Page 117 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

104 Direct methods

ν
while (3.14), the fact that u u in W 1,p and Exercise 3.5.4 show that we can
find γ> 0 so that
ν
kv − v k L p/(p−2) kdet ∇u − det ∇uk L p/2 ≤ γ .
Since is arbitrary we have indeed obtained that (3.8) holds for v ∈ L p/(p−2)
ν
and for u u in W 1,p . The lemma is therefore proved.
We can now proceed with the proof of Theorem 3.19.
Proof. We will prove the theorem under the further following hypotheses
(for a general proof see Theorem 4.2.10 in [31])

f (x, u, ξ)= g (x, u, ξ)+ h (x, det ξ)
¢
where g satisfies (H1) and (H2), with p> 2, of Theorem 3.3 and h ∈ C 1 ¡ Ω × R ,
h ≥ 0, δ → h (x, δ) is convex for every x ∈ Ω and there exists γ> 0 so that
³ ´
(p−2)/2
|h δ (x, δ)| ≤ γ 1+ |δ| . (3.15)
The proof is then identical to the one of Theorem 3.3, except the second step
(the weak lower semicontinuity), that we discuss now. We have to prove that
u ν u in W 1,p ⇒ lim infI (u ν ) ≥ I (u)
ν→∞
where I (u)= G (u)+ H (u) with
Z Z
G (u)= g (x, u (x) , ∇u (x)) dx, H (u)= h (x, det ∇u (x)) dx .
Ω Ω
We have already proved in Theorem 3.3 that

lim infG (u ν ) ≥ G (u)
ν→∞
and therefore the result will follow if we can show

lim infH (u ν ) ≥ H (u) .
ν→∞
1
Since h is convex and C we have
h (x, det ∇u ν ) ≥ h (x, det ∇u)+ h δ (x, det ∇u)(det ∇u ν − det ∇u) . (3.16)

¡ ¢
2
We know that u ∈ W 1,p Ω; R , which implies that det ∇u ∈ L p/2 (Ω),and
hence using (3.15) we deduce that
h δ (x, det ∇u) ∈ L p/(p−2) = L (p/2) 0 , (3.17)