Boundary homogenization for a triharmonic intermediate problem

Abstract

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.

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

“…The analysis of eigenvalue problems for differential operators on thin domains has attracted noticeable interest in recent years, see e.g., [4,5,6,9,11,12,13,20,23,29,30,36,38,40] and references therein. A somehow complementary point of view is adopted in the asymptotic analysis of domains with small holes or perforations, see e.g., [1,16,19,33,39].…”

Spectral stability of the $curl curl$ operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

“…Thus, another method has to be used in the analysis of the stability problem for α = 3/2. For example, in the case of nonconstant periodic functions b one could use the unfolding method as in [11], adopted also in [2,3,20,21]: in those papers, for α = 3/2 we have spectral instability in the sense that the limiting problem differs from the given problem in Ω by a strange term appearing in the boundary conditions (as often happens in homogenization problems). We plan to discuss the details of this problem for the curlcurl operator in a forthcoming paper, but we can already mention that a preliminary formal analysis would indicate that no strange limit appears in the limiting problem for α = 3/2.…”

“…where E, H denote the spatial parts of the electric and the magnetic field respectively and ω > 0 is the angular frequency. Indeed, taking the curl in the first equation of (3) and setting λ = ω 2 , one immediately obtains problem (2). Note that here the medium filling Ω is homogeneous and isotropic and for simplicity the corresponding electric permittivity ε and magnetic permeability µ have been normalized by setting ε = µ = 1.…”

“…The aim of the present paper is to prove spectral stability results under less stringent assumptions on the families of domain perturbations. For this purpose, we adopt the approach of [3] further developed in [2,19,20,21].…”

“…In this model case, the question is whether one can get spectral stability for α < 2. Note that a profile of the form (5) is typical in the study of boundary homogenization problems and thin domains, see for example [2,3,4,9,11,20,21,22]. This problem was solved for the biharmonic operator with intermediate boundary conditions (modelling an elastic hinged plate) in [3] where condition (4) is relaxed by introducing a suitable notion of weighted convergence which allows to prove spectral stability for α > 3/2 in the model problem above.…”