Conformally Invariant Variational Problems. - SAM

in the middle of this arc : |p − p 0 | = |q − p 0 | < δ/2. Consider
ρ ∈ [δ, √ δ] given by the Courant lemma V.3 and satisfying
‖u(x)−u(y)‖ 2 L ∞ ((∂B ρ (p 0 )∩D 2 ) 2 ) ≤ 4π ∫
log 1 |∇u| 2 dx
δ D 2
Let p ′ and q ′ be the twopoints given by the intersectionbetween
∂D 2 and ∂B ρ (p 0 ). Since u is continuous up to the boundary we
deduce that
|u(p ′ )−u(q ′ )| ≤ 4π
log 1 δ
We fix now δ in such a way that
∫
4π C
log 1 δ
D 2 |∇u| 2 dx ≤ 4π C
log 1 δ
< η
(V.34)
Because of (V.31) one of the two arcs in Γ connecting u(p ′ )
and u(q ′ ) has to be contained in a Bε m −ball. Since ε has been
chosen small enough satisfying (V.32) this arc connecting u(p ′ )
and u(q ′ ) and contained in a Bε m −ball can contain at most one
of the Q i . In the mean time δ has been chosen small enough in
such a way that B √ δ (p 0)∩∂D 2 contains also at most one of the
P i , thus, since u in monotonic on ∂D 2 , the arc connecting u(p ′ )
and u(q ′ ) and contained in a Bε m −ball has to be u(∂B ρ (p 0 )∩D 2 )
and we have than proved that
|u(p)−u(q)| ≤ |u(p ′ )−u(q ′ )| < ε .
This concludes the proof of claim (V.30) and lemma V.2 is
proved.
✷
In order to establish that E posses a minimizer in C ∗ (Γ) we
are going to establish the following result.
Lemma V.4. Let u be the weak limit of a minimizing sequence
of E in C ∗ (Γ), then u ∈ C 0 (D 2 ,R m ).
✷
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