On the spectral instability for weak intermediate triharmonic problems

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.