Page 210 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 3: Direct methods 197

Invoking Poincaré inequality (cf. Theorem 1.47) we can find γ ,γ ,γ ,sothat
3
5
4
p p q
m +1 ≥ γ ku ν k W 1,p − γ ku 0 k W 1,p − γ ku ν k W 1,p − γ 5
3
1
4
and hence, γ being a constant,
6
p q
m +1 ≥ γ ku ν k 1,p − γ ku ν k 1,p − γ .
3 W 1 W 6
Since 1 ≤ q< p,wecan find γ ,γ so that
7
8
p
m +1 ≥ γ ku ν k W 1,p − γ 8
7
which, combined with the fact that m< ∞, leads to the claim, namely
ku ν k W 1,p ≤ γ .
9
Exercise 3.3.2. This time it is the lower semicontinuity step in Theorem 3.3
that has to be changed. Let therefore u ν u in W 1,p and let
I (u)= I 1 (u)+ I 2 (u)

where Z Z
I 1 (u)= h (x, ∇u (x)) dx , I 2 (u)= g (x, u (x)) dx .
Ω Ω
It is clear that, by the proof of the theorem, lim inf I 1 (u ν ) ≥ I 1 (u).The result
will thereforebeprovedifwecan show

lim I 2 (u ν )= I 2 (u) .
i→∞
Case 1: p> n. From Rellich theorem (Theorem 1.43) we obtain that u ν → u
∞
in L ; in particular there exists R> 0 such that ku ν k ∞ , kuk ∞ ≤ R.The
L L
result then follows since
Z
|I 2 (u ν ) − I 2 (u)| ≤ |g (x, u ν ) − g (x, u)| dx ≤ γ ku ν − uk ∞ meas Ω.
L
Ω
Case 2: p = n. The estimate
Z
³ ´
q−1 q−1
|I 2 (u ν ) − I 2 (u)| ≤ γ 1+ |u ν | + |u| |u ν − u| dx ,
Ω
combined with Hölder inequality gives us
µZ q ¶ q−1 µZ ¶ 1
³ ´ q q
q−1 q−1 q−1 q
|I 2 (u ν ) − I 2 (u)| ≤ γ 1+ |u ν | + |u| |u ν − u| .
Ω Ω