[Inégalités de Carleman et problèmes inverses pour l'équation de Schrödinger]
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.
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.
Accepté le :
Publié le :
@article{CRMATH_2008__346_1-2_53_0,
author = {Mercado, Alberto and Osses, Axel and Rosier, Lionel},
title = {Carleman inequalities and inverse problems for the {Schr\"odinger} equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {53--58},
publisher = {Elsevier},
volume = {346},
number = {1-2},
year = {2008},
doi = {10.1016/j.crma.2007.11.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/}
}
TY - JOUR AU - Mercado, Alberto AU - Osses, Axel AU - Rosier, Lionel TI - Carleman inequalities and inverse problems for the Schrödinger equation JO - Comptes Rendus. Mathématique PY - 2008 SP - 53 EP - 58 VL - 346 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/ DO - 10.1016/j.crma.2007.11.014 LA - en ID - CRMATH_2008__346_1-2_53_0 ER -
%0 Journal Article %A Mercado, Alberto %A Osses, Axel %A Rosier, Lionel %T Carleman inequalities and inverse problems for the Schrödinger equation %J Comptes Rendus. Mathématique %D 2008 %P 53-58 %V 346 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/ %R 10.1016/j.crma.2007.11.014 %G en %F CRMATH_2008__346_1-2_53_0
Mercado, Alberto; Osses, Axel; Rosier, Lionel. Carleman inequalities and inverse problems for the Schrödinger equation. Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 53-58. doi : 10.1016/j.crma.2007.11.014. http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/
[1] Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, Volume 18 (2002), pp. 1537-1554
[2] Global uniqueness of a class of inverse problems, Sov. Math. Dokl., Volume 24 (1982), pp. 244-247
[3] Geometric control in the presence of a black box, J. Amer. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471
[4] Inverse problem for the Schrödinger operator in an unbounded strip | arXiv
[5] Carleman type estimates in an anisotropic case and applications, J. Differential Equations, Volume 105 (1993), pp. 217-238
[6] Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: H1-estimates, J. Inv. Ill-Posed Problems, Volume 11 (2004) no. 1, pp. 43-123
[7] Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., Volume 35 (1997) no. 5, pp. 1574-1590
[8] A. Mercado, A. Osses, L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, in press
[9] L. Rosier, B.-Y. Zhang, Null controllability of the complex Ginzburg–Landau equation, submitted for publication
Cité par Sources :
