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arXiv · 1905.02432

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

Paolo Tilli·Davide Zucco
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Abstract

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.

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Paolo Tilli, Davide Zucco. 2019-05-07. Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues. https://arxiv.org/abs/1905.02432

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