Homogenization and correctors for the wave equation in non periodic perforated domains

Patrizia Donato; Florian Gaveau; Patrizia Donato; Florian Gaveau

doi:10.3934/nhm.2008.3.97

Networks and Heterogeneous Media

2008, Volume 3, Issue 1: 97-124. doi: 10.3934/nhm.2008.3.97

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Homogenization and correctors for the wave equation in non periodic perforated domains

Patrizia Donato ^{1
,},
Florian Gaveau ^{2
,}

1.
Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Avenue de l’Université, BP 12, 76801 Saint Etienne de Rouvray
2.
Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Université Paris 6 BP 187, 4 Place Jussieu 75252 Paris Cedex 05

Received: 01 May 2007 Revised: 01 October 2007
Primary: 35B27; Secondary: 35L05.

We consider here the wave equation in a (not necessarily periodic) perforated domain, with a Neumann condition on the boundary of the holes. Assuming $H^0$-convergence ([3]) on the elliptic part of the operator, we prove two main theorems: a convergence result and a corrector one. To prove the corrector result, we make use of a suitable family of elliptic local correctors given in [4] whose columns are piecewise locally square integrable gradients. As in the case without holes ([2]), some additional assumptions on the data are needed.
- Homogenization,
- correctors,
- $H^0$-convergence,
- wave equation
Citation: Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains[J]. Networks and Heterogeneous Media, 2008, 3(1): 97-124. doi: 10.3934/nhm.2008.3.97

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Abstract

We consider here the wave equation in a (not necessarily periodic) perforated domain, with a Neumann condition on the boundary of the holes. Assuming $H^0$-convergence ([3]) on the elliptic part of the operator, we prove two main theorems: a convergence result and a corrector one. To prove the corrector result, we make use of a suitable family of elliptic local correctors given in [4] whose columns are piecewise locally square integrable gradients. As in the case without holes ([2]), some additional assumptions on the data are needed.
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