On Carleman estimates with two large parameters
Résumé
A Carleman estimate for a differential operator $P$ is a weighted energy estimate with a weight of exponential form $\exp(\tau \varphi)$ that involves a large parameter, $\tau>0$. The function $\varphi$ and the operator $P$ need to fulfill some sub-ellipticity properties that can be achieved for instance by choosing $\varphi = \exp(\al \psi)$, involving a second large parameter, $\al>0$, with $\psi$ satisfying some geometrical conditions. The purpose of this article is to give the framework to keep explicit the dependency upon the two large parameters in the resulting Carleman estimates. Carleman estimates of various strengths are considered and the associated geometrical conditions for the function $\psi$ are proven necessary and sufficient. Some optimality aspects of the estimates are also presented.
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