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3.9 Direct Boundary Integral Equations 147

Definition 3.9.1. The boundary value problem is called a regular elliptic
boundary value problem if P in (3.9.1) is elliptic and the boundary con-
ditions in (3.9.2) are normal and satisfy the Lopatinski–Shapiro conditions.
In order to have a regular elliptic boundary value problem (see H¨ormander
[130], Wloka [322]) we require in addition to the fundamental assumption,
i.e. normal boundary conditions, that the Lopatinski–Shapiro condition is
fulfilled (see [322, Chap.11]). This condition reads as follows:
Definition 3.9.2. The boundary conditions (3.9.3) for the differential op-
erator in (3.9.1) are said to satisfy the Lopatinski–Shapiro conditions if the
initial value problem defined by the constant coefficient ordinary differential
equations
2m
k
(2m)
d

P (x, iξ ) χ(σ n )=0 for 0 <σ n (3.9.6)
k
dσ n
k=0
together with the initial conditions
k−1
λ d
r jkλ (x)(iξ ) χ | σ n =0 =0 for j =1,...,m (3.9.7)
dσ n
|λ|=µ j −k−1
and the radiation condition χ(σ n )= o(1) as σ n →∞ admits only the trivial

solution χ(σ n )=0 for every fixed x ∈ Γ and every ξ ∈ IR n−1 with |ξ | =1.
Here, the coefficients in (3.9.6) are given by the principal part of P in local
coordinates via (3.4.50) as
(2m) γ

P (x, ξ ):= a α (x)c α,γ,k (σ , 0)(iξ ) ;
k
|γ|≤2m−k |α|=2m
and the coefficients in (3.9.7), from the tangential boundary operators in
(3.9.2) in local coordinates are given by
λ
∂
R jk = r jkλ (x) .
∂σ
|λ|≤µ j −k−1
The case of more general boundary conditions in which the orders µ j
for the different components u of u are different, with corresponding given
components ϕ jt of ϕ j , was treated by Grubb in [109]. Here the boundary
conditions have the form
p µ j +1
t λ
r jkλ (x)(iξ ) γ k−1 u = ϕ jt , (3.9.8)
=1 k=1 |λ|=µ j −k−1
m

t =1,...,q j ≤ p, j =1,...,m and q j = m · p.
j=1