Conformally Invariant Variational Problems. - SAM

control it gives on the image itself u n (D 2 ). Starting from some
given minimizing sequence u n , one could indeed always modify
u n by adding for instance tentacles of any sort filling more
and more the space but without any significant cost for A(u n )
and keeping the sequence minimizing. The ”limiting” object
lim n→+∞ u n (D 2 ) would then be some ”monster” dense in R m .
In order to overcome this lack of coercivity of the area lagrangian
A, Douglas and Radó proposed instead to minimize
the energy of the map u
E(u) = 1 ∫
|∂ x u| 2 +|∂ y u| 2 dx∧dy .
2 D 2
In contrast with A, E has good coercivity properties and lower
semicontinuityintheweaktopologyoftheSobolevspaceW 1,2 (D 2 ,R m ),
unliketheareaOnecrucialobservationisthefollowingpointwise
inequality valid for all u dans W 1,2 (D 2 ,R 3 ),
|∂ x u×∂ y u| ≤ 1 2
[
|∂x u| 2 +|∂ y u| 2] a.e. in D 2
and integrating this pointwise inequality over the disc D 2 gives
A(u) ≤ E(u) ,
with equality if and only if u is weakly conformal, namely:
|∂ x u| = |∂ y u| et ∂ x u·∂ y u = 0 a.e. .
TheinitialideaofDouglasandRadóbearsresemblancetothe
corresponding strategy in 1 dimension while trying to minimize
the length among all immersions γ of the segment [0,1] joining
two arbitrary points a = γ(0) and b = γ(1) in a riemannian
manifold (M m ,g),
∫
L(γ) := |˙γ| g dt .
[0,1]
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