Conformally Invariant Variational Problems. - SAM

Combining this to (VII.2) reveals that u ∞ is a solution of the
CMC equation (VII.1).
Obtaining information on the regularity of weak W 1,2 solutionsof
the CMC equation(VII.2) requiressome more elaborate
work. More precisely, a result from the theory of integration by
compensation due to H. Wente is needed.
Theorem VII.1. [We]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.3)
⎩
ϕ = 0 on ∂D 2 .
Then φ belongs to C 0 ∩W 1,2 (D 2 ) and
‖φ‖ L∞ (D 2 )+‖∇φ‖ L2 (D 2 ) ≤ C 0 ‖∇a‖ L2 (D 2 ) ‖∇b‖ L2 (D 2 ) . (VII.4)
where C 0 is a constant independent of a and b. 12
✷
Proof of theorem VII.1. Weshallfirstassumethataandb
are smooth, so as to legitimize the various manipulations which
we will need to perform. The conclusion of the theorem for
general a and b in W 1,2 may then be reached through a simple
density argument. In this fashion, we will obtain the continuity
of φ from its being the uniform limit of smooth functions.
Observe first that integration by parts and a simple application
of the Cauchy-Schwarz inequality yields the estimate
∫ ∫
|∇φ| 2 = − φ∆φ ≤ ‖φ‖ ∞ ‖∂ x a∂ y b−∂ x b∂ y a‖ 1
D 2 D 2
≤ 2‖φ‖ ∞ ‖∇a‖ 2 ‖∇b‖ 2 .
12 Actually, one shows that theorem VII.1 may be generalized to arbitrary oriented
Riemannian surfaces, with a constant C 0 independent of the surface, which is quite a
remarkable and useful fact. For more details, see [Ge] and [To].
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