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

Published online by Cambridge University Press:  21 October 2011

Jérôme Le Rousseau  and
Gilles Lebeau
Jérôme Le Rousseau
Affiliation:
Universitéd’Orléans, Laboratoire Mathématiques et Applications, Physique Mathématique d’Orléans, CNRS UMR 6628, Fédération Denis Poisson, FR CNRS 2964, B.P. 6759, 45067 Orléans Cedex 2, France. jlr@univ-orleans.fr
Gilles Lebeau
Affiliation:
Universitéde Nice Sophia-Antipolis, Laboratoire Jean Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 02, France; gilles.lebeau@unice.fr
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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.

Keywords

Carleman estimatessemiclassical analysiselliptic operatorsparabolic operatorscontrollabilityobservability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 712 - 747
Copyright
© EDP Sciences, SMAI, 2011

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