On exact Pleijel's constant for some domains

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.