Page 175 -

4. Sobolev Spaces

In order to study the variational formulations of boundary integral equations
and their numerical approximations, one needs proper function spaces. The
Sobolev spaces provide a very natural setting for variational problems. This
chapter contains a brief summary of the basic definitions and results of the
2
L –theory of Sobolev spaces which will suffice for our purposes. A more gen-
eral discussion on these topics may be found in the standard books such as
Adams [1], Grisvard [108], Lions and Magenes [190], Maz‘ya [201] and also
in McLean [203].

s
4.1 The Spaces H (Ω)
p
The Spaces L (Ω)(1 ≤ p ≤∞)
p
We denote by L (Ω)for 1 ≤ p< ∞, the space of equivalence classes of
n p
Lebesgue measurable functions u on the open subset Ω ⊂ IR such that |u|
is integrable on Ω. We recall that two Lebesgue measurable functions u and
v on Ω are said to be equivalent if they are equal almost everywhere in Ω,
i.e. u(x)= v(x) for all x outside a set of Lebesgue measure zero (Kufner et al
p
[173]. The space L (Ω) is a Banach space with the norm

1/p

p

u
L p (Ω) := |u(x)| dx .
Ω
In particular, for p = 2, we have the space of all square integrable functions
2
L (Ω) which is also a Hilbert space with the inner product

2
(u, v) L 2 (Ω) := u(x)v(x)dx for all u, v ∈ L (Ω) .
Ω
A Lebesgue measurable function u on Ω is said to be essentially bounded if
there exists a constant c ≥ 0 such that |u(x)|≤ c almost everywhere (a.e.)
in Ω. We define

ess sup |u(x)| =inf{c ∈ IR ||u(x)|≤ c a.e.in Ω}.
x∈Ω