Conformally Invariant Variational Problems. - SAM

growth conditions
∃C > 0 t.q. ∀z ∈ R m ∀p ∈ R 2 ⊗R m
C −1 |p| 2 ≤ l(X,p) ≤ C|p| 2 .
Let L be the Lagrangian
∫
L(u) = l(u,∇u)(x,y) dx dy
D 2
(VI.19)
(VI.20)
acting on W 1,2 (D 2 ,R m )-maps u. We suppose that L is conformally
invariant: for every conformal application φ positive and
of degree 1, there holds
∫
L(u◦φ) = l(u◦φ,∇(u◦φ))(x,y) dx dy = L(u) .
φ −1 (D 2 )
(VI.21)
Then there exist on R m a C 1 metric g and a C 1 two-form ω such
that
L = E ω g . (VI.22)
Maps taking values in a submanifold of R m .
Up to now, we have restricted our attention to maps from D 2
into a manifold with only one chart (R n ,g). More generally, it
is possible to introduce the Sobolev space W 1,2 (D 2 ,N n ), where
(N n ,g) is an oriented n-dimensional C 2 -manifold. When this
manifold is compact without boundary (which we shall henceforth
assume, for the sake of simplicity), a theorem by Nash
guarantees that it can be isometricallyimmersed into Euclidean
space R m , for m large enough. We then define
W 1,2 (D 2 ,N n ) := { u ∈ W 1,2 (D 2 ,R m ) ; u(p) ∈ N n a.e. p ∈ D 2}
Given on N n a C 1 two-form ω, we may consider the Lagrangian
E ω (u) = 1 ∫
|∇u| 2 dx dy +u ∗ ω (VI.23)
2 D 2
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