Steklov vs. Steklov: A Fourth-Order Affair Related to the Babu\v{s}ka Paradox

TL;DR

This paper investigates two fourth-order Steklov eigenvalue problems, revealing a Babuška paradox in convex domain approximations and establishing eigenvalue continuity under domain perturbations.

Contribution

It extends eigenvalue continuity results to a second-order nonlocal problem and connects these findings to classical paradoxes and degeneration phenomena.

Findings

Eigenvalues depend continuously on convex domain perturbations.

The analysis links the Steklov problems to the Babuška paradox.

Results relate to classical plate problems and degeneration scenarios.

Abstract

We discuss two fourth-order Steklov problems and highlight a Babu\v{s}ka paradox appearing in their approximations on convex domains via sequences of convex polygons. To do so, we prove that the eigenvalues of one of the two problems depend with continuity upon domain perturbation in the class of convex domains, extending a result known in the literature for the first eigenvalue. This is obtained by examining in detail a nonlocal, second-order problem for harmonic functions introduced by Ferrero, Gazzola, and Weth. We further review how this result is connected to diverse variants of the classical Babu\v{s}ka paradox for the hinged plate and to a degeneration result by Maz'ya and Nazarov.