Unique continuation for the solutions of the laplacian plus a drift

Ruiz, Alberto; Vega, Luis

Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 651-663

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

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

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