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

We consider a spectral problem for the Laplacian operator in a planar T-like shaped thin structure , where E; denotes the transversal thickness of both branches. We assume the homogeneous Dirichlet boundary condition on the ends of the branches and the homogeneous Neumann boundary condition on the remaining part of the boundary of . We study the asymptotic behavior, as ; tends to zero, of the high frequencies of such a problem. Unlike the asymptotic behavior of the low frequencies where the limit problem involves only longitudinal vibrations along each branch of the T-like shaped thin structure (i.e. 1D limit spectral problems), we obtain a two dimensional limit spectral problem which allows us to capture other kinds of vibrations. We also give a characterization of the asymptotic form of the eigenfunctions originating these vibrations.This work has partially been supported by MINECO grant MTM2013-44883-P and MICINN grant PGC2018-098178-B-I00. The first author is also member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Estudio asintótico de las vibraciones de un cuerpo con una masa concentrada en una superficie

El problema que aquí consideramos es un modelo matemático sobre las vibraciones de un cuerpo que contienen en su interior una región de pequeño espesor O(ε) limitada por dos planos paralelos y en donde la densidad es de orden O(ε −m) con m > 1. Fuera de dicha región, denominada masa concentrada sobre una superficie, la densidad es de orden O(1). En [4] D. Gómez, M. Lobo, and E. Pérez. Sobre vibraciones de baja frecuencia de un cuerpo con una masa concentrada sobre una superficie, Actas del XIX CEDYA, Universidad Carlos III, 2006 describimos el comportamiento asintótico, cuando ε tiende a cero, de los valores propios de orden O(ε m−1) del problema espectral asociado, bajas frecuencias. Aquí, caracterizamos comportamientos límites de frecuencias propias de otros órdenes de magnitud más grandes, las denominadas altas o medias frecuencias

Spectral gaps in a double-periodic perforated Neumann waveguide

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter ε>0. The periodicity cell itself contains a string of holes at a distance O(ε) between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as ε→0. We demonstrate that, for ε small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a 'fast Floquet variable' and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.The work has been partially supported by MICINN through PGC2018-098178-B-I00, PID2020-114703GB-I00 and Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S)

On correctors for spectral problems in the homogenization of Robin boundary conditions with very large parameters

We obtain estimates for convergence rates of the eigenelements (λ", u") for the Laplace operator in a domain ⊂ R3 periodically perforated along a plane γ = ∩ {x1 = 0}. The boundary conditions are of the Dirichlet type on ∂ and of the Robin type, involving a large parameter O(ε− ), on the boundary of the cavities. The small parameter ε denotes the period while the size of each cavity is O(ε ). Here we consider the most significant case where α = κ = 2

Unilateral problems for the p-Laplace operator in perforated media involving large parameters

We address homogenization problems for variational inequalities issue from unilateral con-straints for the p-Laplacian posed in perforated domains of Rn, with n 3 and p 2 [2; n]. " is a small parameter which measures the periodicity of the structure while a" " measures the size of the perforations. We impose constraints for solutions and their uxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the ux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter " which may be very large, namely, " ! 1 as " ! 0. We rst consider the case where p < n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the di erent parameters of the problem: p, n, ", a" and ". Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for p = n and the case of non periodically distributed isoperimetric perforations. We make it clear that the averaged constants of the problem, the perimeter of the perforations appears for any shape.This work has been partially supported by MINECO:MTM2013-44883-P

Boundary homogenization with large reaction terms on a strainer-type wall

ABSTRACT: We consider a homogenization problem for the Laplace operator posed in a bounded domain of the upper halfspace, a part of its boundary being in contact with the plane {x3 = 0}. On this part, the boundary conditions alternate from Neumann to nonlinear-Robin, being of Dirichlet type outside. The nonlinear-Robin boundary conditions are imposed on small regions periodically placed along the plane and contain a Robin parameter that can be very large. We provide all the possible homogenized problems, depending on the relations between the three parameters: period ε, size of the small regions rε and Robin parameter β(ε). In particular, we address the convergence, as ε tends to zero, of the solutions for the critical size of the small regions rε = O(ε2). For certain β(ε), a nonlinear capacity term arises in the strange term which depends on the macroscopic variable and allows us to extend the usual capacity definition to semilinear boundary conditions.Supported by Grant PGC2018-098178-BBI00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe

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

We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain [Omega] in the plane R2. Here [ ], where [Omega] is a fixed bounded domain with boundary , is a curvilinear band of width O(epsilon), and . The density and stiffness constants are of order mt and t respectively in this band, while they are of order 1 in ; t1, m>2, and is a small positive parameter. We address the asymptotic behavior, as 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 o

Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

We consider a spectral homogenization problem for the linear elasticity system posed in a domain of the upper half-space R3+, a part of its boundary being in contact with the plane {x3=0}. We assume that the surface is traction-free out of small regions T, where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function M(x) and a reaction parameter () that can be very large when 0. The size of the regions T is O(r), where r, and they are placed at a distance between them. We provide all the possible spectral homogenized problems depending on the relations between , r and (), while we address the convergence, as 0, of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on . New capacity matrices are introduced to define these strange terms.This work has been partially supported by Russian Foundation on Basic Research grant 18-01-00325, Spanish MICINN grant PGC2018-098178-B-I00 and the Convenium Banco Santander - Universidad de Cantabria 2018

Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients

To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max?min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes