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4 1. Introduction

c
Vσ(x):= E (x, y)σ(y) ds y , x ∈ Ω ∪ Ω , (1.2.1)
y∈Γ
and the double layer potential
∂ c

Wϕ(x) := ( E(x, y))ϕ(y)ds y ,x ∈ Ω ∪ Ω . (1.2.2)
∂n y
y∈Γ
Here, σ and ϕ are referred to as the densities of the corresponding potentials.
In (1.1.7), for the solution of (1.1.6), these are the Cauchy data which are not
both given for boundary value problems. For their complete determination
we consider the Cauchy data of the left– and the right–hand sides of (1.1.7)
on Γ; this requires the limits of the potentials for x approaching Γ and their
normal derivatives. This leads us to the following definitions of boundary
integral operators, provided the corresponding limits exist. For the potential
equation (1.1.6), this is well known from classical analysis (Mikhlin [213,
p. 360] and G¨unter [113, Chap. II]):
Vσ(x) := lim Vσ(z) for x ∈ Γ, (1.2.3)
z→x∈Γ
1
Kϕ(x) := lim Wϕ(z)+ ϕ(x) for x ∈ Γ, (1.2.4)
2
z→x∈Γ,z∈Ω
1
K σ(x) := lim grad Vσ(z) · n x − σ(x)for x ∈ Γ, (1.2.5)

z
z→x∈Γ,z∈Ω 2
Dϕ(x) := − lim grad Wϕ(z) · n x for x ∈ Γ. (1.2.6)
z
z→x∈Γ,z∈Ω
To be more explicit, we quote the following standard results without proof.
2
Lemma 1.2.1. Let Γ ∈ C and let σ and ϕ be continuous. Then the limits
in (1.2.3)–(1.2.5) exist uniformly with respect to all x ∈ Γ and all σ and ϕ
with sup x∈Γ |σ(x)|≤ 1, sup x∈Γ |ϕ(x)|≤ 1. Furthermore, these limits can be
expressed by

Vσ(x)= E(x, y)σ(y)ds y for x ∈ Γ, (1.2.7)
y∈Γ \{x}
∂E

Kϕ(x) = (x, y)ϕ(y)ds y for x ∈ Γ, (1.2.8)
∂n y
y∈Γ \{x}
∂E

K σ(x)= (x, y)σ(y)ds y for x ∈ Γ. (1.2.9)
∂n x
y∈Γ \{x}
We remark that here all of the above boundary integrals are improper with
weakly singular kernels in the following sense (see [213, p. 158]): The kernel
k(x, y) of an integral operator of the form

k(x, y)ϕ(y)ds y
Γ