Homogenization of the Variational Inequality Corresponding to a Problem with Rapidly Varying Boundary Conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on $\varepsilon$-periodically located subsets $G_\varepsilon$ belonging to the boundary $\partial\Omega$ of the domain $\Omega\subset \mathbb R^3$. We construct a limit (homogenized) problem and prove the strong (in $H_1(\Omega)$) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as $\varepsilon\to0$ in the so-called critical case.