Homogenization of the boundary value problem for the Laplace operator in a domain perforated along (n – 1)-dimensional manifold with nonlinear Robin type boundary condition on the boundary of arbitrary shaped holes: Critical case

Abstract

The asymptotic behavior of the solution to the boundary value problem for the Laplace operator in a domain perforated along an (n − 1)-dimensional manifold is studied. A nonlinear Robin-type condition is assumed to hold on the boundary of the holes. The basic difference of this work from previous ones concerning this subject is that the domain is perforated not by balls, but rather by sets of arbitrary shape (more precisely, by sets diffeomorphic to the ball). A homogenized model is constructed, and the solutions of the original problem are proved to converge to the solution of the homogenized one.