Page 216 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 3: Direct methods 203

ν
ν
q
we deduce that u → u in L , for every q ≥ 1; however the convergence u → u
ν
in L ∞ does not hold. We next show that u u in W 1,p if p ∈ [1, 2).We
will show this only up to a subsequence (it is not difficult to see that the whole
sequence has this property). We readily have
⎛ |x| ⎞
2
1 x + −x 1 x 2
2
ν
∇u = 2 ⎜ ν ⎟
⎠
⎝
|x| (|x| +1/ν) 2 |x|
−x 1 x 2 x +
1
ν
and thus
µ ¶ 1/2
2 2 |x| 2
|x| + + 2
ν
|∇u | = ν ν 2 .
(|x| +1/ν)
We therefore find, if 1 ≤ p< 2,that, γ denoting a constant independent of ν,
³ ´ p/2
2 2
ZZ Z 1 (r +1/ν) +1/ν
ν p
|∇u | dx 1 dx 2 =2π 2p rdr
Ω 0 (r +1/ν)
Z p Z
1 2 p/2 (r +1/ν) 1 rdr
≤ 2π 2p rdr =2 (2+p)/2 πν p p
0 (r +1/ν) 0 (νr +1)
Z ν+1 (s − 1) ds
≤ 2 (2+p)/2 πν p−2 p ≤ γ.
1 s
ν
This implies that, up to the extraction of a subsequence, u u in W 1,p ,as
claimed.
(iii) A direct computation gives
1
ν
ν
det ∇u = |det ∇u | = 3
ν (|x| +1/ν)
and hence
ZZ Z 1 Z ν+1
ν 2 rdr (s − 1) ds
|det ∇u | dx 1 dx 2 =2πν 3 =2π 3
Ω 0 (νr +1) 1 s
∙ ¸ ν+1
1 1
=2π − .
2s 2 s
1
We therefore have ZZ
ν
lim |det ∇u | dx 1 dx 2 = π. (7.14)
ν→∞
Ω
© ª
2
Observe that if Ω δ = x ∈ R : |x| <δ , then for every fixed δ> 0,wehave
ν
ν
det ∇u = |det ∇u | → 0 in L ∞ (ΩÂΩ δ ) . (7.15)