Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities

Abstract

A boundary-value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain with periodic perforation by small cavities arranged along a fixed hypersurface at small distances one from another. The distances are proportional to a small parameter $\varepsilon$, and the linear sizes of the cavities are proportional to $\varepsilon\eta(\varepsilon)$, where $\eta(\varepsilon)$ is a function taking values in the interval $[0,1]$. The main result is a complete asymptotic expansion for the solution of the perturbed problem. The asymptotic expansion is a combination of an outer and an inner expansion; it is constructed using the method of matched asymptotic expansions. Both outer and inner expansions are power expansions in $\varepsilon$ with coefficients depending on $\eta$. These coefficients are shown to be infinitely differentiable with respect to $\eta\in(0,1]$ and uniformly bounded in $\eta\in[0,1]$. Bibliography: 38 titles.