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

Bauyrzhan Derbissaly; Pier Domenico Lamberti; Bauyrzhan Derbissaly; Pier Domenico Lamberti

doi:10.3934/mine.2026009

Mathematics in Engineering

2026, Volume 8, Issue 2: 269-307. doi: 10.3934/mine.2026009

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Research article

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

Bauyrzhan Derbissaly ¹,
Pier Domenico Lamberti ^{2
,
,}

1.
Institute of Mathematics and Mathematical Modeling, 125 Pushkin, Almaty, 050010, Kazakhstan
2.
Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università degli Studi di Padova, 3 Stradella S. Nicola, Vicenza, 36100, Italy

Received: 27 December 2025 Revised: 17 March 2026 Accepted: 18 March 2026 Published: 30 March 2026

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.
- Steklov boundary conditions,
- multi-parameter eigenvalue problems,
- biharmonic Steklov eigenvalues,
- domain perturbations,
- spectral stability
Citation: Bauyrzhan Derbissaly, Pier Domenico Lamberti. Domain perturbation for biharmonic Steklov problems: spectral stability, boundary homogenization and degeneration[J]. Mathematics in Engineering, 2026, 8(2): 269-307. doi: 10.3934/mine.2026009

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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.

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