Asymptotic treatment of perforated domains without homogenization

Abstract

As a main result of the paper, we construct and justify an asymptotic approximation of Green's function in a domain with many small inclusions. Periodicity of the array of inclusions is not required. We start with an analysis of the Dirichlet problem for the Laplacian in such a domain to illustrate a method of mesoscale asymptotic approximations for solutions of boundary value problems in multiply perforated domains. The asymptotic formula obtained involves a linear combination of solutions to certain model problems whose coefficients satisfy a linear algebraic system. The solvability of this system is proved under weak geometrical assumptions, and both uniform and energy estimates for the remainder term are derived. In the second part of the paper, the method is applied to derive an asymptotic representation of the Green's function in the same perforated domain. The important feature is the uniformity of the remainder estimate with respect to the independent variables. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.