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148 3. Representation Formulae

Here, for each fixed we have µ j ∈{0,..., 2m − 1} which are ordered as
µ j <µ j+1, , then we also need complementing orders
2m−1
"
ν j ∈{0,..., 2m−1}\ {µ j } with ν j >ν j+1, and complementing bound-
j=1
t
ary conditions S . For a system of normal boundary conditions S the bound-
jk
ary conditions in R together with the complementing boundary conditions
should be such that the rearranged triangular matrix M =((M jk )) 2m×2m
1
with M jk =0 for j< k and tangential operators
t
M jk =((M )) q j ×p
jk
now define a complete system of boundary conditions
p
t
M γ k−1 u = ϕ jt ,t = 1,q j ,j = 1, 2m
jk
=1 k≤j
where the diagonal matrices M jj =((M jj )) q j × p, whose entries are functions,
define surjective mappings onto the q j –vector valued functions, and where
q j ≤ p. Then there exists the right inverse N to the triangular matrix M
1
and, hence, to M.
−1
If we require the stronger fundamental assumption that M exists, the
boundary conditions still will be a normal system. In this case, in the Shapiro–
Lopatinski condition, the initial conditions (3.9.7) are now to be replaced by
p µ j +1
k−1
t λ d
r jkλ (x)(iξ ) χ | σ n =0 =0
dσ n (3.9.9)
=1 k=1 |λ|=µ j −k−1
for t =1,...,q j with q j ≤ p and j =1,...,m.
In regard to the Shapiro–Lopatinski condition for general elliptic systems
in the sense of Agmon–Douglis–Nirenberg, we refer to the book by Wloka
et al [323, Sections 9.3,10.1], where one can find an excellent presentation of
these topics.
Now, for the special cases for m = 1 we consider the following
second order systems:
In this case, only the following two combinations of boundary conditions
are possible:
i)
R =(R 00 , 0) with order (R 00 )= 0
(3.9.10)
S =(S 00 ,S 01 ) with order (S 00 ) = 1 and order (S 01 )=0 .

−1
For (3.9.5) to be satisfied, we require that the inverse matrices R 00 and
−1
S 01 exist along Γ. Then M and N are given by