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3.9 Direct Boundary Integral Equations 145

which follows from (3.7.12), we find

C Ω γM = γM = γ c M.
Applying C Ω c to this equation yields (3.8.25) since C Ω cC Ω = 0 because of
(3.8.20).

Remark 3.8.1: From the definition of Q Ω we now define the finite part
multiple layer potentials on Γ,

∂ ∂
j−1 k−1
D jk φ k := p.f. E(x, y) φ k (y)ds y ,
∂n x ∂n y
Γ
j, k =1,..., 2m (3.8.26)
and
2m 2m

Dφ := D 1,k φ k ,..., D 2m,k φ k
k=1 k=1
1 −1 1 −1
= Q Ω φ + P φ = Q Ω cφ − P φ on Γ (3.8.27)
2 2

∞
for any given φ =(φ 1 ,...,φ 2m ) ∈ C (Γ). Correspondingly, we have
1 1
D(Pψ)= −C Ω ψ + ψ = C Ω cψ − ψ on Γ. (3.8.28)
2 2
As we will see, (3.8.27) or (3.8.28) are the jump relations for multilayer po-
tentials.

3.9 Direct Boundary Integral Equations

As we have seen in the examples in elasticity and the biharmonic equation
in Chapter 2, the boundary conditions are more general than parts of the
Cauchy γu data appearing in the Calder´on projector directly. Correspond-
ingly, for more general boundary conditions, the boundary potentials in ap-
plications will also be more complicated.

3.9.1 Boundary Value Problems

Let us consider a more general boundary value problem for strongly elliptic
c
equations associated with (3.6.1) in Ω (and later on also in Ω ). As before,
∞
Ω is assumed to be a bounded domain with C –boundary Γ = ∂Ω.The
linear boundary value problem here consists of the uniformly strongly elliptic
system of partial differential equations