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

Abstract

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.

Keywords:

• homogenization
• thick junction
• rapidly oscillating boundary
• random boundary

AMS Subject Classifications:

• 35B27
• 76M50
• 35P05

Acknowledgements

This article was mostly written during the stay of Gregory A. Chechkin, Tatiana P. Chechkina and Taras A. Mel'nyk at the Università degli Studi di Salerno in July 2008. The authors want to express deep thanks for the hospitality, their wonderful working conditions and for the support. The work of the first and the second authors was partially supported by the program ‘Leading Scientific Schools’ (project HIII-1698.2008.1) and was also supported in part by RFBR (project 09-01-00353).