Homogenization of the parabolic Signorini boundary-value problem in a thick junction of type 3:2:1

Abstract

We consider a parabolic Signorini boundary-value problem in a thick junction $\Omega_{\varepsilon}$ which is the union of a domain $\Omega_0$ and a large number of $\varepsilon$-periodically situated thin cylinders. The Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as $\varepsilon\to0,$ i.e., when the number of the thin cylinders infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as $\varepsilon\to0)$ in differential inequalities in the region that is filled up by the thin cylinders.