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Asymptotic treatment of perforated domains without homogenization
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2010 (English)In: MATHEMATISCHE NACHRICHTEN, ISSN 0025-584X, Vol. 283, no 1, 104-125 p.Article in journal (Refereed) Published
Abstract [en]

As a main result of the paper, we construct and justify an asymptotic approximation of Greens 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 Greens function in the same perforated domain. The important feature is the uniformity of the remainder estimate with respect to the independent variables.

Place, publisher, year, edition, pages
2010. Vol. 283, no 1, 104-125 p.
Keyword [en]
Singular perturbations, mesoscale approximations, multiply perforated domains, Greens function
National Category
URN: urn:nbn:se:liu:diva-54062DOI: 10.1002/mana.200910045ISI: 000274138500008OAI: diva2:298310
Available from: 2010-02-22 Created: 2010-02-22 Last updated: 2010-02-22

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Maz´ya, Vladimir
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Applied MathematicsThe Institute of TechnologyDepartment of Mathematics

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