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146 3. Representation Formulae

α
Pu = a α (x)D u = f in Ω (3.9.1)
|α|≤2m
with real C ∞ coefficient p × p matrices a α (x) (which automatically is prop-
erly elliptic as mentioned before, see Wloka [322, Sect. 9]) together with the
boundary conditions
Rγu = ϕ on Γ (3.9.2)
where R =(R jk ) with j =1,...,m and k =1,..., 2m is a rectangular matrix
of tangential differential operators along Γ having orders
order (R jk )= µ j − k + 1 where 0 ≤ µ j ≤ 2m − 1
and µ j <µ j+1 for 1 ≤ j ≤ m − 1 .

If µ j − k +1 < 0 then R jk := 0. Hence, component–wise, the boundary
conditions (3.9.2) are of the form

2m µ j +1

R jk γ k−1 u = R jk γ k−1 u = ϕ j for j =1,...,m. (3.9.3)
k=1 k=1
For p–vector–valued functions u each R jk is a p × p matrix of operators. In
the simplest case µ j is constant for all components of u =(u 1 ,...,u p ) ,and

let us consider here first this simple case. We shall return to the more general
case later on.
In order to have normal boundary conditions (see e.g. Lions and Magenes
[190]), we require the following crucial assumption.
Fundamental assumption: There exist complementary tangential
boundary operators S =(S jk ) with j =1,...,m and k =1,..., 2m,
having the orders

order (S jk )= ν j − k +1 , where ν j = µ
for all j, =1,...,m, 0 ≤ ν j ≤ 2m − 1; ν j >ν j+1 for 1 ≤ j ≤ m − 1
such that the square matrix of tangential differential operators

R
M := (3.9.4)
S
admits an inverse
−1
N := M , (3.9.5)
which is a matrix of tangential differential operators.
Since M is a rearranged triangular matrix this means that the operators
on the rearranged diagonal, i.e. R j,µ j +1 and S j,ν j +1 are multiplications with
non vanishing coefficients.