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

摘要

We consider a semilinear variation inequality in a thick multi-level junction Ω ε $Omega_{arepsilon}$ , 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 $u_{arepsilon}$ of this variation inequality as ε → 0 $arepsilono0$ , 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 ${ arepsilon^{lpha_{k}}}_{k=1}^{m}$ 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 ${{lpha_{k}}}_{k=1}^{m}$ . For each case we prove a convergence theorem. As a consequence, we see that u ε $u_{arepsilon}$ converges (as