  The existence of Dirichlet minimizing multiple-valued functions for given
boundary data has been known since pioneering work of F. Almgren. Here we prove
a multiple-valued analogue of the classical Plateau problem of the existence of
area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb
N,$ $k_1,...,k_K\in \mathbb N$ with sum $Q$ and any collection of $K$ disjoint
Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary
data with these multiplicities which extends to a Dirichlet minimizing
$Q$-valued function with minimal Dirichlet energy among all possible monotone
parameterizations of the boundary curves. Under a condition analogous to the
Douglas condition for minimizers from planar domains, conformality of the
minimizer follows from topological methods and some complex analysis. Finally,
we analyze two particular cases: in contrast to single-valued Douglas
solutions, we first give a class of examples for which our multiple-valued
Plateau solution has branch points. Second, we give examples of a degenerate
behavior, illustrating the weakness of the multiple-valued maximum principle
and provide motivation for our analogous Douglas condition.
