On Vibrating Membranes With Very Heavy Thin Inclusions

Golovaty, Yu. D.; Gómez, D.; Лобо, М.; Pérez, Eugenia

doi:10.1142/s0218202504003520

Cited by 23 publications

(6 citation statements)

References 14 publications

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“…First, in Section 6, the main feature is that the damping is concentrated and, therefore, the limit problem is a wave equation with boundary feedback boundary condition; in particular, the damping acts only on the boundary in the limit. In such a situation the type of approximating problems we consider, appear naturally in control theory/stabilization of waves, see [15,44,59,47,48], or in homogenization of vibration problems with inclusions near the boundary, see [51,28,29,17] and references therein. On the other hand, the limit problem appears in the boundary control theory, see [40,41,39,67,50,19] and references therein.…”

Section: Pde Problems With Concentrating Terms Near the Boundary 2149mentioning

confidence: 99%

PDE problems with concentrating terms near the boundary

Jiménez-Casas¹,

Rodriguez-Bernal²

2020

Communications on Pure &Amp; Applied Analysis

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In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions. 1. Introduction. In this paper we consider several types of PDE problems in which certain terms in the equations concentrate, as a parameter ε → 0. Typically, near the boundary of the domain. This implies that the problem under consideration are subjected to singular perturbations that drastically change the nature of the problem, when passing to the limit. In all the problems considered the goal is then to identify the form of the limit problem and to describe the process of convergence of solutions, as ε → 0. To make the notations apparent, we will consider Ω, an open bounded smooth set in IR N with a C 2 boundary ∂Ω. Let Γ = ∂Ω and consider the subset of Ω ω ε = {x ∈ Ω : dist(x, Γ) < ε} for sufficiently small ε, say 0 < ε < ε 0. Notice that this set has measure |ω ε | ∼ ε|Γ| for small values of ε. For sufficiently small σ ≥ 0 we can define the "parallel" interior boundary Γ σ = {y − σ n(y), y ∈ Γ} where n(x) denotes the outward normal vector. Note that Γ 0 = Γ and that for small ε, the set ω ε is a neighborhood of Γ in Ω, that collapses to the boundary when the parameter ε goes to zero.

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Section: Pde Problems With Concentrating Terms Near the Boundary 2149mentioning

confidence: 99%

PDE problems with concentrating terms near the boundary

Jiménez-Casas¹,

Rodriguez-Bernal²

2020

Communications on Pure &Amp; Applied Analysis

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“…be the eigenvalues of (8) with the usual convention of repeated eigenvalues. We assume that the corresponding eigenfunctions {V k } ∞ k=0 are subject to the orthogonality condition…”

Section: Asymptotics For the Low Frequencies The Weak Formulation Of ...mentioning

confidence: 99%

“…For each fixed k ∈ N, the sequence λ ε k /ε m converges towards the eigenvalue µ k of (8) as ε → 0. Moreover, for any eigenvalue µ k of (8)…”

Section: Asymptotics For the Low Frequencies The Weak Formulation Of ...mentioning

confidence: 99%

“…It should be mentioned that asymptotics for low, middle and high frequencies have been considered in [8] for a problem different from (1); also, the results obtained are very different. In [8] a Dirichlet problem is considered in Ω ε ≡ Ω (that is, ω ε ⊂ Ω and ∂ω ε ∩ ∂Ω = ∅), the width of ω ε is constant (that is, ω ε ≡ εω) and the band ω ε is only heavy (that is, t = 0). As a consequence, for the value of m = 3 considered in [8], different limiting problems arise and the localization phenomena has not been considered.…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

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

Gómez¹,

Назаров²,

Pérez³

2011

Networks &Amp; Heterogeneous Media

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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.

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“…Finally, similar eigenvalue problems to the ones associated to (1) appear in homogenization of vibration problems with inclusions near the boundary, see [25,12,13,10] and references therein.…”

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confidence: 98%