Eigenvalue variations of the Neumann Laplace operator due to perturbed boundary conditions
Nursultanov M. Trad W. Tzou J. Tzou L.
March 2025Springer Science and Business Media Deutschland GmbH
Research in Mathematical Sciences
2025#12Issue 1
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.
Eigenvalues , Neumann Laplacian , Singular perturbation
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Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
School of Mathematics and Statistics, University of Sydney, Sydney, Australia
School of Mathematical and Physical Sciences, Macquarie University, Sydney, Australia
University of Amsterdam, Amsterdam, Netherlands
Department of Mathematics and Statistics
Institute of Mathematics and Mathematical Modeling
School of Mathematics and Statistics
School of Mathematical and Physical Sciences
University of Amsterdam
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