﻿ 完备黎曼流形上椭圆方程的局部梯度估计

# 完备黎曼流形上椭圆方程的局部梯度估计Local Derivative Estimates for an Elliptic Equation on Complete Riemannian Manifolds

Abstract: The main purpose of this paper is to derive local gradient estimates for a second-order elliptic equation of Δfu=plogu+qu with smooth functions f, p and q on a complete Riemannian man-ifold.

1. 研究背景

$\Delta u+au\mathrm{log}u+bu=0,$ (1.1)

$f$ -Laplacian是Laplace-Beltrami算子的自然类比，它由 ${C}^{2}$ 函数 $u$ 定义：

${\Delta }_{f}u=\Delta u-〈\nabla u,\nabla f〉,$

Bakry-Émery Ricci曲率定义为：

$Ri{c}_{f}:=Ric+Hessf.$

K. Brighton [4] 研究了下列方程的正解的梯度估计：

${\Delta }_{f}u=0,$ (1.2)

${\Delta }_{f}u=pu\mathrm{log}u+qu,$ (1.3)

$\begin{array}{c}\frac{|\nabla u|}{u}\le \sqrt{\frac{2n}{1-\epsilon }}{\left[\begin{array}{c}\frac{|\alpha |{C}_{1}+\left(n-1\right)K\left(2\rho -1\right){C}_{1}}{2\rho }+\frac{{C}_{2}}{2{\rho }^{2}}+\frac{\left(2n+\epsilon \right){C}_{1}^{2}}{\epsilon {\rho }^{2}}\\ +\frac{D{\theta }_{1}+{\sigma }_{1}+{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}+{\sigma }_{2}}{2}\end{array}\right]}^{\frac{1}{2}}\\ +\sqrt[4]{\frac{n\left({\theta }_{2}+{\sigma }_{2}\right)}{1-\epsilon }},\end{array}$ (1.4)

$\frac{|\nabla u|}{u}\le \sqrt{2n}{\left[\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}\right]}^{\frac{1}{2}}+\sqrt[4]{n\left(D{\theta }_{2}+{\sigma }_{2}\right)}.$ (1.5)

$B\left(\stackrel{¯}{x},2\rho \right)$ 上，对于正常值 $D$${\theta }_{1}$${\theta }_{2}$${\sigma }_{1}$${\sigma }_{2}$${\eta }_{1}$ ，我们假设 $\alpha :={\mathrm{max}}_{\partial B\left(\stackrel{¯}{x},1\right)}{\Delta }_{f}r$$u\le {e}^{D}$$|p|\le {\theta }_{1}$$|\nabla p|\le {\theta }_{2}$$|q|\le \sigma$$|\nabla q|\le {\sigma }_{2}$$|\nabla f|\le {\eta }_{1}$

2. 理论基础

$-{C}_{1}\le \frac{{\xi }^{\prime }\left(s\right)}{\sqrt{\xi \left(s\right)}}\le 0$

${\xi }^{″}\left(s\right)\ge -{C}_{2}.$

$\frac{{|\nabla \phi |}^{2}}{\phi }\le \frac{{C}_{1}^{2}}{{\rho }^{2}},$ (2.1)

${\Delta }_{f}\phi =\frac{{\xi }^{\prime }{\Delta }_{f}r}{\rho }+\frac{{\xi }^{″}{|\nabla r|}^{2}}{{\rho }^{2}}.$ (2.2)

${\Delta }_{f}r\left(x\right)\le \alpha +\left(n-1\right)K\left(2\rho -1\right),$ (2.3)

${\Delta }_{f}\phi \left(x\right)\ge -\frac{{C}_{1}}{\rho }\left[\alpha +\left(n-1\right)K\left(2\rho -1\right)\right]-\frac{{C}_{2}}{{\rho }^{2}}.$ (2.4)

3. 梯度估计

${\Delta }_{f}h=ph+q-{|\nabla h|}^{2},$ (3.1)

$\begin{array}{c}\frac{1}{2}{\Delta }_{f}{|\nabla h|}^{2}\ge \frac{1}{2n}{\left({\Delta }_{f}h\right)}^{2}-\frac{1}{n}{〈\nabla h,\nabla f〉}^{2}+p{|\nabla h|}^{2}+Ri{c}_{f}\left(\nabla h,\nabla h\right)\\ +h〈\nabla p,\nabla h〉+〈\nabla q,\nabla h〉-〈\nabla h,\nabla {|\nabla h|}^{2}〉.\end{array}$ (3.2)

${\Delta }_{f}h=\frac{{\Delta }_{f}u}{u}-\frac{{|\nabla u|}^{2}}{{u}^{2}}=ph+q-{|\nabla h|}^{2}.$

