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.

On Pleijel's nodal domain theorem for the Robin problem

“…The value of the Pleijel constant for a given domain is generally unknown. Bobkov [3] calculated Pl(Δ 𝐷 Ω ) for some simple domains, including the Euclidean disk in ℝ 2 . When Ω = 𝔹 is a Euclidean disk in ℝ 2 , he showed that Pl(Δ 𝐷 𝔹 ) = 0.461 ⋯…”

“…The value of the Pleijel constant for a given domain is generally unknown. Bobkov [3] calculated $$\operatorname{Pl}(\Delta _\Omega ^{D})$$ for some simple domains, including the Euclidean disk in $$\mathbb {R}^2$$. When $$\Omega ={{\mathbb {B}}}$$ is a Euclidean disk in $$\mathbb {R}^2$$, he showed that $$\operatorname{Pl}(\Delta _{{\mathbb {B}}}^{D})=0.461\cdots$$…”