2023/06/01 by Nursultanov, Medet, Trad, William, Tzou, Justin +1
#35B25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35J25 Secondary: 35P20 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2306.00491
This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold (M,g,∂ M) under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive a sharp asymptotic 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 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. 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.