Partial Differential Equations
Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation
Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 555-560.

We establish geometrical conditions for the inverse problem of determining a stationary potential in the wave equation with Dirichlet data from a Neumann measurement on a suitable part of the boundary. We present the stability results when we measure on a part of the boundary satisfying a rotated exit condition. The proofs rely on global Carleman estimates with angle type dependence in the weight functions.

On établit des conditions geométriques pour le problème inverse consistant à déterminer un potentiel stationnaire dans l'équation des ondes avec une donnée de Dirichlet et à partir d'une mesure de Neumann sur une partie appropriée de la frontière. On présente des résultats de stabilité quand on mesure sur une partie de la frontière satisfaisant une condition sortante à directions variables. Les démonstrations réposent sur des inégalités de Carleman globales avec des fonctions poids qui dépendent d'un angle comme paramètre.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.09.022
Doubova, Anna 1; Osses, Axel 2

1 Departamento E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Departamento de Ingenería Matemática, University of Chile, Casilla 170/3, correo 3, Santiago, Chili
@article{CRMATH_2005__341_9_555_0,
     author = {Doubova, Anna and Osses, Axel},
     title = {Application of global {Carleman} estimates with rotated weights to an inverse problem for the wave equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {555--560},
     publisher = {Elsevier},
     volume = {341},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.022},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.09.022/}
}
TY  - JOUR
AU  - Doubova, Anna
AU  - Osses, Axel
TI  - Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 555
EP  - 560
VL  - 341
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.09.022/
DO  - 10.1016/j.crma.2005.09.022
LA  - en
ID  - CRMATH_2005__341_9_555_0
ER  - 
%0 Journal Article
%A Doubova, Anna
%A Osses, Axel
%T Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation
%J Comptes Rendus. Mathématique
%D 2005
%P 555-560
%V 341
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.09.022/
%R 10.1016/j.crma.2005.09.022
%G en
%F CRMATH_2005__341_9_555_0
Doubova, Anna; Osses, Axel. Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation. Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 555-560. doi : 10.1016/j.crma.2005.09.022. http://www.numdam.org/articles/10.1016/j.crma.2005.09.022/

[1] Bardos, C.; Lebeau, G.; Rauch, J. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control. Optim., Volume 30 (1992), pp. 1024-1465

[2] A. Doubova, A. Osses, Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation, in preparation

[3] Dautray, R.; Lions, J.-L. Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 2000

[4] Imanuvilov, O.Yu. On Carleman estimates for hyperbolic equations, Asymptot. Anal., Volume 32 (2002) no. 3–4, pp. 185-220

[5] Imanuvilov, O.Yu.; Yamamoto, M. Global uniqueness and stability in determining coefficients of wave equations, Commun. Partial Differential Equations, Volume 26 (2001) no. 7–8, pp. 1409-1425

[6] Imanuvilov, O.Yu.; Yamamoto, M. Determination of a coefficient in an acoustic equation with a single measurement, Inverse Problems, Volume 19 (2003), pp. 157-171

[7] Lions, J.-L. Contrôlabilité exacte, perturbation et stabilisation de systèmes distribués, 1, Masson, Paris, 1988

[8] Osses, A. A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control, SIAM J. Control Optim., Volume 40 (2001) no. 3, pp. 777-800

[9] J.-P. Puel, Application of Carleman inequalities to controllability and inverse problems, Textos de Metodos Matematicos de l'Instituto de Matematica de l'UFRJ, in preparation

[10] Yamamoto, M.; Puel, J.-P. Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques, C. R. Acad. Sci. Paris, Sér. I, Volume 320 (1995), pp. 1171-1776

Cited by Sources: