Homogenization of variational inequalities for the \(p\)-Laplace operator in perforated media along manifolds (Q2422356)

In the paper under review, the authors study the homogenization of variational inequalities for the \(p\)-Laplace equation in a domain \(\Omega_\epsilon\) of \(\mathbb{R}^n\) (\(n\geq 3\), \(p \in [2,n)\)) periodically perforated by balls of radius \(O(\epsilon^\alpha)\) where \(\alpha >1\), \(\epsilon\) is the size of the period, and the perforations \(G_\epsilon\) are distributed along a \((n-1)\) dimensional manifold \(\gamma\). In such a domain they consider the equation \[ -\Delta_p u_\epsilon=f, \] where the \(p\)-Laplace operator is defined as \(\Delta_p u \equiv \mathrm{div}\Big(|\nabla u|^{p-2}\nabla u\Big)\). Moreover, they pose a Dirichlet condition on the exterior boundary, whereas on \(\partial G_\epsilon\) they require the solution \(u_\epsilon\) to satisfy the following inequalities \[ u_\epsilon\geq 0, \quad \partial_{\nu_p} u_\epsilon \geq -\epsilon^{-\kappa} \sigma(x,u_\epsilon), \quad u_\epsilon(\partial_{\nu_p} u_\epsilon +\epsilon^{-\kappa} \sigma(x,u_\epsilon))=0,\] where \[ \partial_{\nu_p}u\equiv |\nabla u|^{p-2}(\nabla u,\nu), \] with \(\nu\) denoting the outward unit normal to \(\Omega_\epsilon\) on the boundary of the perforations. The authors investigate the asymptotic behavior of the solutions and describe the homogenized problems for different values of the parameters \(p\), \(n\), \(\epsilon\), \(\alpha\), and \(\kappa\). Furthermore, they characterize the relations giving rise to the appearance of the so-called \textit{strange term}.