Domain perturbation for biharmonic Steklov problems: spectral stability, boundary homogenization and degeneration

Abstract

In this paper, we investigate the spectral stability of two biharmonic Steklov problems under domain perturbation. We provide optimal conditions on the boundary perturbations ensuring the stability of both eigenvalues and eigenfunctions. To highlight the optimality of those conditions, we present alternative assumptions on the boundary perturbations that lead to either a degeneration of the spectrum or to the appearance of a strange term in the limiting problem. In particular, these phenomena are discussed for a boundary homogenization problem exhibiting a trichotomy in the asymptotic behaviour.

Keywords

Steklov boundary conditions / multi-parameter eigenvalue problems / biharmonic Steklov eigenvalues / domain perturbations / spectral stability

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