Spectral stability of the Steklov problem

This paper investigates the stability properties of spectrum classical Steklov problem under domain perturbation. We find conditions which guarantee spectral and we show their optimality. emphasize fact that our results also involve convergence eigenfunctions in a suitable sense according with definition connecting system by Vainikko (1979). The can be expressed terms H1 strong convergence. arguments used proofs are based on an appropriate compact resolvent operators associated problems varying domains. In order to optimality present alternative assumptions give rise degeneration or discontinuity eigenvalues converge limit does not coincide limiting domain.