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 and 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.

[2]

E. BécacheL. BourgeoisJ. 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.

[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.

[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.

[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.

[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.

[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.

[8]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Seoul, 1996.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[11]

L. Hörmander, Linear Partial Differential Operators, Fourth Printing, Springer-Verlag, 1976.

[12]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.

[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.

[14]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[15]

R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.

[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.

[17]

J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.

[18]

J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 2, Dunod, Paris, 1968.

[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.

[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.

[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.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[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.

[24]

M. Yamamoto, Carleman estimates for parabolic equations and applications Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013.

show all references

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.

[2]

E. BécacheL. BourgeoisJ. 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.

[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.

[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.

[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.

[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.

[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.

[8]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Seoul, 1996.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[11]

L. Hörmander, Linear Partial Differential Operators, Fourth Printing, Springer-Verlag, 1976.

[12]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.

[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.

[14]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[15]

R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.

[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.

[17]

J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.

[18]

J. -L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 2, Dunod, Paris, 1968.

[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.

[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.

[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.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[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.

[24]

M. Yamamoto, Carleman estimates for parabolic equations and applications Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013.

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