Conformally Invariant Variational Problems. - SAM

which converges as Radon measures to f. We next introduce
[ n∑
]
u n (x,y) := log λ n 2
i log
|(x,y)−a n i=1
i | .
On D 2 , we find that
n∑
v n = λ n 2
n∑
i log
|(x,y)−a n i | > λ n i log 8 5 = log 8 5
i=1
i=1
On the other hand, there holds
∫ ∫ ∫
|∇u n | 2 ∂u n
= − ∆u n = −
D 2 D 2 ∂D ∂r
∫
2 |∇v n |
≤ ≤ 1 ∫
∂D |v 2 n | log 8 |∇v n | ≤ C
5 ∂D 2
. (VI.13)
for some constant C independent of n . Hence, (u n ) n is a sequence
of solutionsto (VI.10)uniformlybounded in W 1,2 . Since
the sequence (f n ) converges as Radon measures to f, it follows
that for any p < 2, the sequence (v n ) converges strongly in W 1,p
to
v := log 2 r ∗f .
The uniform upper bounded (VI.13) paired to the aforementioned
strong convergence shows that for each p < 2, the sequence
u n = logv n converges strongly in W 1,p to
u := log
[log 2 ]
r ∗f
Fromthehypothesessatisfiedbyf,weseethat∆(e u ) = −2πf ≠
0. As f is regular, so is thus e u , and therefore, owing to (VI.11),
u cannot fullfill (VI.10).
Accordingly, we have constructed a sequence of solutions to
(VI.10) which converges weakly in W 1,2 to a map that is not a
solution to (VI.10).
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