Conformally Invariant Variational Problems. - SAM

VII Integrabilitybycompensationtheoryapplied
to some conformally invariant Lagrangians
VII.1 Constant mean curvature equation (CMC)
Let H ∈ R be constant. We study the analytical properties of
solutions in W 1,2 (D 2 ,R 3 ) of the equation
∆u−2H ∂ x u×∂ y u = 0 .
(VII.1)
The Jacobian structure of the right-hand side enable without
muchtrouble,interalia,toshowthatPalais-Smale sequences
converge weakly:
Let F n be a sequence of distributions converging to zero in
H −1 (D 2 ,R 3 ), and let u n be a sequence of functions uniformly
bounded in W 1,2 and satisfying the equation
∆u n −2H ∂ x u n ×∂ y u n = F n → 0 strongly in H −1 (D 2 ) .
We use the notation
(∂ x u n ×∂ y u n ) i = ∂ x u i+1
n ∂ yun i−1 −∂ x un i−1∂
yu i+1
n
= ∂ x (u i+1
n
∂ y u i−1
n )−∂ y (u i+1
n ∂ x u i−1
n ) .
(VII.2)
The uniform bounded on the W 1,2 -norm of u n enables the extraction
of a subsequence u n
′ weakly converging in W 1,2 to some
limit u ∞ . With the help of the Rellich-Kondrachov theorem, we
seethatthesequenceu n isstronglycompactinL 2 . Inparticular,
we can pass to the limit in the following quadratic terms
and
u i+1
n
u i+1
n
∂ y u i−1
n → u i+1
∞ ∂ yu i−1
∞ in D ′ (D 2 )
∂ x u i−1
n → u i+1
∞ ∂ x u i−1
∞ in D ′ (D 2 ) .
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