[Inégalités fines de Carleman pour des problèmes paraboliques singuliers et application à des problèmes inverses]
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 avec où 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 when where 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.
Accepté le :
Publié le :
@article{CRMATH_2010__348_13-14_801_0,
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},
publisher = {Elsevier},
volume = {348},
number = {13-14},
year = {2010},
doi = {10.1016/j.crma.2010.06.001},
language = {en},
}
TY - JOUR AU - Judith Vancostenoble TI - Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems JO - Comptes Rendus. Mathématique PY - 2010 SP - 801 EP - 805 VL - 348 IS - 13-14 PB - Elsevier DO - 10.1016/j.crma.2010.06.001 LA - en ID - CRMATH_2010__348_13-14_801_0 ER -
%0 Journal Article %A Judith Vancostenoble %T Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems %J Comptes Rendus. Mathématique %D 2010 %P 801-805 %V 348 %N 13-14 %I Elsevier %R 10.1016/j.crma.2010.06.001 %G en %F CRMATH_2010__348_13-14_801_0
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/
[1] Remarks on the inverse square potential in quantum mechanics, Birmingham, Ala., 1983 (North-Holland Math. Stud.), Volume vol. 92, North-Holland, Amsterdam (1984), pp. 31-35
[2] Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., Volume 46 (2007) no. 5, pp. 1849-1881
[3] Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., Volume 47 (2008) no. 1, pp. 1-19
[4] Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Acad. Sci. Sér. I Math., Volume 347 (2009) no. 3–4, pp. 147-152
[5] P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications, in press
[6] P. Cannarsa, J. Tort, M. Yamamoto, Determination of source terms in a degenerate parabolic equation, in press
[7] Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. in PDE 33, Volume 11 (2008), pp. 1996-2019
[8] Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Seoul, Korea, 1996
[9] Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, Volume 14 (1998) no. 5, pp. 1229-1245
[10] Inverse Source Problems, Mathematical Surveys and Monographs, vol. 34, American Mathematical Society, Providence, RI, 1990
[11] Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2004 (iv+282 pp)
[12] Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., Volume 6 (2006) no. 2, pp. 325-362
[13] On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, Volume 12 (1996) no. 6, pp. 995-1002
[14] J. Tort, Determination of source terms in a degenerate parabolic equation from a locally distributed observation, C. R. Acad. Sci. Sér., in press
[15] Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., Volume 254 (2008) no. 7, pp. 1864-1902
[16] Hardy inequalities, observability and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1508-1532
[17] The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Volume 173 (2000) no. 1, pp. 103-153
Cité par Sources :
Commentaires - Politique
Jacques Tort
C. R. Math (2010)
Carleman estimates and null controllability for boundary-degenerate parabolic operators
Piermarco Cannarsa; Partick Martinez; Judith Vancostenoble
C. R. Math (2009)
Nulle contrôlabilité régionale pour des équations de la chaleur dégénérées
Piermarco Cannarsa; Patrick Martinez; Judith Vancostenoble
C. R. Méca (2002)