Conformally Invariant Variational Problems. - SAM

Therefore, for m = 3, any minimizer u of (V.3) is an immersion
in the interior of D 2 . A thorough presentation of the Plateau
problem can be found for instance in [DHKW1] and [DHKW2].
We first establish some elementary properties of minimizers
of (V.3) assuming it exists. We are going to first establish (V.4)
postponing to later the existence question.
We have the following elementary proposition
Proposition V.1. Let u be the weak limit in W 1,2 of a minimizing
sequence of E in C(Γ). Then u satisfies the Laplace
equation
∀i = 1···m ∆u i = 0 in D ′ (D 2 ) .
where u i are the components of u in R m .
✷
Proof of proposition V.1.
Let u n be a minimizing sequence. Modulo extraction of a
subsequence we can always assume that u n weakly converges to
a limit u ∈ W 1,2 (D 2 ,R m ). Let φ ∈ C ∞ 0 (D2 ,R m ) be a smooth
function, compactly supported in D 2 .
It is clear that for every t ∈ R and for every n ∈ N
Hence we have for any t ∈ R
u n +tφ ∈ C(Γ) .
inf E(u) = lim E(u n) ≤ liminf E(u n+tφ) .
u∈C(Γ) n→+∞ n→+∞
(V.5)
Observe that we have
E(u n +tφ) = E(u n )
∫
+t < ∇u n ·∇φ > dx+ t2
D 2
2
∫
D 2 |∇φ| 2 dx
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