Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into n subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-04, Vol.268 (8), p.4900-4919
Main Authors: Tilli, Paolo, Zucco, Davide
Format: Article
Language:English
Subjects:
Fair division problem
Minimax problem
Optimal partitioning
Set function
Sturm-Liouville eigenvalue
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Summary:We study the optimal partitioning of a (possibly unbounded) interval of the real line into n subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as n tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2019.12.026