On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations∗,∗∗
Universitéd’Orléans, Laboratoire Mathématiques et Applications,
Physique Mathématique d’Orléans, CNRS UMR 6628, Fédération Denis Poisson, FR CNRS
2964, B.P. 6759,
Orléans Cedex 2,
2 Universitéde Nice Sophia-Antipolis, Laboratoire Jean Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 02, France
Received: 13 July 2009
Revised: 25 January 2011
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.
Mathematics Subject Classification: 35B60 / 35J15 / 35K05 / 93B05 / 93B07
Key words: Carleman estimates / semiclassical analysis / elliptic operators / parabolic operators / controllability / observability
The CNRS Pticrem project facilitated the writting of these notes. The first author was partially supported by l’Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01.
M. Bellassoued’s handwritten notes of  were very valuable to us and we wish to thank him for letting us use them. The authors wish to thank L. Robbiano for many discussions on the subject of these notes and L. Miller for discussions on some of the optimality results. We also thank M. Léautaud for his corrections.
© EDP Sciences, SMAI, 2011