Conformally Invariant Variational Problems. - SAM

Lemma VII.1. Let v be a harmonic function on D 2 . For every
point p in D 2 , the function
ρ ↦−→ 1 ∫
|∇v| 2
ρ 2
is increasing.
B ρ (p)
Proof. Note first that
[ ]
d 1
|∇v|
dρ ρ
∫B 2 = − 2 ∫
|∇v| 2 + 1 ∫
|∇v| 2 .
2
ρ (p) ρ 3 B ρ (p) ρ 2 ∂B ρ (p)
(VII.11)
Denotebyv theaverageofv on∂B ρ (p): v := |∂B ρ (p)| −1∫ ∂B ρ (p) v.
Then, there holds
∫
0 =
B ρ (p)
This implies that
∫
(v −v) ∆v = −
B ρ (p)
∫
|∇v| 2 +
∂B ρ (p)
(v −v) ∂v
∂ρ
∫ ( )1 (
2 ∫ 1
|∇v| 2 1
≤ |v −v|
ρ B ρ (p) ρ
∫∂B 2 ∂v
2 )1 2
2 ∣
ρ (p) ∂B ρ (p) ∂ρ∣
.
(VII.12)
In Fourier space, v satisfies v = ∑ ∑
n∈Z a ne inθ and v − v =
n∈Z
a ∗ n e inθ . Accordingly,
1
2πρ
∫
∂B ρ (p)
|v−v| 2 = ∑ n∈Z ∗ |a n | 2 ≤ ∑ n∈Z ∗ |n| 2 |a n | 2 ≤ 1
2π
Combining the latter with (VII.12) then gives
∫ ( ∫
1
|∇v| 2 ≤
ρ B ρ (p) ∂B ρ (p)
1∂v
∣ρ∂θ∣
2 )1 2
( ∫ ∂v
∣
∂B ρ (p) ∂ρ∣
∫ 2π
0
2 )1 2
∂v
∣∂θ∣
.
2
.
✷
(VII.13)
dθ .
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