[Identification de deux coefficients indépendants avec une seule observation pour l'opérateur de Schrödinger dans une bande non bornée]
Dans cet article, il s'agit de prouver un résultat de stabilité pour deux coefficients indépendants (chacun dépendant d'une seule variable) pour un opérateur de Schrödinger avec une seule observation sur une partie non bornée du bord. Nous rappelons l'estimation de Carleman globale prouvée dans Cardoulis et al. (2008) [3]. En utilisant une estimation de type Carleman pour un opérateur différentiel du premier ordre (cf. Immanuvilov et Yamamoto (2005) [4]) ainsi qu'une estimation d'énergie, nous obtenons l'identification simultanée du coefficient de diffusion et du potentiel avec une seule observation.
This article is devoted to prove a stability result for two independent coefficients (each one depending on only one variable) for a Schrödinger operator in an unbounded strip with only one observation on an unbounded subset of the boundary. For that, we first use the global Carleman estimate proved in Cardoulis et al. (2008) [3]. Then, with a Carleman-type estimate for a first order differential operator (cf. Immanuvilov and Yamamoto (2005) [4]) and an energy estimate, we prove the simultaneous identification of the diffusion coefficient and the potential with only one observation.
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@article{CRMATH_2010__348_3-4_149_0,
author = {Laure Cardoulis and Patricia Gaitan},
title = {Identification of two independent coefficients with one observation for the {Schr\"odinger} operator in an unbounded strip},
journal = {Comptes Rendus. Math\'ematique},
pages = {149--153},
publisher = {Elsevier},
volume = {348},
number = {3-4},
year = {2010},
doi = {10.1016/j.crma.2009.10.030},
language = {en},
}
TY - JOUR AU - Laure Cardoulis AU - Patricia Gaitan TI - Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip JO - Comptes Rendus. Mathématique PY - 2010 SP - 149 EP - 153 VL - 348 IS - 3-4 PB - Elsevier DO - 10.1016/j.crma.2009.10.030 LA - en ID - CRMATH_2010__348_3-4_149_0 ER -
%0 Journal Article %A Laure Cardoulis %A Patricia Gaitan %T Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip %J Comptes Rendus. Mathématique %D 2010 %P 149-153 %V 348 %N 3-4 %I Elsevier %R 10.1016/j.crma.2009.10.030 %G en %F CRMATH_2010__348_3-4_149_0
Laure Cardoulis; Patricia Gaitan. Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 149-153. doi : 10.1016/j.crma.2009.10.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.030/
[1] Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, Volume 18 (2002), pp. 1537-1554
[2] L. Cardoulis, P. Gaitan, Simultaneous identification of the diffusion coefficient and the potential for the Schrödinger operator with one observation, Inverse Problems, submitted for publication
[3] Inverse problem for the Schrödinger operator in an unbounded strip, J. Inverse Ill-Posed Probl., Volume 16 (2008) no. 2, pp. 127-146
[4] Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., Volume 11 (2005) no. 1, pp. 1-56
[5] Inverse Problems for Partial Differential Equations, Springer-Verlag, 1998
[6] Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2004
[7] Lipshitz stability for an inverse problem for an acoustic equation, Appl. Anal., Volume 85 (2006), pp. 515-538
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