Conformally Invariant Variational Problems. - SAM

where [p 1 ,p 2 ] is the segment joining the two points p 1 and p 2 in
D 2 and dσ is the length element along this segment.
The mean-value theorem implies then that there exists a
point p 0 ∈ [p 1 ,p 2 ] such that
λ(p 0 ) ≥ log |⃗ Φ(p 1 )− ⃗ Φ(p 2 )|
|p 2 −p 1 |
.
Thus in particular we have
ν(p 0 ) = (λ−µ)(p 0 ) ≥ log |⃗ Φ(p 1 )−Φ(p ⃗ 2 )|
|p 2 −p 1 |
∫
− C |∇⃗n ⃗Φ | 2 .
D 2
(X.172)
Let ρ < 1 such that B ρ (0) contains the segment [p 1 ,p 2 ]. Let
r := (1+ρ)/2. The explicit expression of the Poisson Kernel 67
gives
ν(p 0 ) = r2 −|p 0 | 2
2πr
∫
∂B 2 r (0) ν(z)
|z −p 0 | dσ(z) ,
where dσ(z) is the volume form on ∂Br 2(0). Let ν+ = sup{0,ν}
and ν − = inf{0,ν}. We then have
∫
ν − (z) 2πr
dσ(z) ≥
∂Br 2(0) |z −p 0 | r 2 −|p 0 | ν(p 0)
2
∫
ν + (X.173)
(z)
−
|z −p 0 | dσ(z)
∂B 2 r(0)
We apply (X.165) on Br 2 (0) and we have
ν + ≤ 1 [ ∫ ]
2 log+ C r e 2λ +‖µ‖ ∞ .
D 2
67 See for instance [GT] theorem 2.6.
(X.174)
178