A Numerical study of the Dirichlet and Neumann eigenvalue problem of the Laplacian on cusp domains

Abstract

Consider a planar annulus. A result of Ramm and Shivakumar states that as the inner circle moves toward the outer circle, the principal Dirichlet eigenvalue of the Laplacian decreases. Numerical experiments in that paper clearly verify this result. The purpose of this short note is to fill in a small gap in that paper: the numerical calculation of the principal eigenvalue when the two circles touch. This is a non-trivial numerical problem because the domain has a cusp which is a strong singularity. Adaptive finite element methods have difficulty converging in the presence of such singularities. Our method is to perform a transformation taking the domain to a rectangle, where it is relatively straightforward to compute the principal eigenvalue. We also calculate the minimal non-zero eigenvalue of the Neumann problem. Numerically, the Neumann eigenvalue has no such monotonicity property.