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Asymptotic treatment of perforated domains without homogenization
Author(s) -
Maz'ya V.,
Movchan A.
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910045
Subject(s) - remainder , mathematics , homogenization (climate) , asymptotic expansion , algebraic number , boundary value problem , mathematical analysis , asymptotic formula , biodiversity , ecology , arithmetic , biology
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 ofthe array ofinclusions 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)