On the spectral instability for weak intermediate triharmonic problems

TL;DR

This paper investigates how weak intermediate boundary conditions for the triharmonic operator are affected by domain perturbations, revealing non-preservation and emergence of various boundary conditions, including a novel microscopic energy condition.

Contribution

It introduces a new analysis of boundary condition sensitivity for the triharmonic operator under domain perturbations, including the construction of specific perturbations that alter boundary conditions.

Findings

Weak intermediate boundary conditions are not preserved under domain perturbations.

Multiple limit boundary conditions can emerge depending on domain convergence.

A novel microscopic energy boundary condition is identified in certain limits.

Abstract

We define the weak intermediate boundary conditions for the triharmonic operator $- \Delta^3$. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation $(\Omega_\epsilon)_{\epsilon > 0}$ of a smooth domain $\Omega$ of $\mathbb{R}^N$ for which the weak intermediate boundary conditions on $\partial \Omega_\epsilon$ are not preserved in the limit on $\partial \Omega$, analogously to the Babu\v{s}ka paradox for the hinged plate. Four different boundary conditions can be produced in the limit, depending on the convergence of $\partial \Omega_\epsilon$ to $\partial \Omega$. In one particular case, we obtain a ``strange'' boundary condition featuring a microscopic energy term related to the shape of the approaching domains. Many aspects of our analysis could be generalised to an arbitrary order elliptic differential operator of order $2m$ and to more general domain perturbations.