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1.2 Boundary Potentials and Calder´on’s Projector 5

is called weakly singular if there exist constants c and λ<n − 1 such that
−λ
|k(x, y)|≤ c|x − y| for all x, y ∈ Γ. (1.2.10)
2
For the Laplacian, for Γ ∈ C and E(x, y) given by (1.1.2), one even has
−λ
|E(x, y)|≤ c λ |x − y| for any λ> 0for n =2 and λ =1 for n =3 ,
(1.2.11)
∂E 1 (x − y) · n y
(x, y) = , (1.2.12)
∂n y 2(n − 1)π |x − y| n
∂E 1 (y − x) · n x
(x, y) = for x, y ∈ Γ. (1.2.13)
∂n x 2(n − 1)π |x − y| n
In case n = 2, both kernels in (1.2.12), (1.2.13) are continuously extendable
0
to a C -function for y → x (see Mikhlin [213]), in case n = 3 they are weakly
singular with λ =1 (see G¨unter [113, Sections II.3 and II.6]). For other
differential equations, as e.g. for elasticity problems, the boundary integrals
in (1.2.7)–(1.2.9) are strongly singular and need to be defined in terms of
Cauchy principal value integrals or even as finite part integrals in the sense
of Hadamard. In the classical approach, the corresponding function spaces
are the H¨older spaces which are defined as follows:

m
C m+α (Γ):= {ϕ ∈ C (Γ) ||ϕ|| C m+α (Γ ) < ∞}

where the norm is defined by
β β
β |∂ ϕ(x) − ∂ ϕ(y)|
||ϕ|| C m+α (Γ ) := sup |∂ ϕ(x)| + sup α
x∈Γ x,y∈Γ |x − y|
|β|≤m |β|=m x =y
β
for m ∈ IN 0 and 0 <α< 1. Here, ∂ denotes the covariant derivatives
β
β n−1
∂ := ∂ β 1 ··· ∂ n−1
1
n−1
on the (n − 1)–dimensional boundary surface Γ where β ∈ IN 0 is a multi–
index and |β| = β 1 + ... + β n−1 (see Millman and Parker [216]).
2
Lemma 1.2.2. Let Γ ∈ C and let ϕ be a H¨older continuously differentiable
function. Then the limit in (1.2.6) exists uniformly with respect to all x ∈ Γ
and all ϕ with
ϕ
C 1+α≤ 1. Moreover, the operator D can be expressed as
a composition of tangential derivatives and the simple layer potential opera-
tor V :
d dϕ
Dϕ(x) = − V ( )(x) for n = 2 (1.2.14)
ds x ds
and
Dϕ(x) = −(n x ×∇ x ) · V (n y ×∇ y ϕ)(x) for n =3 . (1.2.15)