A localized domain perturbation which splits the spectrum of the Laplacian
Alexander Dabrowski · Complex Variables and Elliptic Equations · 2020
For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The second one consists in the deformation of the boundary of the domain itself, and requires more technical properties of the bilinear form of the variational problem.