Conformally Invariant Variational Problems. - SAM

where f(z,p) is an arbitrarycontinuousfunction for which there
exists constants C 0 > 0 and C 1 > 0 satisfying
∀z ∈ R m ∀p ∈ R 2 ⊗R m
f(z,p) ≤ C 1 |p| 2 +C 0 . (VI.8)
In dimension two, these equations are critical for the Sobolev
space W 1,2 . Indeed,
u ∈ W 1,2 ⇒ Γ(∇u,∇u) ∈ L 1 ⇒ ∇u ∈ L p loc (D2 ) ∀p < 2 .
In other words, from the regularitystandpoint, the demand that
∇u be square-integrable provides the information that 7 ∇u belongs
to L p loc
for all p < 2. We have thus lost a little bit of
information! Had this not been the case, the problem would
be “boostrapable”, thereby enabling a successful study of the
regularity of u. Therefore, in this class of problems, the main
difficulty lies in the aforementioned slight loss of information,
which we symbolically represent by L 2 → L 2,∞ .
Therearesimpleexamplesofequationswithquadraticgrowth
in two dimensionsfor which the answersto the questions(i)-(iii)
are all negative. Consider 8
∆u+|∇u| 2 = 0 .
(VI.10)
This equation has quadratic growth, and it admits a solution in
W 1,2 (D 2 ) which is unbounded in L ∞ , and thus discontinuous. It
7 Actually, one can show that ∇u belongs to the weak-L 2 Marcinkiewicz space L 2,∞
loc
comprising those measurable functions f for which
supλ 2 |{p ∈ ω ; |f(p)| > λ}| < +∞ ,
λ>0
(VI.9)
where |·| is the standard Lebesgue measure. Note that L 2,∞ is a slightly larger space than
L 2 . However, it possesses the same scaling properties.
8 This equation is conformally invariant. However, as shown by J. Frehse [Fre], it is also
the Euler-LagrangeequationderivedfromaLagrangianwhichisnotconformallyinvariant:
L(u) =
∫D 2 (
1+
)
1
1+e 12u (log1/|(x,y)|) −12
|∇u| 2 (x,y) dx dy .
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