# Homogenization in perforated domains with rapidly pulsing perforations

Doina Cioranescu; Andrey L. Piatnitski

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

- Volume: 9, page 461-483
- ISSN: 1292-8119

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topCioranescu, Doina, and Piatnitski, Andrey L.. "Homogenization in perforated domains with rapidly pulsing perforations." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 461-483. <http://eudml.org/doc/90706>.

@article{Cioranescu2010,

abstract = {
The aim of this paper is to study a class of domains whose
geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains
with rapidly pulsing (in time) periodic
perforations, with a homogeneous Neumann condition on the boundary of the holes.
We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero.
Since standard conservation laws do not
hold in this model, a first difficulty is to get
a priori estimates of the solutions. We obtain them in a weighted space
where the
weight is the principal eigenfunction of an “adjoint” periodic
time-dependent eigenvalue problem. This problem is not a
classical one, and its investigation is an important
part of this work. Then, by using the multiple scale method,
we construct the
leading terms of a formal expansion (with respect to ε) of the solution and give the limit
“homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes,
a large convection term may appear in the limit equation.
},

author = {Cioranescu, Doina, Piatnitski, Andrey L.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Homogenization; perforated domains; pulsing perforations; multiple scale method.; multiple scale method; homogeneous Neumann condition; weighted space},

language = {eng},

month = {3},

pages = {461-483},

publisher = {EDP Sciences},

title = {Homogenization in perforated domains with rapidly pulsing perforations},

url = {http://eudml.org/doc/90706},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Cioranescu, Doina

AU - Piatnitski, Andrey L.

TI - Homogenization in perforated domains with rapidly pulsing perforations

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 461

EP - 483

AB -
The aim of this paper is to study a class of domains whose
geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains
with rapidly pulsing (in time) periodic
perforations, with a homogeneous Neumann condition on the boundary of the holes.
We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero.
Since standard conservation laws do not
hold in this model, a first difficulty is to get
a priori estimates of the solutions. We obtain them in a weighted space
where the
weight is the principal eigenfunction of an “adjoint” periodic
time-dependent eigenvalue problem. This problem is not a
classical one, and its investigation is an important
part of this work. Then, by using the multiple scale method,
we construct the
leading terms of a formal expansion (with respect to ε) of the solution and give the limit
“homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes,
a large convection term may appear in the limit equation.

LA - eng

KW - Homogenization; perforated domains; pulsing perforations; multiple scale method.; multiple scale method; homogeneous Neumann condition; weighted space

UR - http://eudml.org/doc/90706

ER -

## References

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- M.L. Kleptsyna and A.L. Piatnitski, Homogenization of random parabolic operators. Gakuto International Series. Math. Sci. Appl.9 (1997), Homogenization and Appl. to Material Sciences, 241-255. Zbl0892.35019
- M.L. Kleptsyna and A.L. Piatnitski, Averaging of non selfadjoint parabolic equations with random evolution (dynamics), Preprint INRIA. J. Funct. Anal. (submitted).
- M.A. Krasnosel'skii, E.A. Lifshits and E.A. Sobolev, Positive linear systems. The method of positive linear operators.Heldermann Verlag , Sigma Ser. Appl. Math.5 (1989)
- A.L. Piatnitsky, Parabolic equations with rapidly oscillating coefficients (In Russian). English transl. Moscow Univ. Math. Bull.3 (1980) 33-39.

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