Spectral stability of the Steklov problem

TL;DR

This paper analyzes how the spectrum of the Steklov problem remains stable under domain changes, establishing optimal conditions for spectral stability and eigenfunction convergence.

Contribution

It provides new conditions ensuring spectral stability of the Steklov problem under domain perturbations, including eigenfunction convergence in the $H^1$ sense.

Findings

Spectral stability conditions are optimal.

Eigenfunctions converge in $H^1$ norm.

Spectral discontinuities occur under alternative assumptions.

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 \cite{Vainikko}. 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.