Conformally Invariant Variational Problems. - SAM

The identity (X.234) becomes
∫ ∫ ∫
∇c·∇φ = a ∆ψ = − ∇a·∇ψ .
D 2 D 2 D 2
(X.237)
Combining (X.233) with (X.236) and (X.237) gives
∫
‖∇φ‖ L2 (D 2 ) = sup X ·∇φ ≤ C‖∇a‖ L
2,∞ ‖∇b‖ L
2 ,
‖X‖ L 2≤1 D 2 (X.238)
which is the identity (X.231) and the proof of theorem X.17 is
complete.
✷
It remains to prove theorem X.16 and the present subsection
will be complete.
Proof of theorem X.16. Combining corollary X.5 and theorem
X.16 gives in a straightforwardway that ∇S and ∇ ⃗ R given
respectively by the first and the second line of (X.208) and satisfying
the elliptic system (X.223) are in L 2 loc (D2 ).
Argueingexactlylikeintheproofoftheregularityofsolutions
to CMC surfaces we obtain the existence of α > 0 such that
sup r −α |∆S|+|∆R| x 0 ∈B
∫B ⃗ < +∞
1/2 2 (0) ; r 2 such that ∇S and
∇R ⃗ are in L p loc (D2 ). Bootstrapping this information again in
the two first lines of (X.223), using lemma VIII.1, gives that
∇S and ∇R ⃗ are in L p loc (D2 ) for any p < +∞. Bootstarpping
the latest in the third equation of (X.223) gives that Φ ⃗ ∈
W 2,p
loc (D2 ,R m ) for any p < +∞, from which we can deduce that
⃗n ∈ W 1,p
loc (D2 ,Gr m−2 (R m )) for any p < +∞, information that
we inject back in the two first lines of the system (X.223)...etc
and one obtains after iterating again and again that Φ ⃗ is in
W k,p
loc (D2 ,R m ) for any k ∈ N and 1 ≤ p ≤ +∞. This gives that
⃗Φ is in Cloc ∞(D2 ) and theorem X.16 is proved. ✷
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