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 Babuška 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.

accepted for publication in 'Mathematical Methods in the Applied Sciences'

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