On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Jérôme Le Rousseau; Gilles Lebeau

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 712-747
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Le Rousseau, Jérôme, and 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 18.3 (2012): 712-747. <http://eudml.org/doc/272793>.

@article{LeRousseau2012,
abstract = {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.},
author = {Le Rousseau, Jérôme, Lebeau, Gilles},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carleman estimates; semiclassical analysis; elliptic operators; parabolic operators; controllability; observability; observability inequality},
language = {eng},
number = {3},
pages = {712-747},
publisher = {EDP-Sciences},
title = {On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations},
url = {http://eudml.org/doc/272793},
volume = {18},
year = {2012},
}

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
PB - EDP-Sciences
VL - 18
IS - 3
SP - 712
EP - 747
AB - 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.
LA - eng
KW - Carleman estimates; semiclassical analysis; elliptic operators; parabolic operators; controllability; observability; observability inequality
UR - http://eudml.org/doc/272793
ER -

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