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

Abstract This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $$(M,g,\partial M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$x^*\in \partial M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>∈</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> 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 $$x^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> . 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.