Conformally Invariant Variational Problems. - SAM

tangent frame field to T 2 . Define the energy
∫
min |(e 1 ,∇e 2 )| 2 dxdy ,
ψ∈W 1,2 (D 2 ,R) D 2
where (·,·) is the standard scalar product on R m , and
(VII.26)
e 1 (x,y)+ie 2 (x,y) := e iψ(x,y) (ẽ 1 (x,y)+iẽ 2 (x,y)) .
We seek to optimize the map (ẽ) by minimizing this energy over
the W 1,2 (D 2 )-maps taking values in the space of rotationsof the
plane R 2 ≃ T u(x,y) T 2 . Our goal is to seek a frame field as regular
as possible in which the harmonic map equation will be recast.
The variational problem (VII.26) is well-posed, and it further
admits a solution in W 1,2 . Indeed, there holds
|(e 1 ,∇e 2 )| 2 = |∇ψ +(ẽ 1 ,∇ẽ 2 )| 2 .
Hence, there exists a unique minimizer in W 1,2 which satisfies
0 = div(∇ψ +(ẽ 1 ,∇ẽ 2 )) = div((e 1 ,∇e 2 )) . (VII.27)
A priori, (e 1 ,∇e 2 ) belongs to L 2 . But actually, thanks to the
careful selection brought in by the variational problem (VII.26),
we shall discover that the frame field (e 1 ,∇e 2 ) over D 2 lies in
W 1,1 , thereby improving the original L 2 belongingness 16 . Because
the vector field (e 1 ,∇e 2 ) is divergence-free, there exists
some function φ ∈ W 1,2 such that
(e 1 ,∇e 2 ) = ∇ ⊥ φ . (VII.29)
16 Further yet, owing to a result of Luc Tartar [Tar2], we know that W 1,1 (D 2 ) is continuously
embedded in the Lorentz space L 2,1 (D 2 ), whose dual is the Marcinkiewicz weak-L 2
space L 2,∞ (D 2 ), whose definition was recalled in (VI.9). A measurable function f is an
element of L 2,1 (D 2 ) whenever
∫ +∞
0
∣ { p ∈ D 2 ; |f(p)| > λ }∣ ∣ 1 2
dλ .
(VII.28)
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