We construct and justify an efficient model arising when homogenizing the Poisson equation in an ε -periodically perforated domain along an ( n − 1)-dimensional manifold by sets of an arbitrary shape and critical size on the boundary of which we impose a nonlinear dynamic condition containing an absorption coefficient of the form ε − k , where k takes the critical value ( n − 1)/( n − 2), n ≥ 3. We show that the transmission conditions on the manifold contain a nonlocal monotone operator.

A homogenized model is constructed (with rigorous justification) for a boundary-value problem for the Poisson equation in a periodically perforated domain with a nonlinear Robin condition on the boundary of the cavities. This condition contains a parameter depending on the period of the structure and a function σ(x, u) responsible for the nonlinearity. The cavities can have an arbitrary shape and the parameters of the problem have “critical values,” which results in a homogenized problem with a different type of nonlinearity.

A strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is considered. A problem with dynamic boundary conditions given on the union of some boundary subsets of critical size arranged ε-periodically along the boundary and with homogeneous Neumann conditions given on the rest of the boundary is studied. It is proved that the homogenized boundary condition is a Robin-type containing a nonlocal term depending on the trace of the solution u(x, t) on the boundary $$\partial \Omega $$ .

Изучена задача усреднения для уравнения диффузии в области, перфорированной вдоль ( n - 1)-мерного многообразия с динамическими краевыми условиями на границе перфораций. Построена усреднённая модель, являющаяся задачей сопряжения для уравнения диффузии, причём условия сопряжения содержат слагаемое с памятью. Доказана теорема о сходимости решений исходной задачи к решению усреднённой.

The problem of homogenization the diffusion equation in a domain perforated along an (n - 1)-dimensional manifold with dynamic boundary conditions on the boundary of the perforations is studied. A homogenization model is constructed that is a transmission problem for the diffusion equation with the transmission conditions containing a term with memory. A theorem on the convergence of solutions of the original problem to the solution of the homogenized one is proved.

The main goal of this paper is to characterize the change of structural behavior (i.e. the appearance of the so-called "strange terms') arising in the homogenization process when applied to distributed microscopic chemical reactions taking place on fixed-bed nanoreactors, at the microscopic level, on the boundary of the particles of critical size. The presence of non-homogeneous distributed functions b(epsilon)(j)(x) of the reaction kinetics may be originated by many different reasons. The case of quick oscillation is often due to own structure of the fixed bed reactor since the flux of the fluid acts on each particle in a non-homogeneous way. In some other cases, the non-homogeneous distributed functions b(epsilon)(j)(x) of the reaction kinetics is artificially provoked in order to control a certain desired global effect. Our main result gives a complete classification of the strange terms according the assumed periodicity on the distributed functions b(epsilon)(j)(x) of the reaction kinetic.

We construct and justify a homogenized model of the variational inequality with the Laplace operator and a nonlinear boundary constraint on the flow on arbitrary shaped cavities generating perforation of the domain with critical values of parameters.

The asymptotic behavior of solutions to a boundary value problem in a domain periodically perforated by small holes with a rapidly oscillating nonhomogeneous Robin-type condition on their boundaries is investigated in the case of critical parameter values.

The main objective of this article is to get a complete characterization of the homogenized global absorption term, and to give a rigorous proof of the convergence, in a class of diffusion processes with a reaction on the boundary of periodically "microscopic" distributed particles (or holes) given through a nonlinear microscopic reaction (i.e. under nonlinear Robin microscopic boundary conditions). We introduce new techniques to deal with the case of non necessarily symmetric particles (or holes) of critical size which leads to important changes in the qualitative global homogenized reaction (such as it happens in many problems of the Nanotechnology). Here we shall merely assume that the particles (or holes) G(epsilon)(j) in the n-dimensional space, are diffeomorphic to a ball (of diameter a(epsilon) = C-0 epsilon(gamma), gamma = n/n-2 for some C-0 > 0). To define the corresponding "new strange term" we introduce a one-parametric family of auxiliary external problems associated to canonical cellular problem associated to the prescribed asymmetric geometry G(0) and the nonlinear microscopic boundary reaction sigma(s) (which is assumed to be merely a Holder continuous function). We construct the limit homogenized problem and prove that it is a well-posed global problem, showing also the rigorous convergence of solutions, as epsilon -> 0, in suitable functional spaces. This improves many previous papers in the literature dealing with symmetric particles of critical size.

The main objective of this article is to get a complete characterization of the homogenized global absorption term, and to give a rigorous proof of the convergence, in a class of diffusion processes with a reaction on the boundary of periodically “microscopic” distributed particles (or holes) given through a nonlinear microscopic reaction (i.e. under nonlinear Robin microscopic boundary conditions). We introduce new techniques to deal with the case of non necessarily symmetric particles (or holes) of critical size which leads to important changes in the qualitative global homogenized reaction (such as it happens in many problems of the Nanotechnology). Here we shall merely assume that the particles (or holes) Gε, in the n-dimensional space, are diffeomorphic to a ball (of diameter aε = C0ε , γ = n n−2 for some C0 > 0). To define the corresponding “new strange term” we introduce a one-parametric family of auxiliary external problems associated to canonical cellular problem associated to the prescribed asymmetric geometry G0 and the nonlinear microscopic boundary reaction σ(s) (which is assumed to be merely a Hölder continuous function). We construct the limit homogenized problem and prove that it is a well-posed global problem, showing also the rigorous convergence of solutions, as ε → 0, in suitable functional spaces. This improves many previous papers in the literature dealing with symmetric particles of critical size.

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. Gómez, M. Lobo, E. Pérez, 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.

Let uϵ be the solution of the Poisson equation in a domain perforated by thin tubes with a nonlinear Robin‐type boundary condition on the boundary of the tubes (the flux here being β(ϵ)σ(x,uϵ)), and with a Dirichlet condition on the rest of the boundary of Ω. ϵ is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is O(aϵ) with aϵ≪ϵ. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number a. Here, is a strictly monotonic function of the second argument, and the adsorption parameter β(ϵ) > 0 can converge toward infinity. Depending on the relations between the three parameters ϵ, aϵ, and β(ϵ), the effective equations in volume are obtained. Among the multiple possible relations, we provide critical relations, which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as ϵ→0 for the associated spectral problems in the case of σ a linear function of uϵ. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the asymptotic behavior of solutions u ε of the Poisson equation in the ε-periodically perforated domain Ωε ⊂ \( {{\mathbb{R}}^n} \) , n ≥ 3, with the third nonlinear boundary condition of the form ∂ ν u ε + ε−γσ(x, u ε) = ε −γ g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order εα with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u ε , corresponding to different relations between parameters α and γ, are studied.

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.