Page 95 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

82 Direct methods

Step 1 (Compactness). We start with the observation that since u 0 ∈ u 0 +
1,2
W (Ω) we have
0
0 ≤ m ≤ I (u 0 ) < ∞.
Let u ν ∈ u 0 + W 1,2 (Ω) be a minimizing sequence of (D), this means that
0
I (u ν ) → inf {I (u)} = m,as ν →∞.

Observe that by Poincaré inequality (cf. Theorem 1.47) we can find constants
γ ,γ > 0 so that
1 2
p
2I (u ν )= k∇u ν k L 2 ≥ γ ku ν k W 1,2 − γ ku 0 k W 1,2 .
1
2
Since u ν is a minimizing sequence and m< ∞ we deduce that there exists γ > 0
3
so that
ku ν k W 1,2 ≤ γ .
3
1,2
Applying Exercise 1.4.5 we deduce that there exists u ∈ u 0 + W (Ω) and a
0
subsequence (still denoted u ν )sothat
u ν u in W 1,2 ,as ν →∞.

Step 2 (Lower semicontinuity). We now show that I is (sequentially) weakly
lower semicontinuous; this means that

u ν u in W 1,2 ⇒ lim infI (u ν ) ≥ I (u) .
ν→∞
This step is independent of the fact that {u ν } is a minimizing sequence. We
trivially have that

2 2 2
|∇u ν | = |∇u| +2 h∇u; ∇u ν −∇ui + |∇u ν −∇u|
2
≥ |∇u| +2 h∇u; ∇u ν −∇ui .
Integrating this expression we have
Z
I (u ν ) ≥ I (u)+ h∇u; ∇u ν −∇ui dx .
Ω
To conclude we show that the second term in the right hand side of the inequality
2
2
tends to 0. Indeed since ∇u ∈ L and ∇u ν −∇u 0 in L this implies, by
2
definition of weak convergence in L ,that
Z
lim h∇u; ∇u ν −∇ui dx =0 .
ν→∞
Ω