Conformally Invariant Variational Problems. - SAM

Since u n weakly converges to u in W 1,2 (D 2 ) we have
∫
∫
< ∇u n·∇φ > dx −→ < ∇u·∇φ > dx .
D 2 D 2
This fact combined with the previous identity gives
lim E(u n +tφ) = inf E(u)
n→+∞ u∈C(Γ)
∫
∫
+t < ∇u·∇φ > dx+ t2 |∇φ| 2 dx .
D 2
2 D 2
the inequality (V.5) together with the identity (V.6) imply
∫
∫
∀t ∈ R t < ∇u·∇φ > dx+ t2 |∇φ| 2 dx ≥ 0
D 2
2 D 2
(V.6)
Asaconsequenceofthisinequalitybydividingby|t|andmaking
t tend to zero respectively from the right and from the left we
have obtained
∫
∀φ ∈ C0 ∞ (D2 ,R m ) < ∇u·∇φ > dx = 0 . (V.7)
D 2
which is the desired result and proposition V.1 is proved.
Observe that (V.7) is equivalent to
∀φ ∈ C ∞ 0 (D2 ,R m )
d
dt E(u+tφ) | t=0
= 0 .
✷
(V.8)
This is the Euler-Lagrange equation associated to our variational
problem, the Laplace equation in the present case, and
the proposition we just proved can be summarized by saying
that every weak limit of our minimizing sequence is a critical
point of the lagrangian for variations in the target : variations
of the form u + tφ. In order to prove the second line of
(V.4), the conformality of u, we will exploit instead that the
minimizer u - once we will prove it’s existence - has to be a
28