Conformally Invariant Variational Problems. - SAM

is explicitly given by
u(x,y) := loglog
2
√
x2 +y 2 .
The regularity issue can thus be answered negatively. Similarly,
for the equation (VI.10), it takes little effort to devise counterexamplestothe
weaklimitquestion(i),andthustothe question
(ii). To this end, it is helpful to observe that C 2 maps obey the
general identity
∆e u = e u [ ∆u+|∇u| 2] .
(VI.11)
One easily verifies that if v is a positive solution of
∆v = −2π ∑ i
λ i δ ai ,
where λ i > 0 and δ ai are isolated Dirac masses, then u := logv
provides a solution 9 in W 1,2 of (VI.10). We then select a strictly
positive regular function f with integral equal to 1, and supported
on the ball of radius 1/4 centered on the origin. There
exists a sequence of atomic measures with positive weights λ n i
such that
n∑
n∑
f n = λ n i δ a
n
i
and λ n i = 1 , (VI.12)
i=1
9 Indeed, per (VI.11), we find ∆u + |∇u| 2 = 0 away from the points a i . Near these
points, ∇u asymptotically behaves as follows:
i=1
|∇u| = |v| −1 |∇v| ≃ ( |(x,y)−a i | log|(x,y)−a i | ) −1
∈ L
2
.
Hence, |∇u| 2 ∈ L 1 , so that ∆u + |∇u| 2 is a distribution in H −1 + L 1 supported on the
isolated points a i . From this, it follows easily that
∆u+|∇u| 2 = ∑ i
µ i δ ai .
Thus, ∆u is the sum of an L 1 function and of Dirac masses. But because ∆u lies in H −1 ,
the coefficients µ i must be zero. Accordingly, u does belong to W 1,2 .
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