Consistency of Dirichlet Partitions

A Dirichlet k-partition of a domain U subseteq ℝd 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 Sigmak subseteq ℝk. In this paper, we extend results of N. Garc'ia Trillos and D. Slepvcev 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 xi_i ∈ mathbbN from U is sampled i.i.d. with respect to a given probability measure nu on U and for all n ∈ mathbbN, a geometric graph Gn is constructed from the first n points x1, x2, ldots, xn and the pairwise distances between the points. With probability one with respect to the choice of points xi_i ∈ mathbbN, we show that as n → ∞ the discrete Dirichlet energies for functions Gn → Sigmak Gamma-converge to (a scalar multiple of) the continuum Dirichlet energy for functions U → Sigmak 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.