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Gradient estimates for positive solutions of the Laplacian with drift

Authors:	Benito J Gonzá lez Emilio R Negrin

Institution:	Departamento de Análisis Matemático, Universidad de La Laguna, 38271 Canary Islands, Spain ; Departamento de Análisis Matemático, Universidad de La Laguna, 38271 Canary Islands, Spain

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).

Keywords:	Gradient estimate Laplacian with drift Bochner-Lichn\`erowicz-Weitzenb\"ock formula Liouville theorem

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