Singularities at the tip of a plane angular sector

Abstract

Solutions of the Helmholtz and Laplace equations in three dimensions which vanish, or have vanishing normal derivative on an angular sector of opening angle β, are considered. The solutions are required to be functions of distance from the tip of the sector multiplied by functions of the angular coordinates. The angular functions are eigenfunctions of the Laplace–Beltrami operator on the unit sphere, which vanish or have vanishing normal derivative, on a great circle arc of length β. It is shown that the Dirichlet eigenvalues are nondecreasing functions of β, and the Neumann eigenvalues are nonincreasing. Furthermore, each Dirichlet eigenvalue of a sector of angle β is a Neumann eigenvalue of a sector of angle 2π−β and conversely. The eigenvalues for β=0, π, and 2π are found explicitly. These results lead to a qualitative description of the eigenvalues as functions of β. The eigenvalues determine the singular behavior of the solutions at the tip.