Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.
Mots clés : Carleman estimates, semiclassical analysis, elliptic operators, parabolic operators, controllability, observability
@article{COCV_2012__18_3_712_0,
author = {Le Rousseau, J\'er\^ome and Lebeau, Gilles},
title = {On {Carleman} estimates for elliptic and parabolic operators. {Applications} to unique continuation and control of parabolic equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {712--747},
publisher = {EDP-Sciences},
volume = {18},
number = {3},
year = {2012},
doi = {10.1051/cocv/2011168},
mrnumber = {3041662},
zbl = {1262.35206},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2011168/}
}
TY - JOUR AU - Le Rousseau, Jérôme AU - Lebeau, Gilles TI - On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 712 EP - 747 VL - 18 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2011168/ DO - 10.1051/cocv/2011168 LA - en ID - COCV_2012__18_3_712_0 ER -
%0 Journal Article %A Le Rousseau, Jérôme %A Lebeau, Gilles %T On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 712-747 %V 18 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2011168/ %R 10.1051/cocv/2011168 %G en %F COCV_2012__18_3_712_0
Le Rousseau, Jérôme; Lebeau, Gilles. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 712-747. doi : 10.1051/cocv/2011168. http://archive.numdam.org/articles/10.1051/cocv/2011168/
[1] , Lectures on Elliptic Boundary Values Problems. Van Nostrand (1965). | MR | Zbl
[2] and , Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser. Éditions du CNRS (1991). | Zbl
[3] and , Applied Non Linear Analysis. John Wiley & Sons, New York (1984). | MR | Zbl
[4] , Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. | MR | Zbl
[5] , Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal. 35 (2003) 257-279. | MR | Zbl
[6] and , Null controllability of a thermoelastic plate. Abstr. Appl. Anal. 7 (2002) 585-599. | MR | Zbl
[7] , and , Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. J. Math. Anal. Appl. 336 (2007) 865-887. | MR | Zbl
[8] , and , On the controllability of linear parabolic equations with an arbitrary control location for stratified media. C. R. Acad. Sci. Paris, Ser. I 344 (2007) 357-362. | MR | Zbl
[9] and , Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM Control Optim. Calc. Var. 14 (2008) 1-42. | Numdam | MR | Zbl
[10] , Analyse Fonctionnelle. Masson, Paris (1983). | MR | Zbl
[11] , Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26B (1939) 1-9. | Zbl
[12] , Insensitizing controls for a semilinear heat equation. Comm. Partial Differential Equations 25 (2000) 39-72. | MR | Zbl
[13] and , Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series 268. Cambridge University Press, Cambridge (1999). | MR | Zbl
[14] , , and , On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798-819. | MR | Zbl
[15] , and , Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM : COCV 8 (2002) 621-661. | Numdam | MR | Zbl
[16] and , Global Carleman inequalities for parabolic systems and application to controllability. SIAM J. Control Optim. 45 (2006) 1395-1446. | MR | Zbl
[17] and , The cost of approximate controllability for heat equations : the linear case. Adv. Differential Equations 5 (2000) 465-514. | MR | Zbl
[18] and , Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré, Analyse non linéaire 17 (2000) 583-616. | Numdam | MR | Zbl
[19] , , and , Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR | Zbl
[20] , , and , Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N − 1 scalar controls. SIAM J. Control Optim. 45 (2006) 146-173. | MR | Zbl
[21] and , Prolongement unique des solutions de l'equation de Stokes. Comm. Partial Differential Equations 21 (1996) 573-596. | MR | Zbl
[22] and , Controllability of evolution equations, Lecture Notes 34. Seoul National University, Korea (1996). | MR | Zbl
[23] and , Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptotic Anal. 46 (2006) 123-162. | MR | Zbl
[24] and , Microlocal Analysis for Differential Operators. Cambridge University Press, Cambridge (1994). | MR | Zbl
[25] , Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). | Zbl
[26] , The Analysis of Linear Partial Differential Operators IV. Springer-Verlag (1985). | Zbl
[27] , The Analysis of Linear Partial Differential Operators III. Springer-Verlag (1985). 2nd printing 1994. | Zbl
[28] , The Analysis of Linear Partial Differential Operators I. 2nd edition, Springer-Verlag (1990). | Zbl
[29] , Remarks on the exact controllability of Navier-Stokes equations. ESAIM : COCV 6 (2001) 39-72. | Numdam | MR | Zbl
[30] and , Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. 87 (2007) 408-437. | MR | Zbl
[31] and , Harmonic analysis and partial differential equations (Chicago, IL, 1996). chapter Nodal sets of sums of eigenfunctions, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago (1999) 223-239. | MR | Zbl
[32] , and , Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928-943. | MR | Zbl
[33] , , and , Null-controllability of some systems of parabolic type by one control force. ESAIM : COCV 11 (2005) 426-448. | Numdam | MR | Zbl
[34] , Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differential Equations 233 (2007) 417-447. | MR | Zbl
[35] , and , Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. Arch. Rational Mech. Anal. 105 (2010) 953-990. | MR | Zbl
[36] and , Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183 (2011) 245-336. | MR | Zbl
[37] , Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J. Funct. Anal. 258 (2010) 2739-2778. | MR | Zbl
[38] , Cours sur les inégalités de Carleman, Mastère Equations aux Dérivées Partielles et Applications. Faculté des Sciences de Tunis, Tunisie (2005).
[39] and , Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335-356. | MR | Zbl
[40] and , Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR | Zbl
[41] and , Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297-329. | MR | Zbl
[42] , An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag (2002). | MR | Zbl
[43] and , On the lack of null-controllability of the heat equation on the half space. Port. Math. (N.S.) 58 (2001) 1-24. | MR | Zbl
[44] , On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129 (2005) 175-185. | MR | Zbl
[45] , On the controllability of anomalous diffusions generated by the fractional laplacian. Mathematics of Control, Signals, and Systems 3 (2006) 260-271. | MR | Zbl
[46] , Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones. Preprint (2008). http://hal.archives-ouvertes.fr/hal-00411840/fr.
[47] , A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1465-1485. | MR | Zbl
[48] , Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations 16 (1991) 789-800. | Zbl
[49] , Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10 (1995) 95-115. | MR | Zbl
[50] , Autour de l'Approximation Semi-Classique, Progress in Mathematics 68. Birkhäuser Boston, Boston, MA (1987). | MR | Zbl
[51] and , Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | MR | Zbl
[52] , Pseudodifferential Operators and Spectral Theory. 2nd edition, Springer-Verlag, Berlin Heidelberg (2001). | MR | Zbl
[53] , Carleman estimates and unique continuation for the Schroedinger equation. Differential Integral Equations 8 (1995) 901-905. | MR | Zbl
[54] , Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem. Comm. Partial Differential Equations 20 (1995) 855-884. | MR | Zbl
[55] , Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981). | MR | Zbl
[56] , Partial Differential Equations 2 : Qualitative Studies of Linear Equations, Applied Mathematical Sciences 116. Springer-Verlag, New-York (1996). | MR | Zbl
[57] and , On the null-controllability of diffusion equations. preprint (2009).
[58] , Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967). | MR | Zbl
[59] , Uniqueness and Non Uniqueness in the Cauchy Problem. Birkhäuser, Progress in mathematics (1983). | MR | Zbl
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