Conformally Invariant Variational Problems. - SAM

Proof of the regularity of the solutions of the CMC
equation.
Our first aim will be to establish the existence of a positive
constant α such that
∫
sup ρ −α |∇u| 2 < +∞ . (VII.9)
ρ 0. There exists some radius ρ 0 > 0 such that for
every r < ρ 0 and every point p in B 1/2 (0)
∫
|∇u| 2 < ε 0 .
B r (p)
We shall in due time adjust the value ε 0 to fit our purposes. In
the sequel, r < ρ 0 . On B r (p), we decompose u = φ+v in such
a way that
⎧⎨
⎩
∆φ = H ∂ x u×∂ y u
in B r (p)
φ = 0
on ∂B r (p)
Applying theorem VII.1 to φ yields
∫ ∫ ∫
|∇φ| 2 ≤ C 0 |H| |∇u| 2
B r (p)
B r (p)
∫
≤ C 0 |H| ε 0 |∇u| 2 .
B r (p)
B r (p)
|∇u| 2
(VII.10)
The function v = u−φ is harmonic. To obtain useful estimates
on v, we need the following result.
13 See for instance [Gi].
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