Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions
The authors investigate the asymptotic behavior of a solution \(\;u_\varepsilon\in K_\varepsilon\) of a variational inequality \[ \int_\Omega \nabla u_\varepsilon\cdot \nabla(v-u_\varepsilon)dx\geq \int_\Omega f(v-u_\varepsilon)dx\;\forall v\in K_\varepsilon, \] where \newline \(K_\varepsilon=\{g\in H_1(\Omega,\Gamma_2):g\geq 0\;\text{ on}\;G_\varepsilon\}\;\) with \(\varepsilon\)-periodically located subsets \(G_\varepsilon\) belonging to the boundary \(\partial\Omega\) of the domain \(\Omega \subset \{x\in \mathbb{R}^3:x_1>0\}.\;\) They construct a limit (homogenized) problem and prove the strong in \(H_1(\Omega)\) convergence of the solutions of the original variational inequality to the solution of the limit nonlinear boundary-value problem as \(\varepsilon\rightarrow 0\) in the so-called critical state.
domain with periodically bounded subsets
scientific article; zbMATH DE number 5319862