$\begin{array}{c}{\left({\Delta }_{f}h\right)}^{2}={\left(\Delta h-〈\nabla h,\nabla f〉\right)}^{2}\\ \le 2{\left(\Delta h\right)}^{2}+2\nabla {〈h,\nabla f〉}^{2}\\ \le 2n{|\text{Hessh}|}^{2}+2{〈\nabla h,\nabla f〉}^{2},\end{array}$

${|\text{Hessh}|}^{2}\ge \frac{1}{2n}{\left({\Delta }_{f}h\right)}^{2}-\frac{1}{n}{〈\nabla h,\nabla f〉}^{2}.$ (3.3)

$\begin{array}{c}\frac{1}{2}{\Delta }_{f}{|\nabla h|}^{2}={|\text{Hessh}|}^{2}+〈\Delta \nabla h,\nabla h〉-\frac{1}{2}〈\nabla {|\nabla f|}^{2},\nabla f〉\\ ={|\text{Hessh}|}^{2}+〈\nabla \Delta h,\nabla h〉+Ric\left(\nabla h,\nabla h\right)-\text{Hessh}\left(\nabla h,\nabla f\right)\\ ={|\text{Hessh}|}^{2}+〈\nabla {\Delta }_{f}h,\nabla h〉+Ri{c}_{f}\left(\nabla h,\nabla h\right)\\ \ge {|\text{Hessh}|}^{2}+〈\nabla \left(ph+q-{|\nabla h|}^{2}\right),\nabla h〉+Ri{c}_{f}\left(\nabla h,\nabla h\right)\\ ={|\text{Hessh}|}^{2}+\left[p-\left(n-1\right)K\right]{|\nabla h|}^{2}+h〈\nabla p,\nabla h〉+〈\nabla q,\nabla h〉-〈\nabla h,\nabla {|\nabla h|}^{2}〉\\ \ge \frac{1}{2n}{\left({\Delta }_{f}h\right)}^{2}-\frac{1}{n}{〈\nabla h,\nabla f〉}^{2}+p{|\nabla h|}^{2}+Ri{c}_{f}\left(\nabla h,\nabla h\right)\\ +h〈\nabla p,\nabla h〉+〈\nabla q,\nabla h〉-〈\nabla h,\nabla {|\nabla h|}^{2}〉.\end{array}$

$\begin{array}{c}\frac{|\nabla u|}{u}\le \sqrt{\frac{2n}{1-\epsilon }}{\left[\begin{array}{c}\frac{|\alpha |{C}_{1}+\left(n-1\right)K\left(2\rho -1\right){C}_{1}}{2\rho }+\frac{{C}_{2}}{2{\rho }^{2}}+\frac{\left(2n+\epsilon \right){C}_{1}^{2}}{\epsilon {\rho }^{2}}\\ +\frac{D{\theta }_{1}+{\sigma }_{1}+{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}+{\sigma }_{2}}{2}\end{array}\right]}^{\frac{1}{2}}\\ +\sqrt[4]{\frac{n\left({\theta }_{2}+{\sigma }_{2}\right)}{1-\epsilon }},\end{array}$ (3.4)

$\frac{|\nabla u|}{u}\le \sqrt{2n}{\left[\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}\right]}^{\frac{1}{2}}+\sqrt[4]{n\left(D{\theta }_{2}+{\sigma }_{2}\right)}.$ (3.5)

$B\left(\stackrel{¯}{x},2\rho \right)$ 上，对于正常值 $D$${\theta }_{1}$${\theta }_{2}$${\sigma }_{1}$${\sigma }_{2}$${\eta }_{1}$ ，我们假设 $\alpha :={\mathrm{max}}_{\partial B\left(\stackrel{¯}{x},1\right)}{\Delta }_{f}r$$u\le {e}^{D}$$|p|\le {\theta }_{1}$$|\nabla p|\le {\theta }_{2}$$|q|\le {\sigma }_{1}$$|\nabla q|\le m{a}_{2}$$|\nabla f|\le {\eta }_{1}$

$\begin{array}{c}\frac{1}{2}{\Delta }_{f}\left(\phi {|\nabla h|}^{2}\right)=\frac{1}{2}\left({\Delta }_{f}\phi \right){|\nabla h|}^{2}+\frac{\phi }{2}{\Delta }_{f}{|\nabla h|}^{2}+〈\nabla \phi ,\nabla {|\nabla h|}^{2}〉\\ \ge \frac{1}{2}\left({\Delta }_{f}\phi \right){|\nabla h|}^{2}+\frac{\phi }{2n}{\left(ph+q-{|\nabla h|}^{2}\right)}^{2}-\frac{\phi }{n}{〈\nabla h,\nabla f〉}^{2}+\left[p-\left(n-1\right)K\right]\phi {|\nabla h|}^{2}\right]\\ +\phi h〈\nabla p,\nabla h〉+\phi 〈\nabla q,\nabla h〉-〈\nabla h,\nabla \left(\phi {|\nabla h|}^{2}\right)〉+{|\nabla h|}^{2}〈\nabla \phi ,\nabla h〉\\ +\frac{1}{\phi }〈\nabla \phi ,\nabla \left(\phi {|\nabla h|}^{2}\right)〉-\frac{{|\nabla h|}^{2}}{\phi }{|\nabla h|}^{2},\end{array}$ (3.6)

