Unique continuation for the solutions of the laplacian plus a drift
Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 651-663.

Nous prouvons l’unicité du prolongement pour les solutions de l’inégalité $|\Delta u\left(x\right)|\le V\left(x\right)|\nabla u\left(x\right)|$, $x\in \Omega$$\Omega$ est une partie connexe de ${\mathbf{R}}^{n}$ et $V$ appartient aux espaces de Morrey ${F}^{\alpha ,p}$, avec $p\ge \left(n-2\right)/2\left(1-\alpha \right)$ et $\alpha <1$. Ces espaces contiennent ${L}^{q}$ pour $q\ge \left(3n-2\right)/2$ (voir L. Hörmander, Comm. PDE, 8 (1983, 21-64 et Barceló, Kenig, Ruiz, Sogge, Ill. J. of Math., 32-2 (1988), 230-245).

We prove unique continuation for solutions of the inequality $|\Delta u\left(x\right)|\le V\left(x\right)|\nabla u\left(x\right)|$, $x\in \Omega$ a connected set contained in ${\mathbf{R}}^{n}$ and $V$ is in the Morrey spaces ${F}^{\alpha ,p}$, with $p\ge \left(n-2\right)/2\left(1-\alpha \right)$ and $\alpha <1$. These spaces include ${L}^{q}$ for $q\ge \left(3n-2\right)/2$ (see [H], [BKRS]). If $p=\left(n-2\right)/2\left(1-\alpha \right)$, the extra assumption of $V$ being small enough is needed.

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author = {Ruiz, Alberto and Vega, Luis},
title = {Unique continuation for the solutions of the laplacian plus a drift},
journal = {Annales de l'Institut Fourier},
pages = {651--663},
publisher = {Institut Fourier},
volume = {41},
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year = {1991},
doi = {10.5802/aif.1268},
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Ruiz, Alberto; Vega, Luis. Unique continuation for the solutions of the laplacian plus a drift. Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 651-663. doi : 10.5802/aif.1268. http://www.numdam.org/articles/10.5802/aif.1268/

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