arXiv subjects

Delfina Gómez

Publications and source records attributed to Delfina Gómez.

5 recordsLinked to original sources

arXiv2025

Asymptotics for the spectrum of the Laplacian in thin bars with varying cross sections

We consider spectral problems for Laplace operator in 3D rod structures with a small cross section of diameter $O(\varepsilon)$, $\varepsilon$ being a positive parameter. The boundary conditions are Dirichlet (Neumann, respectively) on the bases of this structure and Neumann on the lateral boundary. As $\varepsilon\to 0$, we show the convergence of the spectrum with conservation of the multiplicity towards that of a 1D spectral model with Dirichlet (Neumann, respectively) boundary conditions. This 1D model may arise in diffusion or vibrations models of nonhomogeneous media with different physical characteristics and it takes into account the geometry of the 3D domain. We deal with the low frequencies and the approach to eigenfunctions in the suitable Sobolev spaces is also outlined.

math.AP↗

arXiv2025

Longitudinal oscillations for eigenfunctions in rod like structures

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the lateral boundary. We focus on the low frequencies. We study the asymptotic behavior of the eigenvalues and associated eigenfunctions, which are approached as $\varepsilon\to 0$ by those of a 1D model with Dirichlet boundary conditions, but which takes into account the geometry of the domain. Explicit and numerical computations enlighten the interest of this study, when the parameter becomes smaller. At the same time they show that in order to capture oscillations in the transverse direction we need to deal with the high frequencies. For prism like domains, we show the different asymptotic behavior of the spectrum depending on the boundary conditions.

math.AP↗

arXiv2025

On eigenvibrations of branched structures with heterogeneous mass density

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $Ω$ which is composed of smooth surfaces joined along a line $γ$, the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for normal derivatives, and Neumann conditions on the rest of the boundary of the surfaces. Assuming that the density is $O(\varepsilon^{-m})$ along small bands of width $O(\varepsilon)$, which collapse into the line $γ$ as $\varepsilon$ tends to zero, and it is $O(1)$ outside these bands, we address the asymptotic behavior, as $\varepsilon\to 0$, of the eigenvalues and of the corresponding eigenfunctions for a parameter $m\geq 1$. We also study the asymptotics for high frequencies when $m\in(1,2)$.

math.SP↗

arXiv2023

Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height $H$. The perforations are periodically placed along the ordinate axis at a distance $O(ε)$ between them, where $ε$ is a parameter that converges towards zero. Another parameter $η$, the Floquet-parameter, ranges in the interval $[-π, π]$. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters $ε$ and $η$ and strongly depend on $H$. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.

math.AP↗

arXiv2021

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 $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ 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 $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ 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.

math.AP↗