Singularities at the tip of a plane angular sector

Solutions of the Helmholtz and Laplace equations in three dimensions which vanish, or have vanishing normal derivative on an angular sector of opening angle beta, 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-Beltra mi operator on the unit sphere, which vanish or have vanishing normal deriv ative, on a great circle arc of length beta. It is shown that the Dirichlet eigenvalues are nondecreasing functions of beta, and the Neumann eigenvalu es are nonincreasing. Furthermore, each Dirichlet eigenvalue of a sector of angle beta is a Neumann eigenvalue of a sector of angle 2 pi-beta and conv ersely. The eigenvalues for beta=0, pi, and 2 pi are found explicitly. Thes e results lead to a qualitative description of the eigenvalues as functions of beta. The eigenvalues determine the singular behavior of the solutions at the tip. (C) 1999 American Institute of Physics. [S0022-2488(99)02002-2] .