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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2011168/
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