# American Institute of Mathematical Sciences

September  2017, 7(3): 347-367. doi: 10.3934/mcrf.2017012

## Quantification of the unique continuation property for the heat equation

 Laboratoire POEMS, Ensta ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

Received  April 2016 Revised  November 2016 Published  July 2017

In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method.

Citation: Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
##### References:
 [1] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004.Google Scholar [2] E. Bécache, L. Bourgeois, J. Dardé and L. Franceschini, Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case, Inverse Problems and Imaging, 9 (2015), 971-1002. doi: 10.3934/ipi.2015.9.971. Google Scholar [3] M. Boulakia, Quantification of the unique continuation property for the nonstationary Stokes problem, Mathematical Control and Related Fields, 6 (2016), 27-52. doi: 10.3934/mcrf.2016.6.27. Google Scholar [4] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: The case of $C^{1, 1}$, M2AN Math. Model. Numer. Anal., 44 (2010), 715-735. doi: 10.1051/m2an/2010016. Google Scholar [5] L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains, Applicable Analysis, 89 (2010), 1745-1768. doi: 10.1080/00036810903393809. Google Scholar [6] L. Bourgeois and J. Dardé, The "exterior approach" applied to the inverse obstacle problem for the heat equation, preprint, https://arxiv.org/abs/1609.05682.Google Scholar [7] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. doi: 10.1137/S0363012904439696. Google Scholar [8] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Seoul, 1996. Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Fourth Printing, Springer-Verlag, 1976. Google Scholar [12] V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006. Google Scholar [13] M. V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data, Inverse Problems, 22 (2006), 495-514. doi: 10.1088/0266-5611/22/2/007. Google Scholar [14] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549. Google Scholar [15] R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967. Google Scholar [16] M. M. Lavrentiev, V. G. Romanov and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, RI, 1986.Google Scholar [17] J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. Google Scholar [18] J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 2, Dunod, Paris, 1968. Google Scholar [19] K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace, ESAIM Control Optim. Calc. Var., 9 (2003), 621-635. doi: 10.1051/cocv:2003030. Google Scholar [20] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168. Google Scholar [21] T. Takeuchi and M. Yamamoto, Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation, SIAM J. Sci. Comput., 31 (2008), 112-142. doi: 10.1137/070684793. Google Scholar [22] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. Google Scholar [23] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates Inverse Problems, 24 (2008), 023001, 81pp. doi: 10.1088/0266-5611/24/2/023001. Google Scholar [24] M. Yamamoto, Carleman estimates for parabolic equations and applications Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013. Google Scholar

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##### References:
 [1] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004.Google Scholar [2] E. Bécache, L. Bourgeois, J. Dardé and L. Franceschini, Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case, Inverse Problems and Imaging, 9 (2015), 971-1002. doi: 10.3934/ipi.2015.9.971. Google Scholar [3] M. Boulakia, Quantification of the unique continuation property for the nonstationary Stokes problem, Mathematical Control and Related Fields, 6 (2016), 27-52. doi: 10.3934/mcrf.2016.6.27. Google Scholar [4] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: The case of $C^{1, 1}$, M2AN Math. Model. Numer. Anal., 44 (2010), 715-735. doi: 10.1051/m2an/2010016. Google Scholar [5] L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains, Applicable Analysis, 89 (2010), 1745-1768. doi: 10.1080/00036810903393809. Google Scholar [6] L. Bourgeois and J. Dardé, The "exterior approach" applied to the inverse obstacle problem for the heat equation, preprint, https://arxiv.org/abs/1609.05682.Google Scholar [7] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. doi: 10.1137/S0363012904439696. Google Scholar [8] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Seoul, 1996. Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Fourth Printing, Springer-Verlag, 1976. Google Scholar [12] V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006. Google Scholar [13] M. V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data, Inverse Problems, 22 (2006), 495-514. doi: 10.1088/0266-5611/22/2/007. Google Scholar [14] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549. Google Scholar [15] R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967. Google Scholar [16] M. M. Lavrentiev, V. G. Romanov and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, RI, 1986.Google Scholar [17] J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. Google Scholar [18] J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 2, Dunod, Paris, 1968. Google Scholar [19] K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace, ESAIM Control Optim. Calc. Var., 9 (2003), 621-635. doi: 10.1051/cocv:2003030. Google Scholar [20] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168. Google Scholar [21] T. Takeuchi and M. Yamamoto, Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation, SIAM J. Sci. Comput., 31 (2008), 112-142. doi: 10.1137/070684793. Google Scholar [22] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. Google Scholar [23] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates Inverse Problems, 24 (2008), 023001, 81pp. doi: 10.1088/0266-5611/24/2/023001. Google Scholar [24] M. Yamamoto, Carleman estimates for parabolic equations and applications Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013. Google Scholar
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