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 \((M,g,\partial M)\) $(M,g,\partial M)$ 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 \(x^*\in \partial M\) $x^{\ast }\in \partial M$ 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 \(x^*\) $x^{\ast }$ . 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.