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On the spectral instability for weak intermediate triharmonic problems

We define the weak intermediate boundary conditions for the triharmonic operator $- Δ^3$. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation $(Ω_ε)_{ε> 0}$ of a smooth domain $Ω$ of $\mathbb{R}^N$ for which the weak intermediate boundary conditions on $\partial Ω_ε$ are not preserved in the limit on $\partial Ω$, analogously to the Babuška paradox for the hinged plate. Four different boundary conditions can be produced in the limit, depending on the convergence of $\partial Ω_ε$ to $\partial Ω$. 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.