Calculus of Variations and Geometric Measure Theory

P. Tilli - D. Zucco

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

created by zucco on 08 May 2019
modified on 29 Dec 2019


Accepted Paper

Inserted: 8 may 2019
Last Updated: 29 dec 2019

Journal: Journal of Differential Equations
Year: 2019

ArXiv: 1905.02432 PDF


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.