On exact Pleijel's constant for some domains

TL;DR

This paper derives explicit formulas for Pleijel's constant for specific domains like disks, sectors, and rectangles, providing precise numerical values and characterizations for rings and annular sectors.

Contribution

It offers the first explicit expressions and numerical values for Pleijel's constant for these domains, advancing understanding of eigenfunction nodal domain counts.

Findings

Pleijel constant for the disk is approximately 0.4613

Explicit formulas for Pleijel constants of sectors and rectangles

Characterization of Pleijel constants for rings and annular sectors

Abstract

We provide an explicit expression for the Pleijel constant for the planar disk and some of its sectors, as well as for $N$-dimensional rectangles. In particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we characterize the Pleijel constant for some rings and annular sectors in terms of asymptotic behavior of zeros of certain cross-products of Bessel functions.