Asymptotic behaviour of the Steklov problem on dumbbell domains

Bucur, Dorin; Henrot, Antoine; Michetti, Marco

doi:10.48550/arxiv.2007.04844

Cited by 1 publication

(1 citation statement)

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“…Sufficient conditions ensuring the stability of the resolvent operators (associated with the corresponding Dirichlet-to-Neumann map) in a class of star-shaped domains are given in [20] where the question whether one could obtain the same results in a general setting is considered "out of reach". We also cite the very recent paper [9] concerning the asymptotic behaviour of the Steklov problem on dumbbell domains.…”

Section: Introductionmentioning

confidence: 99%

Spectral stability of the Steklov problem

Ferrero¹,

Lamberti²

2021

Preprint

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This paper investigates the stability properties of the spectrum of the classical Steklov problem under domain perturbation. We find conditions which guarantee the spectral stability and we show their optimality. We emphasize the fact that our spectral stability results also involve convergence of eigenfunctions in a suitable sense according with the definition of connecting system by [21]. The convergence of eigenfunctions can be expressed in terms of the H 1 strong convergence. The arguments used in our proofs are based on an appropriate definition of compact convergence of the resolvent operators associated with the Steklov problems on varying domains.In order to show the optimality of our conditions we present alternative assumptions which give rise to a degeneration of the spectrum or to a discontinuity of the spectrum in the sense that the eigenvalues converge to the eigenvalues of a limit problem which does not coincide with the Steklov problem on the limiting domain.

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Section: Introductionmentioning

confidence: 99%