Conformally Invariant Variational Problems. - SAM

For any such a vector field X satisfying ‖X‖ L
2 ≤ 1 there exists
a unique Hodge decomposition 77
X := ∇c+∇ ⊥ d ,
where c ∈ W 1,2
0 . Moreover, one easily verifies that
1 = ‖X‖ 2 L 2 (D 2 ) = ‖∇c‖2 L 2 (D 2 ) +‖∇d‖2 L 2 (D 2 ) . (X.232)
Indeed one has
∫ ∫ ∫
∇ ⊥ ∂d
d·∇c =
D 2 ∂D ∂τ c− div [ ∇ ⊥ d ] c = 0
2 D 2
Similarly, replacing c by φ the same argument gives
∫ ∫
X ·∇φ = ∇c·∇φ . (X.233)
D 2 D 2
Using again the fact that c = 0 on ∂D 2 , we have
∫ ∫ ∫
∇c·∇φ = − c∆φ = − c div[a ∇ ⊥ b]
D 2 D 2 D
∫
2 (X.234)
= a ∇ ⊥ b·∇c .
D 2
Let ψ be the solution of
⎧
⎪⎨ −∆ψ = ∂ x b∂ y c−∂ y b∂ x c in D 2
(X.235)
⎪⎩
∂ψ
∂ν = 0 on ∂D2 .
Using the Neuman version of theorem VII.3 together with the
embedding of W 1,1 (D 2 ) into L 2,1 (D 2 ) we obtain the existence of
a constant C independent of b and c such that
‖∇ψ‖ L
2,1
(D 2 ) ≤ C ‖∇b‖ L2 (D 2 ) ‖∇c‖ L2 (D 2 ) ≤ C ‖∇b‖ L2 (D 2 ) .
(X.236)
77 c is the minimizer of the following convex problem
∫
min |X −∇c| 2 .
c∈W 1,2
0 (D 2 ) D 2
197