Carleman estimates for a subelliptic operator and unique continuation

Garofalo, Nicola; Shen, Zhongwei

doi:10.5802/aif.1392

Garofalo, Nicola ; Shen, Zhongwei

Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 129-166.

Résumé
Abstract

Nous démontrons une inéqualité du type de Carleman pour l’opérateur sous-elliptique de la forme $ℒ = Δ_{z} + | z |^{2} \partial_{t}^{2}$ dans $ℝ^{n + 1}$ avec $n \geq 2$ , $z \in ℝ^{n}$ , et $t \in ℝ$ . On en déduit que $- ℒ + V$ possède la propriété d’unicité stricte du prolongement des solutions aux points $(0, t)$ , $t \in ℝ$ , si le potentiel $V$ appartient localement à des espaces $L^{p}$ particuliers.

We establish a Carleman type inequality for the subelliptic operator $ℒ = Δ_{z} + | x |^{2} \partial_{t}^{2}$ in $ℝ^{n + 1}$ , $n \geq 2$ , where $z \in ℝ^{n}$ , $t \in ℝ$ . As a consequence, we show that $- ℒ + V$ has the strong unique continuation property at points of the degeneracy manifold ${(0, t) \in ℝ^{n + 1} | t \in ℝ}$ if the potential $V$ is locally in certain $L^{p}$ spaces.

MR Zbl

DOI : 10.5802/aif.1392

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     title = {Carleman estimates for a subelliptic operator and unique continuation},
     journal = {Annales de l'Institut Fourier},
     pages = {129--166},
     publisher = {Institut Fourier},
     address = {Grenoble},
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     number = {1},
     year = {1994},
     doi = {10.5802/aif.1392},
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     zbl = {0791.35017},
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Garofalo, Nicola; Shen, Zhongwei. Carleman estimates for a subelliptic operator and unique continuation. Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 129-166. doi : 10.5802/aif.1392. https://www.numdam.org/articles/10.5802/aif.1392/

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