Page 102 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

A general existence theorem 89

1,p
Let u ν ∈ u 0 + W (Ω) be a minimizing sequence of (P), i.e.
0
I (u ν ) → inf {I (u)} = m,as ν →∞.
Wethereforehavefrom (H2+) that for ν large enough
p
m +1 ≥ I (u ν ) ≥ α 1 k∇u ν k p − |α 3 | meas Ω
L
and hence there exists α 4 > 0 so that
k∇u ν k L p ≤ α 4 .
Appealing to Poincaré inequality (cf. Theorem 1.47) we can find constants
α 5 ,α 6 > 0 so that

α 4 ≥ k∇u ν k L p ≥ α 5 ku ν k W 1,p − α 6 ku 0 k W 1,p
and hence we can find α 7 > 0 so that

ku ν k 1,p ≤ α 7 .
W
Applying Exercise 1.4.5 (it is only here that we use the fact that p> 1)we
1,p
deduce that there exists u ∈ u 0 + W (Ω) and a subsequence (still denoted u ν )
0
so that
u ν u in W 1,p ,as ν →∞.
Step 2 (Lower semicontinuity). We now show that I is (sequentially) weakly
lower semicontinuous; this means that
u ν u in W 1,p ⇒ lim infI (u ν ) ≥ I (u) . (3.2)
ν→∞
This step is independent of the fact that {u ν } is a minimizing sequence. Using
1
the convexity of f and the fact that it is C we get
f (x, u ν , ∇u ν ) ≥
(3.3)
f (x, u, ∇u)+ f u (x, u, ∇u)(u ν − u)+ hf ξ (x, u, ∇u); ∇u ν −∇ui .
Before proceeding further we need to show that the combination of (H3) and
u ∈ W 1,p (Ω) leads to

p
0
n
0
p
f u (x, u, ∇u) ∈ L (Ω) and f ξ (x, u, ∇u) ∈ L (Ω; R ) (3.4)
0
where 1/p +1/p =1 (i.e. p = p/ (p − 1)). Indeed let us prove the first
0
statement, the other one being shown similarly. We have (β being a constant)
1
Z Z p
³ ´
p 0 p 0 p−1 p−1 p−1
|f u (x, u, ∇u)| dx ≤ β 1+ |u| + |∇u| dx
Ω Ω
¡ p ¢
≤ β 1+ kuk
1 W 1,p < ∞ .