Conformally Invariant Variational Problems. - SAM

norm. The analytical difficulties relative to this nonlinear system
are thus, a priori, of the same nature as those arising from
the harmonic map equation.
Example 4. In this last example, we combine the settings of
Examples 2 and 3 to produce a mixed problem. Given on R m
a metric g and a two-form ω, both C 1 with uniformly bounded
Lipschitz norm, consider the Lagrangian
Eg ω (u) = 1 ∫
〈∇u,∇u〉
2 g
dxdy +u ∗ ω .
D 2
As before, it is a coercive conformally invariant Lagrangian
with quadratic growth. Its critical points satisfy the Euler-
Lagrangian equation
∆u i +
m∑
Γ i kl (u)∇uk ·∇u l −2
k,l=1
m∑
Hkl i (u)∇⊥ u k ·∇u l = 0 ,
k,l=1
(VI.18)
for i = 1···m.
Once again, this elliptic system admits a geometric interpretation
which generalizes the ones from Examples 2 and 3. Whenever
a conformal map u satisfies (VI.18), then u(D 2 ) is a surface
in (R m ,g) whose meancurvaturevectorisgivenby (VI.17). The
equation (VI.18) also forms an elliptic system with quadratic
growth, and critical in dimension two for the W 1,2 norm.
Interestingly enough, M. Grüter showed that any coercive
conformally invariant Lagrangian with quadratic growth is of
the form Eg ω for some appropriately chosen g and ω.
Theorem VI.1. [Gr] Let l(z,p) be a real-valued function on
R m × R 2 ⊗ R m , which is C 1 in its first variable and C 2 in its
second variable. Supposethat l obeys the coercivityandquadratic
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