Page 114 -

3.1. Variational Equations and Sobolev Spaces 97

The mapping that restricts v to ∂Ω,
d 2
∞
γ 0 : C (R )| Ω , · 1 → L (∂Ω), · 0 ,
v → v| ∂Ω ,
is continuous.
Thus there exists a unique, linear, and continuous extension
1 2
γ 0 : H (Ω), · 1 → L (∂Ω), · 0 .

Proof: See, for example, [37].

2
Therefore, in short form, γ 0 (v) ∈ L (∂Ω), and there exists some constant
C> 0 such that
1
for all v ∈ H (Ω) .
γ 0(v) 0 ≤ C v 1
1
2
Here γ 0 (v) ∈ L (∂Ω) is called the trace of v ∈ H (Ω).

1
The mapping γ 0 is not surjective; that is, γ 0 (v) v ∈ H (Ω) is a real

d
2
∞
subset of L (∂Ω). For all v ∈ C (R )| Ω we have
γ 0 (v)= v| ∂Ω .
In the following we will use again v| ∂Ω or “v on ∂Ω” for γ 0 (v), but in
the sense of Theorem 3.5. According to this theorem, definition (2.20) is
well-defined with the interpretation of u on ∂Ω as the trace:
1
1
Definition 3.6 H (Ω) := v ∈ H (Ω) γ 0 (v) = 0 (as a function on ∂Ω) .

0
Theorem 3.7 Suppose Ω ⊂ R d is a bounded Lipschitz domain. Then
1
∞
C (Ω) is dense in H (Ω).
0
0
Proof: See [37].
1
d
∞
The assertion of Theorem 3.5, that C (R )| Ω is dense in H (Ω), has
1
severe consequences for the treatment of functions in H (Ω) which are in
general not very smooth. It is possible to consider them as smooth functions
if at the end only relations involving continuous expressions in · 1 (and not
requiring something like ∂ i v ∞ ) arise. Then, by some “density argument”
1
the result can be transferred to H (Ω) or, as for the trace term, new terms
1
can be defined for functions in H (Ω). Thus, for the proof of Lemma 3.4
1
it is necessary simply to verify estimate (3.9), for example for v ∈ C [a, b].
1
By virtue of Theorem 3.7, analogous results hold for H (Ω).
0
1
Hence, for v ∈ H (Ω) integration by parts is possible:
d
Theorem 3.8 Suppose Ω ⊂ R is a bounded Lipschitz domain. The outer
d
unit normal vector ν =(ν i ) i=1,...,d : ∂Ω → R is defined almost everywhere
and ν i ∈ L (∂Ω).
∞