Spectral stability for a class of fourth order Steklov problems under domain perturbations

TL;DR

This paper investigates the spectral stability of two fourth order Steklov problems under domain perturbations, establishing resolvent and eigenvalue stability without domain shape restrictions, and explores related boundary value problems.

Contribution

It introduces weak domain convergence conditions ensuring spectral stability for Steklov problems, including a sharp condition for a variant of the DBS problem, and extends analysis to Navier boundary problems.

Findings

Proved resolvent operator stability under weak domain convergence.

Established eigenvalue and eigenfunction stability in strong $H^2$-norms.

Provided stability and instability results for biharmonic boundary value problems.

Abstract

We study the spectral stability of two fourth order Steklov problems upon domain perturbation. One of the two problems is the classical DBS - Dirichlet Biharmonic Steklov - problem, the other one is a variant. Under a comparatively weak condition on the convergence of the domains, we prove the stability of the resolvent operators for both problems, which implies the stability of eigenvalues and eigenfunctions. The stability estimates for the eigenfunctions are expressed in terms of the strong $H^2$-norms. The analysis is carried out without assuming that the domains are star-shaped. Our condition turns out to be sharp at least for the variant of the DBS problem. In the case of the DBS problem, we prove stability of a suitable Dirichlet-to-Neumann type map under very weak conditions on the convergence of the domains and we formulate an open problem. As bypass product of our analysis, we provide some stability and instability results for Navier and Navier-type boundary value problems for the biharmonic operator.