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.
Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues / Tilli, Paolo; Zucco, Davide. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - (2020).
|Titolo:||Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues|
|Data di pubblicazione:||2020|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.jde.2019.12.026|
|Appare nelle tipologie:||1.1 Articolo in rivista|