Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation

SIAM Journal on Control and Optimization

Volumn 46, Issue 5, 2007, Pages 1849-1881

  Benabdallah, Assia ^a   Gaitan, Patricia ^b   Le Rousseau, Jérôme ^a  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Carleman estimates; Discontinuous coefficients; Inverse problem; Parabolic equations; Stability estimate

Indexed keywords

DIFFUSION; HEATING EQUIPMENT; INVERSE PROBLEMS;

ARBITRARY POSITIVE TIME; CARLEMAN ESTIMATES; DIFFUSION CO-EFFICIENT; DISCONTINUOUS COEFFICIENTS; HEAT EQUATIONS; INITIAL CONDITIONS; INVERSE PROBLEM; MONOTONICITY CONDITIONS; PARABOLIC EQUATIONS; STABILITY ESTIMATE; STABILITY RESULTS;

SEMICONDUCTOR DOPING;

EID: 52649171353     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/050640047     Document Type: Article

References (22)

1. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. L. BAUDOUIN AND J.-P. PUEL, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), pp. 1537-1554.
3. A. BENABDALLAH, Y. DERMENJIAN, AND J. Le ROUSSEAU, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications, C. R. Mécanique, 334 (2006), pp. 582-586.
4. A. BENABDALLAH, Y. DERMENJIAN, AND J. Le ROUSSEAU, Carleman estimates for the onedimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., (2007), to appear.
5. A. BENABDALLAH, P. GAITAN, AND J. Le ROUSSEAU, Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation, preprint, LATP, Universités de Marseille, Marseille, France, 2005; available online from http://www.cmi.univ-mrs.fr/̃jlerous/publications.html.
6. H. BREZIS, Analyse Fonctionnelle, Masson, Paris, 1983.
7. R. DAUTRAY AND J.-L. Lions, Analyse Mathématique et Catcul Numérique pour les Sciences et les Techniques, Vol. 7, Masson, Paris, 1984.
8. A. DOUBOVA, A. OSSES, AND J.-P. PUEL, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 621-661.
9. H. EGGER, H. W. ENGL, AND M. V. KLIBANOV, Global uniqueness and Holder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21 (2005), pp. 271-290.
10. E. FERNÁNDEZ-CARA AND S. GUERRERO, Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., 45 (2006), pp. 1395-1446.
11. A. FURSIKOV AND O. Yu. IMANUVILOV, Controllability of Evolution Equations, Lecture Notes vol. 34, Seoul National University, Korea, 1996.
12. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
13. L. HÖRMANDER, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963.
14. L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, Vol. IV, Springer-Verlag, New York, Berlin, 1985.
15. L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, Vol. I, 2nd ed., Springer-Verlag, New York, Berlin, 1990.
16. O. IMANUVILOV AND M. YAMAMOTO, Lipschitz stability in inverse problems by Carleman estimate, Inverse Problems, 14 (1998), pp. 1229-1245.
17. V. ISAKOV, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105 (1993), pp. 217-238.
18. V. ISAKOV, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, 1998.
19. M. V. KLIBANOV, Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate, Inverse Problems, 20 (2004), pp. 1003-1032.
20. L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations, CBMS-NSF Reg. Conf. Ser. Appl. Math. 22, SIAM, Philadelphia, 1975.
21. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
22. M. YAMAMOTO AND J. ZOU, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), pp. 1181-1202.