Conformally Invariant Variational Problems. - SAM

Returning to the disk D 2 , the Whitney extension theorem
yields the existence of ã and ˜b such that
∫
|∇ã| 2 ≤ C 1 |∇a|
∫D 2 , (VII.7)
2
and
∫
C
C
|∇˜b| 2 ≤ C 1
∫D 2 |∇b| 2 . (VII.8)
Let ˜φ be the function in (VII.5). The difference φ− ˜φ satisfies
the equation
⎧
⎨ ∆(φ− ˜φ) = 0 in D 2
⎩
φ− ˜φ = −˜φ on ∂D 2
Themaximumprincipleappliedtotheinequalities(VII.6),(VII.7)
and (VII.8) produces
‖φ− ˜φ‖ L∞ (D 2 ) ≤ ‖˜φ‖ L∞ (∂D 2 ) ≤ C‖∇a‖ 2 ‖∇b‖ 2 .
With the triangle inequality |‖φ‖ ∞ −‖˜φ‖ ∞ | ≤ ‖φ− ˜φ‖ ∞ and the
inequality (VII.6), we reach the desired L ∞ -estimate of φ, and
therefore, per the above discussion, the theorem is proved. ✷
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