Page 50 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 37

Example 1.46 Let I =(0, 2π) and u ν (x)=(1/ν)cos νx. We have already
∗
seen that u 0 in L ∞ (0, 2π) and hence
0
ν
∗
u ν 0 in W 1,∞ (0, 2π) .
It is clear that we also have
u ν → 0 in L ∞ (0, 2π) .
The last theorem that we will often use is (see Corollary IX.19 in Brézis [14]):

n
Theorem 1.47 (Poincaré inequality). Let Ω ⊂ R be a bounded open set
and 1 ≤ p ≤∞. Then there exists γ = γ (Ω,p) > 0 so that
1,p
kuk L p ≤ γ k∇uk L p , ∀u ∈ W 0 (Ω)
or equivalently
kuk W 1,p ≤ γ k∇uk L p , ∀u ∈ W 0 1,p (Ω) .
Remark 1.48 (i)Weneed toimposeacondition ofthe type u =0 on ∂Ω (which
1,p
comes from the hypothesis u ∈ W ) to avoid constant functions u (which imply
0
∇u =0), otherwise the inequality would be trivially false.
(ii) Sometimes the Poincaré inequality appears under the following form (see
n
Theorem 5.8.1 in Evans [43]). If 1 ≤ p ≤∞,if Ω ⊂ R is a bounded connected
open set, with Lipschitz boundary, and if we denote by
Z
1
u Ω = u (x) dx
meas Ω Ω
then there exists γ = γ (Ω,p) > 0 so that

ku − u Ω k L p ≤ γ k∇uk L p , ∀u ∈ W 1,p (Ω) .
(iii) In the case n =1, that will be discussed in the proof, we will not really
use that u (a)= u (b)= 0,but only u (a)=0 (or equivalently u (b)=0). The
theorem remains thus valid under this weaker hypothesis.
(iv) We will often use Poincaré inequality under the following form. If u 0 ∈
W 1,p (Ω) and u ∈ u 0 + W 0 1,p (Ω),thenthere exist γ ,γ > 0 so that
1
2
k∇uk L p ≥ γ kuk W 1,p − γ ku 0 k W 1,p .
2
1
Proof. We will prove the inequality only when n =1 and Ω =(a, b).Since
1,p
u ∈ W 0 (a, b),wehave u ∈ C ([a, b]) and u (a)= u (b)= 0.
We will prove that, for every 1 ≤ p ≤∞,
0
kuk L p ≤ (b − a) ku k L p . (1.13)