D. Gómez

On the eigenfunctions associated with the high frequencies in systems with a concentrated mass

Abstract We consider the vibrations of a system consisting of the domain Ω of R N , N=2,3 , that contains a small region with diameter depending on a small parameter e . The density is of order O (e −m ) in the small region, the concentrated mass, and it is O (1) outside; m is a parameter, m≥2 . We study the asymptotic behaviour, as e→0 , of the eigenvalues of order O (1) , the high frequencies when m>2 , and the corresponding eigenfunctions of the associated spectral problem. We provide information on the structure of these eigenfunctions. We also check theoretical results with explicit calculations for the dimensions N=1 and N=2 and give correcting terms for the eigenfunctions.

On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems

Abstract. Let ue be the solution of the Poisson equation in a domain periodically perforated along a manifold γ = Ω ∩ {x1 = 0}, with a nonlinear Robin type boundary condition on the perforations (the flux here being O(e−κ)σ(x,ue)), and with a Dirichlet condition on ∂Ω. Ω is a domain of R with n 3, the small parameter e, that we shall make to go to zero, denotes the period, and the size of each cavity is O(e) with α 1. The function σ involving the nonlinear process is a C1(Ω × R) function and the parameter κ ∈ R. Depending on the values of α and κ, the effective equations on γ are obtained; we provide a critical relation between both parameters which implies a different average of the process on γ ranging from linear to nonlinear. For each fixed κ a critical size of the cavities which depends on n is found. As e → 0, we show the convergence of ue in the weak topology of H1 and construct correctors which provide estimates for convergence rates of solutions. All this allows us to derive convergence for the eigenelements of the associated spectral problems in the case of σ a linear function.

ON VIBRATING MEMBRANES WITH VERY HEAVY THIN INCLUSIONS

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain Ω of ℝ2. The inclusion occupies a layer-like domain ωe of width 2e and it has a density of order O(e-m). The density is of order O(1) outside this inclusion ωe, the concentrated mass around the curve γ. e and m are positive parameters, e∈(0,1) and m>2. We set m=3 and show that low, middle and high frequency vibrations are necessary in order to describe the asymptotic behavior of the vibrations of the whole membrane. We study the asymptotic behavior, as e→0, of these frequencies and of the corresponding eigenfunctions.

Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

In this article, we consider variational inequalities arising, e.g., in modelling diffusion of substances in porous media. We assume that the media fills a domain Ωϵ of ℝ n with n ≥ 3. We study the case where the number of cavities is large and they are periodically distributed along a (n − 1)-dimensional manifold. ϵ is the period while ϵα is the size of each cavity with α ≥ 1; ϵ is a parameter that converges towards zero. Moreover, we also assume that the nonlinear process involves a large parameter ϵ−κ with κ = (α − 1)(n − 1). Passing to the scale limit, and depending on the value of α, the effective equation or variational inequality is obtained. In particular, we find a critical size of the cavities when α = κ = (n − 1)/(n − 2). We also construct correctors which improve convergence for α ≥ (n − 1)/(n − 2).

Boundary homogenization in perforated domains for adsorption problems with an advection term

We consider a model for the spreading of a substance through an incompressible fluid in a perforated domain , with . The fluid flows in a domain containing a periodical set of perforations () placed along an inner surface . The size of the perforations is much smaller than the size of the characteristic period . An adsorption phenomena can occur on the boundaries of the perforations, where we assume a strongly nonlinear adsorption law with a large adsorption parameter. An advection term appears in the partial differential equation. We obtain the homogenized model which also involves a nonlinear transmission condition for the normal derivative on . The ‘strange term’ arising in this transmission condition is a nonlinear function implicitly defined by a functional equation. We deal with critical relations both for the size of perforations and the adsorption parameter while we use the energy method for variational inequalities to show the convergence.

Homogenization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space

We address the homogenization of a variational inequality posed in perforated media issue from a unilateral problem for the p-Laplacian. We consider the n-Laplace operator in a perforated domain of ℝn, n ≥ 3, with restrictions for the solution and its flux (the flux associated with the n-Laplacian) on the boundary of the perforations which are assumed to be isoperimetric. The solution is assumed to be positive on the boundary of the holes and the flux is bounded from above by a negative, nonlinear monotone function multiplied by a large parameter. A certain non periodical distribution of the perforations is allowed while the assumption that their size is much smaller than the periodicity scale is performed. We make it clear that in the average constants of the problem, the perimeter of the perforations appears for any shape.

Spectral Boundary Homogenization Problems in Perforated Domains with Robin Boundary Conditions and Large Parameters

In this chapter we address asymptotics for spectral problems posed in periodically perforated domains along a plane. The operator under consideration is the Laplacian, and the spectral problem is posed in a three-dimensional domain Ω, outside the cavities. The boundary conditions are of the Dirichlet type on the boundary of Ω and of the Robin type on the boundary of the cavities. The periodicity of the structure is?; it is a small parameter that converges towards zero. The size of the cavities can be of the same order of magnitude as?, namely O(?), or much smaller than?, namely o(?). Also a large?-dependent parameter (adsorption constant) arises in the Robin conditions. Depending on the different values/relations between the three parameters (periodicity, size of cavities and adsorption constant) different homogenized problems are obtained: both critical sizes for cavities and critical relations for parameters are provided. The results complement earlier ones, where the convergence for the spectrum is outlined when dealing with linear problems. Here, we obtain estimates for convergence rates of the eigenvalues and eigenfunctions in terms of the eigenvalue number and the parameter?.

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

We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain

On a vibrating plate with a concentrated mass

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Homogenization in perforated domains: a Stokes grill and an adsorption process

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