Page 115 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

102 Direct methods

and therefore, since v ∈ C 0 ∞ (Ω),wehave
ZZ ZZ
h i
¡ ¢
det ∇wv dxdy = ϕψ v − (ϕψ ) v dxdy
y x x y
Ω Ω
and hence (3.9) after integration by parts.
The result (3.8) then easily follows. Indeed from Rellich theorem (Theorem
ν
ν
∞
1.43) we have, since ϕ ϕ in W 1,p and p> 2,that ϕ → ϕ in L . Combining
ν
p
this observation with the fact that ψ ,ψ ν y ψ ,ψ in L we deduce (cf.
y
x
x
Exercise 1.3.3) that
p
ν
ν
ν
ν
ϕ ψ ,ϕ ψ ϕψ ,ϕψ in L . (3.11)
x y x y
ν
Since v x ,v y ∈ C 0 ∞ ⊂ L p 0 we deduce from (3.9), applied to w = u ,and from
(3.11) that
ZZ ZZ
¤
ν
lim det ∇u vdxdy = − £ ϕψ v x − ϕψ v y dxdy .
x
y
ν→∞
Ω Ω
Using again (3.9), applied to w = u, we have indeed obtained the claimed result
(3.8).
Step 2. We now show that (3.8) still holds under the further hypothesis
¡ ¢
ν
2
v ∈ C ∞ (Ω), but considering now the general case, i.e. u ,u ∈ W 1,p Ω; R .
0
In fact (3.9) continues to hold under the weaker hypothesis that v ∈ C 0 ∞ (Ω)
¡ ¢
2
and w ∈ W 1,p Ω; R ; of course the proof must be different, since this time we
¡ ¢
2
only know that w ∈ W 1,p Ω; R . Let us postpone for a moment the proof of
this fact and observe that if (3.9) holds for w ∈ W 1,p ¡ Ω; R 2 ¢ then, with exactly
the same argument as in the previous step, we get (3.8) under the hypotheses
¡
¢
2
ν
v ∈ C ∞ (Ω) and u ,u ∈ W 1,p Ω; R .
0
We now prove the above claim and we start by regularizing w ∈ W 1,p ¡ Ω; R 2 ¢

appealing to Theorem 1.34. We therefore find for every > 0,afunction w =
¡

¢

(ϕ ,ψ ) ∈ C 2 Ω; R 2 so that

kw − w k W 1,p ≤ and kw − w k L ∞ ≤ .
Since p ≥ 2 we can find (cf. Exercise 3.5.4) a constant α 1 (independent of )so
that

kdet ∇w − det ∇w k p/2 ≤ α 1 . (3.12)
L
It is also easy to see that we have, for α 2 a constant (independent of ),
° °
x L p ≤ α 2
° ϕψ − ϕ ψ ° ≤ α 2 , kϕψ − ϕ ψ k (3.13)
y y L p x
since, for example, the first inequality follows from
° ° ° ° ° °
°
° ϕψ − ϕ ψ ° ≤ kϕk L ∞ ψ − ψ ° + ψ ° kϕ − ϕ k L ∞ .
°
y y L p y y L p y L p