# Homogenization in perforated domains with rapidly pulsing perforations

Doina Cioranescu; Andrey L. Piatnitski

ESAIM: Control, Optimisation and Calculus of Variations (2003)

- 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 (2003): 461-483. <http://eudml.org/doc/244795>.

@article{Cioranescu2003,

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 $\varepsilon $ 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 $\varepsilon $) 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; homogeneous Neumann condition; weighted space},

language = {eng},

pages = {461-483},

publisher = {EDP-Sciences},

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

url = {http://eudml.org/doc/244795},

volume = {9},

year = {2003},

}

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

PY - 2003

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 $\varepsilon $ 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 $\varepsilon $) 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; homogeneous Neumann condition; weighted space

UR - http://eudml.org/doc/244795

ER -

## References

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- [9] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). Zbl0838.35001MR1329546
- [10] 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. Zbl0892.35019MR1473991
- [11] M.L. Kleptsyna and A.L. Piatnitski, Averaging of non selfadjoint parabolic equations with random evolution (dynamics), Preprint INRIA. J. Funct. Anal. (submitted).
- [12] 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) Zbl0674.47036
- [13] A.L. Piatnitsky, Parabolic equations with rapidly oscillating coefficients (In Russian). English transl. Moscow Univ. Math. Bull. 3 (1980) 33-39. Zbl0466.35005

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