On the spectral instability for weak intermediate triharmonic problems

Abstract

We define the weak intermediate boundary conditions for the triharmonic operator −Δ3. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation (Ωϵ)ϵ > 0 of a smooth domain Ω of for which the weak intermediate boundary conditions on ∂Ωϵ are not preserved in the limit on ∂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 ∂Ωϵ to ∂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.