2021/03/07 by Alberto Ferrero, Ferrero, Alberto, Pier Domenico Lamberti +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2103.04991
openalex publication_date 2021/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeVainikko. The convergence of eigenfunctions can be expressed in terms of the H1 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.