M. E. Pérez

Averaging of boundary-value problem in domain perforated along (n − 1)-dimensional manifold with nonlinear third type boundary conditions on the boundary of cavities

The research considers the asymptotic behavior of solutions uɛ of the Poisson equation in a domain ɛ-periodically perforated along manifold γ = ω ∩ {x1 = 0} ≠ Ø with a nonlinear third type boundary condition ∂vuɛ + ɛ−ασ(x, uɛ) = 0 on the boundary of the cavities. It is supposed that the perforations are balls of radius C0ɛα, C0 > 0, α = n − 1 / n − 2, n ≥ 3, periodically distributed along the manifold γ with period ɛ > 0. It has been shown that as ɛ → 0 the microscopic solutions can be approximated by the solution of an effective problem which contains in a transmission conditions a new nonlinear term representing the macroscopic contribution of the processes on the boundary of the microscopic cavities. This effect was first noticed in [1] where the similar problem was investigated for n = 3 and for the case where Ω is a domain periodically perforated over the whole volume. This paper provides a new method for the proof of the convergence of the solutions {uɛ} to the solution of the effective problem is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and more accurate results are obtained with respect to the energy norm proved via a corrector term. Note that this approach can be generalized to achieve results for perforations of more complex geometry.

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?.

Correcting Terms for Perforated Media by Thin Tubes with Nonlinear Flux and Large Adsorption Parameters

We obtain corrector terms for homogenization problems in perforated media. The perforations are thin cylindrical tubes, periodically distributed over a fixed domain of the three dimensional space. The operator under consideration is the Laplacian and we impose nonlinear Robin type boundary conditions on the boundary of the cavities and Dirichlet condition on the rest of the boundary. The period of the structure is given by a small parameter that converges towards zero. The diameter of the transverse sections of the tubes is of an order of magnitude much smaller than the period. Also a very large parameter (compared with the period) arises in the Robin conditions: the adsorption constant. Depending on the different values/relations between the three parameters (periodicity, diameter and adsorption) different homogenized problems have been obtained in [D. Gomez, M. Lobo, E. Perez, T.A. Shaposhnikova, M.N. Zubova, On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems, Math. Meth. Appl. Sci., DOI:10.1002/mma.3246], where convergences for solutions hold in the weak topology of the corresponding Sobolev spaces. The results in this chapter improve these convergences providing estimates for convergence rates.

High-Frequency Vibrations of Systems with Concentrated Masses Along Planes

Let Ω be an open bounded domain of ℝ3 with a smooth boundary \(\partial\Omega\). Weassume that Ω is divided into two parts Ω+ and Ω- by the plane \(\gamma: \Omega = \Omega_+ \cup \Omega_- \cup \gamma\) .For simplicity, we assume that the plane { x 3 = 0} cuts Ω and \(\gamma = \Omega \cap \{x_3 = 0\}\). Let e be a small positive parameter that tends to zero. We denote by ωe the e-neighborhood of γ, i.e., \(\omega_\varepsilon = \Omega \cup \{|x_3| < \varepsilon\}\); for e sufficiently small, we assume that \(\omega_\varepsilon = \gamma \times (-\varepsilon, \varepsilon)\)) (see Figure 15.1). Note that this conditions the geometry of Ω near γ. Let us denote by \(\bar{x}\) the two first components of any x = (x 1, x 2, x 3) e ℝ3, that is,\(\bar{x} = (x_1, x_2)\)