Gregory A. Chechkin

HOMOGENIZATION OF BOUNDARY-VALUE PROBLEM IN A LOCALLY PERIODIC PERFORATED DOMAIN

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.

Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.

Singular Double Porosity Model

We consider the linear parabolic equation describing the transport of a contaminant in a porous media crossed by a net of infinitely thin fractures. The permeability is very high in the fractures but very low in the porous blocks. We derive the homogenized model corresponding to a net of infinitely thin fractures, by means of the singular measures technique. We assume that these singular measures are supported by hyperplanes of codimension one. We prove in a second step that this homogenized model could be obtained indistinctly either by letting the fracture thickness, in the standard double porosity model, tend to zero, or by homogenizing a model with infinitely thin fractures.

Averaging Operators With Boundary Conditions of Fine-Scaled Structure

We consider the averaging of boundary value problems with a small parameter in the boundary conditions. By using new notions, we prove the compactness theorem for a family of solutions. To verify the method proposed, we study a problem with rapidly varying boundary conditions in the case of a probabilistic description of the structure of the domain.

On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure

We construct the asymptotics of the sharp constant in the Friedrich-type inequality for functions, which vanish on the small part of the boundary. It is assumed that consists of pieces with diameter of order. In addition, and as.

Effective membrane permeability: estimates and low concentration asymptotics

The paper deals with a mathematical model of a steady-state diffusion process through a periodic membrane. For a wide class of periodic membranes, we define the effective permeability and obtain upper and lower estimates of the effective permeability. For periodic membranes made from two materials with different absorbing properties, we study the asymptotic behavior of the effective permeability when the fraction of one material tends to zero (low concentration asymptotics). When the low fraction material forms homothetically vanishing disperse periodic inclusions in the host material, low concentration approximations are built by the method of matched asymptotic expansions. We also show that our results are consistent with those which can be obtained by a boundary homogenization. Finally, we analyze formulas used in physical, chemical, and biological investigations to describe effective membrane properties.

Asymptotic analysis of a boundary-value problem in a cascade thick junction with a random transmission zone

In the article we deal with the homogenization of a boundary-value problem for the Poisson equation in a singularly perturbed two-dimensional junction of a new type. This junction consists of a body and a large number of thin rods, which join the body through the random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin rods and with random perturbed coefficients on the boundary of the transmission zone. We prove the homogenization theorems and the convergence of the energy integrals. It is shown that there are three qualitatively different cases in the asymptotic behaviour of the solutions.

Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses

The asymptotic behaviour (as ϵ → 0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is investigated. This cascade junction consists of the junctions body and great number 5N = 𝒪(ϵ−1) of ϵ-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order 𝒪(ϵ). The density of the junction is order 𝒪(ϵ−α) on the rods from the second class (the concentrated masses if α > 0), and 𝒪(1) outside of them. In addition, we study the influence of the concentrated masses on the asymptotic behaviour of these magnitudes in that case α = 1 and α ∈ (0, 1).

On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems

This paper is devoted to the asymptotic analysis of functions depending on a small parameter characterizing the microinhomogeneous structure of the domain on which the functions are defined. We derive the Friedrichs inequality for these functions and prove the convergence of solutions to corresponding problems posed in a domain perforated aperiodically along the boundary. Moreover, we use numerical simulation to illustrate the results.

On the asymptotic behavior of a simple eigenvalue of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem for the Laplace operator in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the exterior boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. We construct and justify the asymptotic expansions of eigenelements of the boundary value problem.