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68 2. Boundary Integral Equations

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having a weakly singular kernel for Γ ⊂ C , and, hence, defines a continu-
α
ous mapping K : C (Γ) → C 1+α (Γ)(seeLadyˇzenskaya [179, p. 35] where
the fundamental solution and the potentials carry the opposite sign). The
hypersingular operator D is now defined by
Dϕ = −T x Wϕ(x)

:= − lim T z (∂ z ,x) Wϕ(z) (2.3.30)
Ω z→x∈Γ

= lim Πϕ(z) n(x) − µ ∇ z Wϕ(z)+ ∇ z Wϕ(z) n(x) .
Ω z→x∈Γ
With the standard regularization this reads

Dϕ(x)= −p.v. T x T E(x, y) ϕ(y) − ϕ(x) ds y (2.3.31)
y
Γ
−µ 1
= p.v. 2 n(y) · ϕ(y) − ϕ(x)
2(n − 1)π |x − y| n
Γ
n !
+ n(y) · n(x) (x − y) · ϕ(y) − ϕ(x) (x − y) ds y
|x − y| n+2
µ 2n(n +2) !
+ (x − y) · n(x)
2(n − 1)π |x − y| n+4
Γ
! !
× (x − y) · n(y) (x − y) · ϕ(y) − ϕ(x) (x − y)
n ! !
− (x − y) · n(x) (x − y) · ϕ(y) − ϕ(x) n(y)
|x − y| n+2
!
+ n(x) · ϕ(y) − ϕ(x) (y − x) · n(y) (x − y)

! !
+ (x − y) · n(x) (x − y) · n(y) ϕ(y) − ϕ(x) ds y .
Again, as in Lemma 2.2.3, the hypersingular operator can be reformulated.
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Lemma 2.3.1. Kohr et al [164] Let Γ ∈ C and let ϕ be a H¨older continu-
ously differentiable function. Then the operator D in (2.3.30) can be expressed
as a composition of tangential differential operators and simple layer poten-
λ+µ
tials as in (2.2.32)–(2.2.34) where in the case n =2 set =1 in (2.2.33)
λ+2µ
and in the case n =3 take E(x, y) from (2.3.10) in (2.2.34).
Now let us assume that the boundary Γ = L " Γ consists of L separate,
=1
mutually non intersecting compact boundary components Γ 1 ,...,Γ L .
Before we exemplify the details of solvability of the boundary integral
equations, we first summarize some basic properties of their eigenspaces.
Theorem 2.3.2. (See also Kohr and Pop [163].) Let n =3. Then we have
the following relations.