Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T18:06:58.027Z Has data issue: false hasContentIssue false

The wave equation with oscillating density:observability at low frequency

Published online by Cambridge University Press:  15 August 2002

Gilles Lebeau*
Affiliation:
Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau Cedex, France; [email protected].
Get access

Abstract

We prove an observability estimate for awave equation with rapidly oscillating density,in a bounded domain with Dirichlet boundary condition.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avellaneda, M., Bardos, C. and Rauch, J., Contrôlabilité exacte, homogénéisation et localisation d'ondes dans un milieu non-homogène. Asymptot. Anal. 5 (1992) 481-484.
Allaire, G. and Conca, C., Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. CrossRef
N. Burq and G. Lebeau, Mesures de défaut de compacité; applications au système de Lamé, preprint.
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1075. CrossRef
C. Castro, Boundary controllability of the one dimensional wave equation with rapidly oscillating density, preprint.
Castro, C. and Zuazua, E., Contrôle de l'équation des ondes à densité rapidement oscillante à une dimension d'espace. C. R. Acad. Sci. Paris 324 (1997) 1237-1242. CrossRef
P. Gérard, Mesures semi-classiques et ondes de Bloch, Séminaire X EDP, exposé 16 (1991).
Gérard, P. and Leichtnam, E., Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993) 559-607. CrossRef
Lebeau, G., Contrôle de l'équation de Schrödinger. J. Math. Pures Appl. 71 (1993) 267-291.
G. Lebeau, Équation des ondes amorties, Algebraic and Geometric Methods in Mathematical Physics, A. Boutet de Monvel and V. Marchenko, Eds. Kluwer Academic Publishers (1996) 73-109.
R. Melrose and J. Sjöstrand, Singularities of boundary value problems I, II. Comm. Pure Appl. Math. 31 (1978) 593-617; 35 (1982) 129-168.