T. P. Chechkina

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.

Nematodynamics and random homogenization

We study the homogenization problem for the system of equations of dynamics of a mixture of liquid crystals with random structure. We consider a simplified form of the Ericksen–Leslie equations for an incompressible medium with inhomogeneous density with random structure. Under the assumption that randomness is statistically homogeneous and ergodic, we construct the limit problem and prove almost sure convergence of solutions of the original problem to the solution of the limit (homogenized) problem.

Homogenization of 3D thick cascade junction with a random transmission zone periodic in one direction

In the paper, we deal with the homogenization problem for the Poisson equation in a singularly perturbed three-dimensional junction of a new type. This junction consists of a body and a large number of thin curvilinear cylinders, joining to body through a random transmission zone with rapidly oscillating boundary, periodic in one direction. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and with random perturbed coefficients on the boundary of the transmission zone. We prove the homogenization theorems and the convergence of the energy integrals.