Parasitic eigenvalues of spectral problems for the Laplacian with third-type boundary conditions

Spectral problems for the Laplacian with Robin and Steklov (third-type) boundary conditions on a smooth boundary of a plane domain are considered. These conditions involve a small parameter and a coefficient of “wrong” sign, giving rise to negative eigenvalues, which are called parasitic. Such problems and eigenvalues arise in numerical schemes when regular variations in boundaries (small nonuniform shifts along the normal) are modeled by perturbations of differential operators in boundary conditions. Asymptotic expansions of some parasitic eigenvalues are constructed and justified, and a priori estimates are obtained, which help to determine their locations on the real axis and the effect exerted on the simulation errors.