On the singular set of free interface in an optimal partition problem

We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper $(n-2)$-dimensional Minkowski content, and consequently, its $(n-2)$-dimensional Hausdorff measure are locally finite. We also show that the singular set is countably $(n-2)$-rectifiable, namely it can be covered by countably many $C^1$-manifolds of dimension $(n-2)$, up to a set of $(n-2)$-dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well.