Page 46 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 33

which combined with Fubini theorem (which allows to permute the integrals),
leads to
b c t b b
Z Z Z Z Z
0 0 0 0 0
v (x) ϕ (x) dx = − u (t) dt ϕ (x) dx + u (t) dt ϕ (x) dx
a a a c t
c b
Z Z
0
0
= − u (t) ϕ (t) dt + u (t)(−ϕ (t)) dt
a c
b
Z
0
= − u (t) ϕ (t) dt
a
which is exactly (1.11). The fact that v is continuous follows from the observation
that if x ≥ y,then
Z x µZ x ¶ 1 µZ x ¶ 1
p
p 0 p 0
0 0 p
|v (x) − v (y)| ≤ |u (t)| dt ≤ |u (t)| dt 1 dt
y y y (1.12)
1
0
≤ |x − y| p 0 ·ku k L p
wherewehaveusedHölderinequality. If p =1, the inequality (1.12) does not
imply that v is continuous; the continuity of v follows from classical results of
Lebesgue integrals, see Exercise 1.3.7.
Step 2. We now are in a position to conclude. Since from (1.11), we have
Z b Z b
0 0 ∞
vϕ dx = − u ϕdx, ∀ϕ ∈ C 0 (a, b)
a a
R R
and we know that u ∈ W 1,p (a, b) (and hence uϕ dx = − u ϕdx), we deduce
0
0
Z b
0
(v − u) ϕ dx =0, ∀ϕ ∈ C ∞ (a, b) .
0
a
Applying Exercise 1.3.6, we find that v − u = γ a.e., γ denoting a constant, and
u
since v is continuous, we have that e = v − γ has all the desired properties.
We are now in a position to state the main results concerning Sobolev spaces.
They give some inclusions between these spaces, as well as some compact imbed-
n
dings. These results generalize to R what has already been seen in Lemma 1.38
for the one dimensional case. Before stating these results we need to define what
n
kind of regularity will be assumed on the boundary of the domains Ω ⊂ R that
we will consider. When Ω =(a, b) ⊂ R, there was no restriction. We will assume,
n
for the sake of simplicity, that Ω ⊂ R is bounded. The following definition ex-
k
presses in precise terms the intuitive notion of regular boundary (C , C or
∞
Lipschitz).