Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Gradient estimates for positive solutions of the Laplacian with drift

Author(s): Benito J. González; Emilio R. Negrin
Journal: Proc. Amer. Math. Soc. 127 (1999), 619-625.
MSC (1991): Primary 58G11
MathSciNet review: 1469407
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Abstract: Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where $K\geq 0.$ If is a vector field such that $\Vert b\Vert \leq \gamma$ and $\nabla b\leq K_{*}$ on for some nonnegative constants $\gamma$ and $K_{*},$ then we show that any positive $\mathcal{C}^{\infty }(M)$ solution of the equation $\Delta u(x)+(b(x)|\nabla u(x))=0$ satisfies the estimate

$\begin{displaymath}{\frac{{\Vert \nabla u\Vert }^2}{u^2}}\leq \frac{n(K+K_{*})}w+\frac{{\gamma }^2}{w(1-w)}\;, \end{displaymath}$

on , for all $w \in (0,1).$ In particular, for the case when $K=K_{*}=0,$ this estimate is advantageous for small values of $\Vert b\Vert$ and when $b\equiv 0$ it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

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