Conformally Invariant Variational Problems. - SAM

oundary, and sending ∂D 2 monotonically onto Γ. This weakening
of the regularityfor the space of maps prevents a-priori to
make use of the uniformization theorem to a weak object such
as u ∗ g R m. However, the following theorem of Morrey gives an
”almost uniformization” result that permits to overcome this
difficulty of the lack of regularity (see theorem 1.2 of [Mor1]
about ǫ−conformal parametrization)
Theorem V.1. [Mor1] Let u be a map in the Sobolev space
C 0 ∩ W 1,2 (D 2 ,R m ) and let ε > 0, then there exists an homeomorphism
Ψ of the disc such that Ψ ∈ W 1,2 (D 2 ,D 2 ),
u◦Ψ ∈ C 0 ∩W 1,2 (D 2 ,R m ) ,
and
E(u◦Ψ) ≤ A(u◦Ψ)+ε = A(u)+ε .
✷
The main difficulty remains to find a C 1 immersion minimizing
the Dirichlet energy E. Postponing to later the requirement
for the map u to realize an immersion of the disc, one could first
try to find a general minimizer of E within the class of Sobolev
maps in W 1,2 (D 2 ,R m ), continuous at the boundary, and sending
∂D 2 monotonically onto Γ. By monotonically we mean the
following relaxation of the homeomorphism condition - which is
too restrictive in the first approach.
Definition V.2. Let Γ be a Jordan closed curve in R m , i.e. a
subset of R m homeomorphic to S 1 , and let γ be an homeomorphism
from S 1 into Γ. We say that a continuous map ψ from S 1
into Γ is weakly monotonic, if there exists a non decreasing continuous
function τ : [0,2π] → R with τ(0) = 0 and τ(2π) = 2π
such that
∀θ ∈ [0,2π] ψ(e iθ ) = γ(e iτ(θ) ) .
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✷