Carleman inequalities and inverse problems for the Schrödinger equation

Alberto Mercado; Axel Osses; Lionel Rosier

doi:10.1016/j.crma.2007.11.014

Optimal Control

Carleman inequalities and inverse problems for the Schrödinger equation

Presented by: Gilles Lebeau

Alberto Mercado ¹; Axel Osses ¹; Lionel Rosier ^1,²

¹ Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática (DIM), Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
² Institut Elie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 53-58.

Abstract
Résumé

In this Note, we derive new Carleman inequalities for the evolution Schrödinger equation under a weak pseudoconvexity condition, which allows us to use weights with a linear spatial dependence. As a result, less restrictive boundary or internal observation regions may be used to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from a single boundary or internal measurement, respectively.

Dans cette Note, nous établissons de nouvelles inégalités de Carleman pour l'équation d'évolution de Schrödinger sous une hypothèse de pseudoconvexité faible, qui permet d'utiliser des poids affines en la variable d'espace. Comme application, nous pouvons définir des régions d'observabilité moins restrictives dans le problème inverse consistant à retrouver un potentiel stationnaire dans l'équation de Schrödinger à partir d'une mesure simple effectuée au bord ou à l'intérieur du domaine.

Received: 2007-07-24
Accepted: 2007-11-22
Published online: 2007-12-26

DOI: 10.1016/j.crma.2007.11.014

Author's affiliations:

Alberto Mercado ¹; Axel Osses ¹; Lionel Rosier ^{1, 2}

¹ Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática (DIM), Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
² Institut Elie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

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     author = {Alberto Mercado and Axel Osses and Lionel Rosier},
     title = {Carleman inequalities and inverse problems for the {Schr\"odinger} equation},
     journal = {Comptes Rendus. Math\'ematique},
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     volume = {346},
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     year = {2008},
     doi = {10.1016/j.crma.2007.11.014},
     language = {en},
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Alberto Mercado; Axel Osses; Lionel Rosier. Carleman inequalities and inverse problems for the Schrödinger equation. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 53-58. doi : 10.1016/j.crma.2007.11.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.014/

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