Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions

Abstract

We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain Ωε of R 2 . Here Ωε is Ω ∪ ωε ∪ Γ, where Ω is a fixed bounded domain with boundary Γ, ωε is a curvilinear band of variable width O(ε), and Γ = Ω ∩ ωε. The density and stiffness constants are of order O(ε −m−1 ) and O(ε −1 ) respectively in this band, while they are of order O(1) in Ω; m is a positive parameter and ε ∈ (0, 1), ε → 0. Considering the range of the low, middle and high frequencies, we provide asymptotics for the eigenvalues and the corresponding eigenfunctions. For m > 2, we highlight the middle frequencies for which the corresponding eigenfunctions may be localized asymptotically in small neighborhoods of certain points of the boundary.

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

“…Finally, notice that the Steklov problem (1.18)-(1.20) appears here associated with the second order approach of the eigenvalues. Let us mention references [1,7,8,9,12,14] which make it clear how Steklov type boundary conditions can appear associated with the first order approach of eigenvalues of singularly perturbed spectral problems which present a high mass concentration along a part of the boundary or at points along the boundary. See [6] for further recent bibliography on Steklov problems.…”

Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure

“…Precisely, in [17] (see also [18]), as ε tends to zero, the authors obtain a limit spectral problem (see (2.11)) composed by two 1D differential equations whose solutions are coupled by a junction condition (see Theorem 2.1). This limit problem is posed on the skeleton of the T -like structure, namely for (x 1 , x 2 ) ∈ (ω × {0}) ({0}×]0, d[), while a Dirichlet condition is imposed on the extremes ∂ω and at x 2 = d. Nevertheless, as it happens in many singularly perturbed problems (see, for instance, [20], [21], [22], [31], [32], and [42]), there are sequences of eigenvalues {λ ε = λ ε,k(ε) } ε of order O(ε −γ ) with k(ε) → ∞ and for some γ > 0, the so-called high frequencies, whose corresponding eigenfunctions U ε = U ε,k(ε) , suitably normalized, do not vanish asymptotically. The goal of this paper is to localize those sequences of eigenvalues giving rise to other kinds of vibrations, such as the transverse vibrations of the T -like shaped structure, and provide information on the structure of the corresponding eigenfunctions.…”

“…Some of these references also address the phenomena of the localization of certain kinds of vibrations concentrating asymptotically its support in regions where the geometry of the problem has some kind of perturbation. Related to the localization of eigenfunctions giving rise to vibrations concentrated at points or along certain regions of the structure we mention [22], [25], [33], [36], [40], [42]. About the study of junctions as considered in this paper but in other contexts, we refer to [4], [9], [12], [14], [15], [16], [19], [26], [27], [28], [30] and [37].…”

The stiff Neumann problem: asymptotic specialty and "kissing" domains

“…In [20], estimates of convergence rates of the spectrum of stiff elasticity problems are obtained. We also mention the papers [12,14], where the authors deal with the asymptotics of spectral stiff problem in domains surrounded by a thin band depending on ε. For a study of asymptotics for vibrating systems containing a stiff region independent of the small parameter ε, we refer to Sections V.7-V.10 in [32] and the papers [13,21,31].…”

“…In Section 2 we introduce the weak formulation of the problem (1)-(4). We deduce the formal asymptotic expansions for the eigenelements in the most interesting case m ∈ (0, 1/2), whose leading terms are determined by the constant c0 (see (14)) in Ω0 and via Neumann spectral problem for Laplacian in Ω1. We also discuss briefly the infinite asymptotic series.…”