Homogenization for the p-Laplacian in an n-dimensional domain perforated by very thin cavities with a nonlinear boundary condition on their Boundary in the case p = n

We investigate the asymptotic behavior, as ε → 0, of the solution uε to the boundary value problem for the equation −Δpuε = f in a domain Ωε ⊂ ℝn perforated by very thin arbitrarily shaped cavities separated by an O(ε) distance in the case of p = n ≥ 3 with a nonlinear third boundary condition of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{v_p } u_\varepsilon \equiv \left| {\nabla u_\varepsilon } \right|^{n - 2} \left( {\nabla u_\varepsilon ,v} \right) = - \beta ^{n - 1} \left( \varepsilon \right)\sigma \left( {x,u_\varepsilon } \right)$$\end{document} specified on their boundary, where ν is the outward unit normal vector on the boundary of the cavities. The adsorption coefficient β(ε) and the perforation radius aε satisfy conditions that are critical to the given problem. A homogenized model is constructed, and the solutions uε are proved to converge weakly, as ε → 0, to the solution of the homogenized problem.