We analyse the spectral convergence of high order elliptic di erential operators subject to singular domain perturbations and homogeneous boundary conditions of intermediate type. We identify sharp assumptions on the domain perturbations improving, in the case of polyharmonic operators of higher order, conditions known to be sharp in the case of fourth order operators. The optimality is proved by analysing in detail a boundary homogenization problem, which provides a smooth version of a polyharmonic Babuška paradox.Recall that this is the weak formulation of the Poisson problem for (−∆) m +I with (SIBC).For u ∈ W m,2 (Ω), de ne T ϵ u = u • Φ ϵ where Φ ϵ is a smooth di eomorphism mapping Ω ϵ into Ω that coincides with the identity on a large part K ϵ of Ω, with |Ω \ K ϵ | → 0 as ϵ → 0, see (3.5). Let u be such that u ϵ − T ϵ u L 2 (Ω ϵ ) → 0 as ϵ → 0. Theorem 7 states that the limit u solves di erent di erential problems according to the values of the parameter α. More precisely, we have the following trichotomy: (i) (Stability) If α > 3/2, then u solves (1.8) in Ω, that is, u satis es (−∆) m u + u = f in Ω and (SIBC) on ∂Ω; (ii) (Degeneration) If α < 3/2, then u satis es (−∆) m u + u = f in Ω, with Dirichlet boundary conditions on W × {0}, that is ∂ l u ∂n l = 0, for all 0 ≤ l ≤ m − 1, and (SIBC) on the rest of the boundary of Ω; (iii) (Strange term) If α = 3/2, then u satis es (−∆) m u + u = f in Ω with the following boundary conditions on W × {0} D l u = 0, for all 0 ≤ l ≤ m − 2, ∂ m u ∂n m + K ∂ m−1 u ∂n m−1 = 0,

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coe cient of the represented plate.

We consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N‐dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems that depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case that provides a typical homogenization problem leading to a strange boundary term in the limit problem. Copyright © 2016 John Wiley & Sons, Ltd.

We show that the spectrum of the Dirichlet problem for the Laplace operator − in the plane R 2 perforated by a double-periodic family of holes contains any a priori number of gaps, for sufficiently large holes. While this result was already known in the case of circular holes, we consider here a more general geometric setting with holes of the shape {|x 1 | μ + |x 2 | μ ≤ r }, 1 < μ < ∞.