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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gradient estimates for positive solutions
of the Laplacian with drift


Authors: Benito J. González and 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 $M$ be a complete Riemannian manifold of dimension $n$ without boundary and with Ricci curvature bounded below by $-K,$ where $K\geq 0.$ If $b$ is a vector field such that $\Vert b\Vert \leq \gamma $ and $\nabla b\leq K_{*}$ on $M,$ 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 $M$, 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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Additional Information

Benito J. González
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 Canary Islands, Spain
Email: bjglez@ull.es

Emilio R. Negrin
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 Canary Islands, Spain
Email: enegrin@ull.es

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04578-5
PII: S 0002-9939(99)04578-5
Keywords: Gradient estimate, Laplacian with drift, Bochner-Lichn\`erowicz-Weitzenb\"ock formula, Liouville theorem
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society