Title: Consistency of Dirichlet Partitions,
Abstract: A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a
collection of $k$ pairwise disjoint open subsets such that the sum of their
first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet
partitions has been posed on graphs with applications in data analysis. Both
versions admit variational formulations: solutions are characterized by
minimizers of the Dirichlet energy of mappings from $U$ into a singular space
$\Sigma_k \subseteq \mathbb{R}^k$. In this paper, we extend results of N.\
García Trillos and D.\ Slepčev to show that there exist solutions of the
continuum problem arising as limits to solutions of a sequence of discrete
problems. Specifically, a sequence of points $\{x_i\}_{i \in \mathbb{N}}$ from
$U$ is sampled i.i.d.\ with respect to a given probability measure $\nu$ on $U$
and for all $n \in \mathbb{N}$, a geometric graph $G_n$ is constructed from the
first $n$ points $x_1, x_2, \ldots, x_n$ and the pairwise distances between the
points. With probability one with respect to the choice of points $\{x_i\}_{i
\in \mathbb{N}}$, we show that as $n \to \infty$ the discrete Dirichlet
energies for functions $G_n \to \Sigma_k$ $\Gamma$-converge to (a scalar
multiple of) the continuum Dirichlet energy for functions $U \to \Sigma_k$ with
respect to a metric coming from the theory of optimal transport. This, along
with a compactness property for the aforementioned energies that we prove,
implies the convergence of minimizers. When $\nu$ is the uniform distribution,
our results also imply the statistical consistency statement that Dirichlet
partitions of geometric graphs converge to partitions of the sampled space in
the Hausdorff sense.