Asymptotic treatment of perforated domains without homogenization

TL;DR

This paper develops an asymptotic approximation method for Green's functions in domains with many small, non-periodic inclusions, providing uniform estimates and solving related boundary value problems without homogenization.

Contribution

It introduces a meso scale asymptotic approximation technique for perforated domains that does not rely on periodicity, including uniform Green's function estimates.

Findings

Constructed asymptotic approximation of Green's function

Proved solvability of the algebraic system for coefficients

Derived uniform and energy estimates for the remainder

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 meso scale 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.