Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues
Abstract
We study the optimal partitioning of a (possibly unbounded) interval of the real line into 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 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.
Keywords
Cite
@article{arxiv.1905.02432,
title = {Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues},
author = {Paolo Tilli and Davide Zucco},
journal= {arXiv preprint arXiv:1905.02432},
year = {2019}
}