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Journal Articles Indiana University Mathematics Journal Year : 2015

On Carleman estimates with two large parameters

Abstract

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 $\varphi = \exp(\alpha \psi)$, involving a second large parameter, $\alpha>0$, 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. The analysis is absed on the introduction of a proper Weyl-Hörmander calculus for pseudo-differential operators. Carleman estimates of various strengths are considered: estimates under the (strong) \pcty condition and estimates under the simple characteristic property. In each case the associated geometrical conditions for the function $\psi$ is proven necessary and sufficient. In addition some optimality results with respect to the power of the large parameters are proven.
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Dates and versions

hal-00738717 , version 1 (05-10-2012)
hal-00738717 , version 2 (29-06-2013)
hal-00738717 , version 3 (19-05-2014)
hal-00738717 , version 4 (22-01-2015)

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Cite

Jérôme Le Rousseau. On Carleman estimates with two large parameters. Indiana University Mathematics Journal, 2015, 64, pp.55-113. ⟨10.1512/iumj.2015.64.5397⟩. ⟨hal-00738717v4⟩
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