Two‐parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach

We consider a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ, and the level of anisotropy of the cell is determined by a diagonal matrix γ with positive...

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Bibliographic Details
Published in:Mathematische Nachrichten 2018-06, Vol.291 (8-9), p.1310-1341
Main Authors: Lanza de Cristoforis, Massimo, Musolino, Paolo
Format: Article
Language:English
Subjects:
31B10
35J25
45A05
47H30
anisotropic homogenization
Asymptotic series
Dirichlet problem
Homogenization
integral equations
Parameters
Perforation
Periodic structures
periodically perforated domain
Poisson equation
real analytic continuation in Banach space
singularly perturbed domain
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Summary:We consider a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ, and the level of anisotropy of the cell is determined by a diagonal matrix γ with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter ε. For a given value γ∼ of γ, we analyze the behavior of the unique solution of the problem as (ε,δ,γ) tends to (0,0,γ∼) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201600414