Spectral stability of the Steklov problem

Abstract

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.

Spectral stability of the $curl curl$ operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

“…In this model case, the question is whether one can get spectral stability for α < 2. Note that a profile of the form (5) is typical in the study of boundary homogenization problems and thin domains, see for example [2,3,4,9,11,20,21,22]. This problem was solved for the biharmonic operator with intermediate boundary conditions (modelling an elastic hinged plate) in [3] where condition (4) is relaxed by introducing a suitable notion of weighted convergence which allows to prove spectral stability for α > 3/2 in the model problem above.…”