Page 169 -

3.9 Direct Boundary Integral Equations 153

Rγ c u = ϕ on Γ (3.9.28)

and the radiation condition — which means that M(x; u) defined by (3.6.15)
is also given. Here, the solution can be represented by (3.7.13),

u(x)= E(x, y)f(y)dy + M(x; u)
Ω c
2m−1 2m− −1 p

∂
+ E(x, y) P p+ +1 γ c u(y)ds y .
∂n y
=0 p=0
Γ
As for the interior problem, we require the fundamental assumption and
obtain with the exterior Cauchy data Rγ c u = ϕ and Sγ c u = λ, the modified
c
representation formula for x ∈ Ω , similar to (3.9.18):
u(x) = F c (x)+ M(x; u) (3.9.29)
2m−1 2m− −1 m

+ K p P p+ +1 N +1,j ϕ j + N +1,m+j λ j (x)
=0 p=0 j=1
where

F c (x)= E(x, y)f(y)dy .
Ω c
In the same manner as for the interior problem, we obtain the set of
boundary integral equations

ϕ ϕ
= C Ω c + Mγ c F c + Mγ c M(•; u) (3.9.30)

λ λ
where
−1
C Ω c = I− C Ω = MC Ω cM (3.9.31)

and C Ω c is given in (3.8.17). These are the ordinary and modified exterior
c
Calder´on projectors, respectively, associated with P, Ω and M.
As before, we obtain two sets of boundary integral equations for λ.
Integral equation of the “first kind”:
A Ω cλ := −RQ Ω cPN 2 λ = −ϕ + RQ Ω cPN 1 ϕ + Rγ c F c + Rγ c M,
(3.9.32)
1
= −RDPN 2 λ = − ϕ + RDPN 1 ϕ + Rγ c F c + Rγ c M.
2
Integral equation of the “second kind”:
B Ω cλ := λ − SQ Ω cPN 2 λ = SQ Ω cPN 1 ϕ + Sγ c F c + Sγ c M,
(3.9.33)
1
= λ − SDPN 2 λ = SDPN 1 ϕ + Sγ c F c + Sγ c M.
2