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We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in RN. Although for n = 1,2 and a positive boundary parameter α it is known that the minimisers do not depend on α, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α. We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with n1/N, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further...
Il est bien connu que les fréquences propres associées à un d'Alembertien amorti sont confinées dans une bande parallèle à l'axe réel. Nous rappelons l'asymptotique de Weyl pour la distribution des parties réelles des fréquences propres, nous montrons que «presque toutes» les fréquences propres appartiennent à une bande déterminée par la limite de Birkhoff du coefficient d'amortissement. Nous montrons aussi que certaines moyennes des parties imaginaires convergent vers la moyenne du coefficient...
We analyze the accuracy and well-posedness of generalized impedance
boundary value problems in the framework of scattering problems
from unbounded highly absorbing media. We restrict ourselves in this first work
to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties
in the analysis for the generalized impedance boundary conditions, since
classical compactness arguments are no...
2000 Mathematics Subject Classification: 35J05, 35C15, 44P05In this paper, we consider the variations of eigenvalues and eigenfunctions
for the Laplace operator with homogeneous Dirichlet boundary conditions
under deformation of the underlying domain of definition. We derive
recursive formulas for the Taylor coefficients of the eigenvalues as functions
of the shape-perturbation parameter and we establish the existence of a set
of eigenfunctions that is jointly holomorphic in the spatial and boundary-variation
...
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