Conformally Invariant Variational Problems. - SAM

First of all, it is clear that theorem VIII.1 is applicable to the
equation (VIII.4) so as to yield the regularity of harmonic maps
taking values in a manifold of codimension 1.
Another rather direct application of theorem VIII.1 deals
with the solutions of the prescribed mean curvature equation
in R 3 ,
∆u = 2H(u) ∂ x u×∂ y u dans D ′ (D 2 ) .
This equation can be recast in the form
∆u = H(u)∇ ⊥ u×∇u ,
Via introducing
⎛
Ω := H(u) ⎜
⎝
0 −∇ ⊥ u 3 ∇ ⊥ u 2
∇ ⊥ u 3 0 −∇ ⊥ u 1
−∇ ⊥ u 2 ∇ ⊥ u 1 0
⎞
⎟
⎠
we observe successively that Ω is antisymmetric, that it belongs
to L 2 whenever H belongs to L ∞ , and that u satisfies (VIII.6).
The hypotheses of theorem VIII.1 are thus all satisfied, and so
we conclude that that u is Hölder continuous.
This last example outlines clearly the usefulness of theoremVIII.1towardssolvingHeinz-Hildebrandt’sconjecture.
Namely,
it enablesus to weakenthe Lipschitzeanassumptionon H found
in previous works ([Hei1], [Hei2], [Gr2], [Bet1], ...), by only requiring
that H be an element of L ∞ . This is precisely the condition
stated in Hildebrandt’s conjecture. By all means, we are
in good shape.
In fact, Hildebrandt’s conjecture will be completely resolved
with the help of the following result.
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