TY - JOUR

T1 - Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands

AU - Gómez, Delfina

AU - Nazarov, Sergei A.

AU - Pérez-Martínez, Maria Eugenia

N1 - Funding Information: This work has partially been supported by the Spanish MICINN grant PGC2018-098178-B-I00 , the Russian Foundation for Basic Research 18-01-00325 and the Convenium Banco Santander - Universidad de Cantabria 2018. Publisher Copyright: © 2020 Elsevier Inc. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Omega(epsilon) in the plane R-2. Here Omega(epsilon) is Omega boolean OR omega(epsilon) boolean OR Gamma, where Omega is a fixed bounded domain with boundary Gamma, omega(epsilon) is a curvilinear band of width O(epsilon), and Gamma = (Omega) over bar boolean AND (omega) over bar (epsilon). The density and stiffness constants are of order epsilon(-m-t) and epsilon(-t) respectively in this band, while they are of order 1 in Omega; t >= 1, m > 2, and s is a small positive parameter. We address the asymptotic behavior, as epsilon -> 0, for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Gamma. (C) 2020 Elsevier Inc. All rights reserved.

AB - We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Omega(epsilon) in the plane R-2. Here Omega(epsilon) is Omega boolean OR omega(epsilon) boolean OR Gamma, where Omega is a fixed bounded domain with boundary Gamma, omega(epsilon) is a curvilinear band of width O(epsilon), and Gamma = (Omega) over bar boolean AND (omega) over bar (epsilon). The density and stiffness constants are of order epsilon(-m-t) and epsilon(-t) respectively in this band, while they are of order 1 in Omega; t >= 1, m > 2, and s is a small positive parameter. We address the asymptotic behavior, as epsilon -> 0, for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Gamma. (C) 2020 Elsevier Inc. All rights reserved.

KW - Asymptotic analysis

KW - Localized eigenfunctions

KW - Spectral analysis

KW - Stiff problem

KW - LAPLACIAN

KW - EIGENVALUES

KW - ASYMPTOTIC EXPANSIONS

KW - SYSTEMS

KW - EIGENFUNCTIONS

KW - SPECTRUM

UR - http://www.scopus.com/inward/record.url?scp=85091247536&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/c4403626-40a9-3ee1-8b88-0d984c40d72e/

U2 - 10.1016/j.jde.2020.09.011

DO - 10.1016/j.jde.2020.09.011

M3 - Article

AN - SCOPUS:85091247536

VL - 270

SP - 1160

EP - 1195

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -