# 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

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topDamlamian, 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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