Conformally Invariant Variational Problems. - SAM

Corollary X.6. Let Φ ⃗ be a lipschitz immersion of a smooth
surface Σ 2 with L 2 −bounded second fundamental form. Assume
moreover that Φ ⃗ is weak Willmore in the sense of definition X.4.
Then Φ ⃗ is C ∞ in conformal parametrization. ✷
In order to prove theorem X.16 we will need the following
consequence of Coifman Lions Meyer and Semmes result, theorem
VII.3, which is due to F.Bethuel (see [Bet1]).
Theorem X.17. [Bet1] Let a be a function such that ∇a ∈
L 2,∞ (D 2 ) and let b be a function in W 1,2 (D 2 ). Let φ be the
unique solution in ∩ p 0 is a constant independent of a and b.
(X.231)
✷
Proof of theorem X.17. We assume first that b is smooth
and, once the estimate (X.231) will be proved we can conclude
by a density argument. For a smooth b, classical elliptic theory
tells us that ∇φ is in L q (D 2 ) for any q < +∞ and one has in
particular
∫
‖∇φ‖ L2 (D 2 ) = sup X ·∇φ .
‖X‖ L 2≤1 D 2
76 The Jacobian ∂ x a∂ y b−∂ y a∂ x b has to be understood in the weak sense
∂ x a∂ y b−∂ y a∂ x b := div [ a∇ ⊥ b ] .
Since ∇a ∈ L 2,∞ (D 2 ) we have that a ∈ L q (D 2 ) for all q < +∞ and hence a∇ ⊥ b ∈ L p (D 2 )
for all p < 2.
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