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arXiv · 1805.03191

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

Onur Alper
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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.

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Onur Alper. 2018-05-08. On the singular set of free interface in an optimal partition problem. https://arxiv.org/abs/1805.03191

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