Carleman estimates for a subelliptic operator and unique continuation

Garofalo, Nicola; Shen, Zhongwei

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

We establish a Carleman type inequality for the subelliptic operator $ℒ={\Delta }_z+|x{|}^2{\partial }_t^2$ in ${ℝ}^{n+1}$, $n\ge 2$, 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 $\left\{\right(0,t)\in {ℝ}^{n+1}|t\in ℝ\}$ if the potential $V$ is locally in certain $L^p$ spaces.

Nous démontrons une inéqualité du type de Carleman pour l’opérateur sous-elliptique de la forme $ℒ={\Delta }_z+|z{|}^2{\partial }_t^2$ dans ${ℝ}^{n+1}$ avec $n\ge 2$, $z\in {ℝ}^n$, 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.

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