On the Singular Set of Free Interface in an Optimal Partition Problem | Zendy

Alper Onur | Zendy

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On the Singular Set of Free Interface in an Optimal Partition Problem

Author(s) -

Alper Onur

Publication year - 2020

Publication title -

communications on pure and applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.12

H-Index - 115

eISSN - 1097-0312

pISSN - 0010-3640

DOI - 10.1002/cpa.21874

Subject(s) - mathematics , hausdorff measure , partition (number theory) , hausdorff dimension , measure (data warehouse) , minkowski–bouligand dimension , pure mathematics , dimension (graph theory) , dirichlet distribution , combinatorics , mathematical analysis , boundary value problem , fractal , fractal dimension , database , computer science

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. © 2019 Wiley Periodicals, Inc.

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