Conformally Invariant Variational Problems. - SAM

We further assume that L is conformally invariant: for each
positive conformal transformation f of degree 1, and for each
map u ∈ W 1,2 (D 2 ,R m ), there holds
∫
L(u◦f) = l(u◦f,∇(u◦f)) dx ′ dy ′
f −1 (D 2 )
∫
(VI.3)
= l(u,∇u) dx dy = L(u) .
D 2
Example 1. The Dirichlet energy described in the Introduction,
∫
E(u) = |∇u| 2 dxdy ,
D 2
whose critical points satisfy the Laplace equation (V.4), which,
owing to the conformalhypothesis, geometricallydescribes minimal
surfaces. Regularity and compactness matters relative to
this equation are handled with the help of the maximum principle.
Example 2. LetanarbitraryinR m begiven,namely(g ij ) i,j∈Nm ∈
C 1 (R m ,S + m), whereS + m denotesthe subset of M m (R), comprising
the symmetric positive definite m×m matrices. We make the
following uniform coercivity and boundedness hypothesis:
∃ C > 0 s. t. C −1 δ ij ≤ g ij ≤ Cδ ij on R m .
Finally, we suppose that
‖∇g‖ L∞ (R m ) < +∞ .
With these conditions, the second example of quadratic, coercive,
conformally invariant Lagrangian is
E g (u) = 1 ∫
〈∇u,∇u〉
2 g
dxdy
D 2
= 1 ∫
∑ m
g ij (u)∇u i·∇u j dxdy .
2
D 2
i,j=1
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