Inverse problem for the Schrödinger operator in an unbounded strip

Laure Cardoulis; Michel Cristofol; Patricia Gaitan

doi:10.1016/j.crma.2008.04.004

Partial Differential Equations

Inverse problem for the Schrödinger operator in an unbounded strip
[Un problème inverse pour l'opérateur de Schrödinger dans une bande]

Présenté par : Gilles Lebeau

Laure Cardoulis ¹ ; Michel Cristofol ² ; Patricia Gaitan ²

¹ CEREMATH/UMR MIP, Université de Toulouse 1, 21, allées de Brienne, 31000 Toulouse, France
² Laboratoire d'Analyse Topologie Probabilités, CNRS UMR 6632, Universités d'Aix-Marseille, 13453 Marseille cedex, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 635-640.

Résumé
Abstract

Nous démontrons une estimation globale de Carleman et une estimation d'énergie pour l'opérateur de Schrödinger $H : = i \partial_{t} + \nabla \cdot (c \nabla)$ dans une bande non bornée. Ces estimations nous permettent de donner un résultat de stabilité pour le coefficient de diffusion $c (x, y)$ à partir de la mesure de la dérivée normale de la solution sur une partie du bord.

We prove an adapted global Carleman estimate and an energy estimate for the Schrödinger operator $H : = i \partial_{t} + \nabla \cdot (c \nabla)$ in an unbounded strip. Using these estimates, we give a stability result for the diffusion coefficient $c (x, y)$ from the measurement of the normal derivative of the solution on a part of the boundary.

Reçu le : 2007-05-24
Accepté le : 2008-04-11
Publié le : 2008-04-29

DOI : 10.1016/j.crma.2008.04.004

Affiliations des auteurs :

Laure Cardoulis ¹ ; Michel Cristofol ² ; Patricia Gaitan ²

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     author = {Laure Cardoulis and Michel Cristofol and Patricia Gaitan},
     title = {Inverse problem for the {Schr\"odinger} operator in an unbounded strip},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {635--640},
     publisher = {Elsevier},
     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.04.004},
     language = {en},
}

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Laure Cardoulis; Michel Cristofol; Patricia Gaitan. Inverse problem for the Schrödinger operator in an unbounded strip. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 635-640. doi : 10.1016/j.crma.2008.04.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.004/

Bibliographie
Cité par

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[2] O.Yu. Immanuvilov; V. Isakov; M. Yamamoto An inverse problem for the dynamical Lamé system with two set of boundary data, Comm. Pure Appl. Math., Volume 56, 1 (2003) no. 17, pp. 1366-1382

[3] O.Yu. Immanuvilov; M. Yamamoto 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

[4] V. Isakov Inverse Problems for Partial Differential Equations, Springer-Verlag, 1998

[5] M.V. Klibanov; M. Yamamoto Lipschitz stability for an inverse problem for an acoustic equation, Appl. Anal., Volume 85 (2006), pp. 515-538

[6] M.V. Klibanov; A. Timonov Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-Posed Series, VSP, Utrecht, 2004

[7] I. Lasiecka; R. Triggiani; X. Zhang Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates, J. Inv. Ill-Posed Problems, Volume 11 (2003) no. 3, pp. 1-96

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