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

2018/05/08 by Alper, Onur · 3 citations
#35R35 #49Q10 #49Q20 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.03191

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 C¹-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.

Cited by

• A survey on the optimal partition problem
• Spectral minimal partitions of unbounded domains
• A nonlinear model for long-range segregation

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