Conformally Invariant Variational Problems. - SAM

On the other hand, φ satisfies by definition
−∆φ = (∇e 1 ,∇ ⊥ e 2 ) =
m∑
∂ y e j 1 ∂ xe j 2 −∂ xe j 1 ∂ ye j 2 . (VII.30)
j=1
The right-hand side of this elliptic equation comprises only Jacobians
of elements of W 1,2 . This configuration is identical to
those previously encountered in our study of the constant mean
curvature equation and of the equation of S n -valued harmonic
maps. In order to capitalize on this particular structure, we call
upon an extension of Wente’s theorem VII.1 due to Coifman,
Lions, Meyer, and Semmes.
Theorem VII.3. [CLMS]Let a andbbe twofunctionsin W 1,2 (D 2 ),
and let φ be the unique solution in W 1,p
0 (D 2 ), for 1 ≤ p < 2 , of
the equation
⎧
⎨ −∆φ = ∂ x a∂ y b−∂ x b∂ y a in D 2
(VII.31)
⎩
φ = 0 on ∂D 2 .
Then φ lies in W 2,1 and
‖∇ 2 φ‖ L1 (D 2 ) ≤ C 1 ‖∇a‖ L2 (D 2 ) ‖∇b‖ L2 (D 2 ) .
where C 1 is a constant independent of a and b. 17
(VII.32)
✷
Applying this result to the solution φ of (VII.30) then reveals
that (e 1 ,∇e 2 ) is indeed an element of W 1,1 .
We will express the harmonic map equation (VI.26) in this
particular Coulomb frame field, distinguished by its increased
17 Theorem VII.1 is a corollary of theorem VII.3 owing to the Sobolev embedding
W 2,1 (D 2 ) ⊂ W 1,2 ∩ C 0 . In the same vein, theorem VII.3 was preceded by two intermediary
results. The first one, by Luc Tartar [Tar1], states that the Fourier transform of
∇φ lies in the Lorentz space L 2,1 , which also implies theorem VII.1. The second one, due
to Stefan Müller, obtains the statement of theorem VII.3 under the additional hypothesis
that the Jacobian ∂ x a∂ y b−∂ x b∂ y a be positive.
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