Page 116 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The vectorial case 103

Returningto(3.9) we have
ZZ ZZ
£ ¤
det ∇wv dxdy + ϕψ v x − ϕψ v y dxdy
y
x
Ω Ω
ZZ ZZ £ ¤

= det ∇w vdxdy + ϕ ψ v x − ϕ ψ v y dxdy
y x
Ω Ω
ZZ

+ (det ∇w − det ∇w ) vdxdy
Ω
ZZ
£¡ ¢ ¤
+ ϕψ − ϕ ψ y v x − (ϕψ − ϕ ψ ) v y dxdy .
y
x
x
Ω

2

Appealing to (3.9) which has already been proved to hold for w =(ϕ ,ψ ) ∈ C ,
to Hölder inequality, to (3.12) and to (3.13) we find that, α 3 being a constant
independent of ,
¯ZZ ZZ ¯
¯ £ ¤ ¯
det ∇w v dxdy + ϕψ v x − ϕψ v y dxdy
¯ ¯
y
x
¯ ¯
Ω Ω
h i
≤ α 3 kvk (L p/2 + kv x k L p 0 + kv y k L p 0 .
)
0
Since is arbitrary we have indeed obtained that (3.9) is also valid for w ∈
¡
¢
2
W 1,p Ω; R .
Step 3. We are finally in a position to prove the lemma, removing the last
unnecessary hypothesis (v ∈ C ∞ (Ω)). We want (3.8) to hold for v ∈ L p/(p−2) .
0
This is obtained by regularizing the function as in Theorem 1.13. This means,

for every > 0 and v ∈ L p/(p−2) ,thatwecan find v ∈ C 0 ∞ (Ω) so that

kv − v k L p/(p−2) ≤ . (3.14)
We moreover have
ZZ ZZ ZZ

ν
ν
ν
det ∇u vdxdy = det ∇u (v − v ) dxdy + det ∇u v dxdy .
Ω Ω Ω
Using, once more, Hölder inequality we find
¯ZZ ¯
¯ ν ¯
(det ∇u − det ∇u) v
¯ ¯
¯ ¯
Ω
¯ZZ ¯
¯ ¯
ν ν
¯
≤ kv − v k L p/(p−2) kdet ∇u − det ∇uk L p/2 + ¯ (det ∇u − det ∇u) v .
¯
Ω ¯
The previous step has shown that
¯ZZ ¯
¯ ν ¯
lim ¯ (det ∇u − det ∇u) v ¯ =0
¯ ¯
ν→∞
Ω