Page 165 -

3.9 Direct Boundary Integral Equations 149

& '
−1
, 0 R , 0
R 00 00
M = ; N = −1 −1 −1 (3.9.11)
S 00 , S 01 −S 01 S 00 R 00 ,S 01
ii)
R =(R 00 ,R 01 ) with order (R 00 ) = 1 and order (R 01 )=0 ,
(3.9.12)
S =(S 00 , 0) with order (S 00 ) = 0 .

Similarly, here for (3.9.5) one needs the existence of R −1 and S −1 . Then M
01 00
and N are given by
& '
−1
, 0 , S
R 00 R 01 ∞
M = ; N = −1 −1 −1 (3.9.13)
S 00 , 0 R , −R R 00 S
01 01 00
In general, the verification of the fundamental assumption is nontrivial
(see e.g. Grubb [109, 110]), however, in most applications, the fundamental
assumption is fulfilled.
Becauseof(3.9.4)onemayexpresstheCauchydataγu =(γ 0 u,...,γ 2m−1 u)
by the given boundary functions ϕ j in (3.9.3) and a set of complementary
boundary functions λ j via

ν j +1

S jk γ k−1 u = λ j with j =1,...,m on Γ. (3.9.14)
k=1
If we require (3.9.5) or normal boundary conditions, the Cauchy data γu can
be recovered from ϕ and λ by the use of N,
m 2m

γ −1 u = N j ϕ j + N p λ p−m for =1,..., 2m ; (3.9.15)
j=1 p=m+1
in short,
γu = N 1 ϕ + N 2 λ (3.9.16)
with rectangular tangential differential operators N 1 and N 2 from (3.9.15).
Then
RN 1 = I, RN 2 =0 ,SN 1 =0 ,SN 2 = I (3.9.17)
since N is the right inverse. This enables us to insert the boundary data ϕ
and λ with given ϕ into the representation formula (3.8.5), i.e., for x ∈ Ω we
obtain:

u(x)= E(x, y) f(y)dy (3.9.18)

Ω
2m−1 2m− −1 m m

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