Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems

Judith Vancostenoble

doi:10.1016/j.crma.2010.06.001

Partial Differential Equations/Optimal Control

Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems
[Inégalités fines de Carleman pour des problèmes paraboliques singuliers et application à des problèmes inverses]

Présenté par : Gilles Lebeau

Judith Vancostenoble ¹

¹ Institut de mathématiques de Toulouse, UMR CNRS 5219, université Paul-Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse cedex 4, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 801-805.

Résumé
Abstract

On étudie la stabilité Lipschitzienne pour des problèmes inverses de détermination d'une source pour l'équation de la chaleur perturbée par un potentiel singulier de la forme $- μ / {| x |}^{2}$ avec $μ ⩽ μ^{⋆} (N) : = {(N - 2)}^{2} / 4$ où $μ^{⋆} (N)$ est la constante optimale de l'inégalité de Hardy. Suivant Immanuvilov et Yamamoto (1998) [9], notre preuve repose sur des inégalités de Carleman telles que celles introduites par Fursikov et Immanuvilov (1996) [8] pour l'équation de la chaleur classique. Cependant, il faut ici tenir compte de la singularité. La première étape de la preuve consiste donc en une amélioration des inegalités de Carleman spécifiquement démontrées pour des equations avec un potentiel singulier par Vancostenoble et Zuazua (2008) [15] puis Ervedoza (2008) [7]. Certaines étapes majeures reposent sur diverses formes améliorées de l'inégalité de Hardy.

We address the question of Lipschitz stability results in inverse source problems for the heat equation perturbed by a singular inverse-square potential $- μ / {| x |}^{2}$ when $μ ⩽ μ^{⋆} (N) : = {(N - 2)}^{2} / 4$ where $μ^{⋆} (N)$ is the optimal constant in the so-called Hardy inequality. Following Immanuvilov and Yamamoto (1998) [9], our proof is based on Carleman inequalities like those developed by Fursikov and Immanuvilov (1996) [8] for the classical heat equation. However, we need here to take into account the singularity. Therefore, the first step of the proof consists in some improvements of the Carleman inequalities specifically developed for equations with inverse-square potentials by Vancostenoble and Zuazua (2008) [15] and next Ervedoza (2008) [7]. Major steps rely on various improved forms of the Hardy inequality.

Reçu le : 2010-02-23
Accepté le : 2010-06-01
Publié le : 2010-06-23

DOI : 10.1016/j.crma.2010.06.001

Affiliations des auteurs :

Judith Vancostenoble ¹

¹ Institut de mathématiques de Toulouse, UMR CNRS 5219, université Paul-Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse cedex 4, France

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     author = {Judith Vancostenoble},
     title = {Sharp {Carleman} estimates for singular parabolic equations and application to {Lipschitz} stability in inverse source problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {801--805},
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     doi = {10.1016/j.crma.2010.06.001},
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Judith Vancostenoble. Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 801-805. doi : 10.1016/j.crma.2010.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.001/

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Xiaoyu Fu; Jiaxin Tian Carleman estimates for the structurally damped plate equations with Robin boundary conditions and applications, Applicable Analysis, Volume 101 (2022) no. 13, p. 4668 | DOI:10.1080/00036811.2020.1866172
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J. Tort; J. Vancostenoble Determination of the insolation function in the nonlinear Sellers climate model, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 29 (2012) no. 5, p. 683 | DOI:10.1016/j.anihpc.2012.03.003
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Judith Vancostenoble Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Continuous Dynamical Systems - S, Volume 4 (2011) no. 3, p. 761 | DOI:10.3934/dcdss.2011.4.761

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