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

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