Eigenvalue problems in a non-Lipschitz domain

In this paper we analyse piecewise linear finite element approximations of the Laplace eigenvalue problem in the plane domain Ω = { (x,y) : 0 < x < 1, 0 < y < x^α}, which gives for 1<α the simplest model of an external cusp. Since Ω is curved and non-Lipschitz, the classical spectral theory cannot be applied directly. We present the eigenvalue problem in a proper setting, and relying on known convergence results for the associated source problem with α<3, we obtain a quasi-optimal order of convergence for the eigenpairs.