”first kind”:
A Ω λ := −RQ Ω PN 2 λ = ϕ + RQ Ω PN 1 ϕ − RγF
(3.9.23)
1
= −RDPN 2 λ = ϕ + RDPN 1 ϕ − RγF on Γ.
2