Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

Alain Damlamian; Patrizia Donato

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 555-585
  • ISSN: 1292-8119

Abstract

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In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H0-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.

How to cite

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Damlamian, Alain, and Donato, Patrizia. "Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 555-585. <http://eudml.org/doc/90660>.

@article{Damlamian2010,
abstract = { In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H0-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified. },
author = {Damlamian, Alain, Donato, Patrizia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Periodic homogenization; perforated domains; H0-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains.; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality},
language = {eng},
month = {3},
pages = {555-585},
publisher = {EDP Sciences},
title = {Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?},
url = {http://eudml.org/doc/90660},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Damlamian, Alain
AU - Donato, Patrizia
TI - Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 555
EP - 585
AB - In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H0-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.
LA - eng
KW - Periodic homogenization; perforated domains; H0-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains.; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality
UR - http://eudml.org/doc/90660
ER -

References

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