Eigenvalue variations of the Neumann Laplace operator due to perturbed boundary conditions

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold (M, g, ∂M) under a singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive an asymptotic expansion of the perturbed eigenvalues as the Dirichlet part shrinks to a point x^∗ ∈ ∂M in terms of the spectral parameters of the unperturbed system. This asymptotic expansion demonstrates the impact of the geometric properties of the manifold at a specific point x^∗. Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic expansion. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green’s function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work.