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.
|Titolo:||Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues|
|Data di pubblicazione:||2020|
|Digital Object Identifier (DOI):||10.1016/j.jde.2019.12.026|
|Appare nelle tipologie:||1.1 Articolo in rivista|