Conformally Invariant Variational Problems. - SAM

Wente’s theorem (VII.1) guarantees that C lies in W 1,2 , and
moreover
∫
|∇C|
∫D 2 ≤ C 0 |∇B|
∫D 2 |∇u| 2 . (VIII.18)
2 2 D 2
By construction, div(A∇u−∇C) = 0. Poincaré’s lemma thus
yields the existence of D in W 1,2 with ∇ ⊥ D := A∇u−∇C, and
∫ ∫
|∇D| 2 ≤ 2 |A∇u| 2 +|∇C| 2
D 2 D
∫
2 ∫ ∫
≤ 2‖A‖ ∞ |∇u| 2 +2C 0 |∇B| 2 |∇u| 2 .
D 2 D 2 D 2 (VIII.19)
The function D satisfies the identity
∆D = −∇A·∇ ⊥ u = ∂ x A∂ y u−∂ y A∂ x u .
Just as we did in the case of the CMC equation, we introduce
the decomposition D = φ+v, with φ fulfilling
⎧
⎨ ∆φ = ∂ x A∂ y u−∂ y A∂ x u in B r (p)
(VIII.20)
⎩
φ = 0 on ∂B r (p) ,
and with v being harmonic. Once again, Wente’s theorem VII.1
gives us the estimate
∫ ∫ ∫
|∇φ| 2 ≤ C 0 |∇A| 2 |∇u| 2 . (VIII.21)
B r (p)
B r (p)
B r (p)
The arguments which we used in the course of the regularity
prooffortheCMCequationmayberecycledheresoastoobtain
the analogous version of (VII.16), only this time on the ball
B δr (p), where 0 < δ < 1 will be adjusted in due time. More
precisely, we find
∫ ∫
|∇D| 2 ≤ 2δ 2 |∇D| 2
B δr (p)
B r (p)
∫
+3 |∇φ| 2 .
B r (p)
84
(VIII.22)