Homogenization of a semilinear variational inequality in a thick multi-level junction

Abstract We consider a semilinear variation inequality in a thick multi-level junction Ω ε $\Omega_{\varepsilon}$ , which is the union of a domain Ω 0 $\Omega_{0}$ (the junction’s body) and a large number of thin cylinders. The thin cylinders are divided into m classes depending on the geometrical characteristics and the semilinear perturbed boundary conditions of the Signorini type given on their lateral surfaces. In addition, the thin cylinders from each class are ε-periodically alternated along some manifold on the boundary of the junction’s body. The purpose is to study the asymptotic behavior of the solution u ε $u_{\varepsilon}$ of this variation inequality as ε → 0 $\varepsilon\to0$ , i.e. when the number of the thin cylinders from each class infinitely increases and their thickness tends to zero. The passage to the limit is accompanied by special intensity factors { ε α k } k = 1 m $\{ \varepsilon^{\alpha_{k}}\}_{k=1}^{m}$ in the boundary conditions. We establish two qualitatively different cases in the asymptotic behavior of the solution depending on the value of parameters { α k } k = 1 m $\{{\alpha_{k}}\}_{k=1}^{m}$ . For each case we prove a convergence theorem. As a consequence, we see that u ε $u_{\varepsilon}$ converges (as ε → 0 $\varepsilon\to0$ ) to the solution of the corresponding nonstandard homogenized problem and show that the semilinear boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders from each class