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

TL;DR

This paper analyzes the structure and measure-theoretic properties of the singular set in optimal partitions related to Dirichlet eigenvalues, establishing its rectifiability and finiteness of certain measures.

Contribution

It proves the local finiteness of the Minkowski content and Hausdorff measure of the singular set, and shows it is countably rectifiable, extending results to Riemannian manifolds and harmonic maps.

Findings

Singular set has locally finite upper Minkowski content.

Singular set is countably $(n-2)$-rectifiable.

Results apply to Riemannian manifolds and harmonic maps.

Abstract

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.