Conformally Invariant Variational Problems. - SAM

The analogousversion of theorem VII.1 with Neuman boundary
conditions yields the estimate
∫
|∇E| ≤ C 0 |∇D|
∫D
∫D 2 |∇P −1 | 2 . (VIII.44)
2 2 D 2
Moreover, because ∇D = ∇ ⊥ E P, there holds |∇D| ≤ |∇E|.
Put into (VIII.44), this shows that if ∫ D 2 |∇P| 2 is chosen sufficiently
small (i.e. for ε 0 (m) in (VIII.25) small enough), then
D ≡ 0. Whence, we find
∇Ã−Ã∇⊥ ξ +∇ ⊥ B P = 0 in D 2 ,
thereby ending the proof of theorem VIII.4.
✷
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