$A:=\frac{{C}_{1}}{2\rho }\left[\alpha +\left(n-1\right)K\left(2\rho -1\right)\right]+\frac{{C}_{2}}{2{\rho }^{2}}.$

$\begin{array}{c}0\ge -AG+\frac{{\phi }^{2}}{2n}{\left(ph+q-{|\nabla h|}^{2}\right)}^{2}-\frac{\phi }{n}{|\nabla f|}^{2}G+\left[p-\left(n-1\right)K\right]\phi G\\ -{\phi }^{\frac{3}{2}}h|\nabla p|{G}^{\frac{1}{2}}-{\phi }^{\frac{3}{2}}|\nabla q|{G}^{\frac{1}{2}}-\frac{|\nabla \phi |}{{\phi }^{1/2}}{G}^{\frac{3}{2}}-\frac{{|\nabla \phi |}^{2}}{\phi }G\\ \ge -AG-\frac{{\phi }^{2}}{n}\left(ph+q\right){|\nabla h|}^{2}+\frac{{G}^{2}}{2n}-\frac{{|\nabla f|}^{2}}{n}G-\left[|p|+\left(n-1\right)K\right]G\\ -\frac{|h||\nabla p|}{2}\left(1+G\right)-\frac{|\nabla q|}{2}\left(1+G\right)-\frac{{C}_{1}}{\rho }{G}^{\frac{3}{2}}-\frac{{C}_{1}^{2}}{{\rho }^{2}}G\\ \ge \frac{1-\epsilon }{2n}{G}^{2}-\left[A+\frac{|ph+q|}{n}+\frac{{|\nabla f|}^{2}}{n}+|p|+\left(n-1\right)K+\frac{|h||\nabla p|}{2}\\ +\frac{|\nabla q|}{2}+\frac{2n{C}_{1}^{2}}{\epsilon {\rho }^{2}}+\frac{{C}_{1}^{2}}{{\rho }^{2}}\right]G-\frac{|h||\nabla p|+|\nabla q|}{2},\end{array}$ (3.7)

$\frac{1-\epsilon }{2n}{G}^{2}\left({x}_{1}\right)\le \left[A+\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}+\frac{2n{C}_{1}^{2}}{\epsilon {\rho }^{2}}+\frac{{C}_{1}^{2}}{{\rho }^{2}}\right]G\left({x}_{1}\right)+\frac{D{\theta }_{2}+{\sigma }_{2}}{2},$ (3.8)

${G}^{2}\left({x}_{1}\right)\le \frac{2n}{1-\epsilon }\left[A+\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}+\frac{2n{C}_{1}^{2}}{\epsilon {\rho }^{2}}+\frac{{C}_{1}^{2}}{{\rho }^{2}}\right]G\left({x}_{1}\right)+\frac{n\left(D{\theta }_{2}+{\sigma }_{2}\right)}{1-\epsilon }.$ (3.9)

${a}_{0}\le \frac{{a}_{2}}{2}+\sqrt{{a}_{1}+\frac{{a}_{2}^{2}}{4}}\le \frac{{a}_{2}}{2}+\sqrt{{a}_{1}}+\frac{{a}_{2}}{2}={a}_{2}+\sqrt{{a}_{1}}.$

$G\left({x}_{1}\right)\le \frac{2n}{1-\epsilon }\left[A+\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}+\frac{2n{C}_{1}^{2}}{\epsilon {\rho }^{2}}+\frac{{C}_{1}^{2}}{{\rho }^{2}}\right]+{\left(\frac{n\left(D{\theta }_{2}+{\sigma }_{2}\right)}{1-\epsilon }\right)}^{\frac{1}{2}}.$ (3.10)

$\underset{x\in \stackrel{¯}{B\left(\stackrel{¯}{x},\rho \right)}}{sup}{|\nabla h|}^{2}=\underset{x\in \stackrel{¯}{B\left(\stackrel{¯}{x},\rho \right)}}{sup}\left(\phi {|\nabla h|}^{2}\right)\le G\left({x}_{1}\right),$

$\frac{|\nabla u|}{u}\le \sqrt{\frac{2n}{1-\epsilon }}{\left[A+\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}+\frac{2n{C}_{1}^{2}}{\epsilon {\rho }^{2}}+\frac{{C}_{1}^{2}}{{\rho }^{2}}\right]}^{\frac{1}{2}}+{\left(\frac{n\left(D{\theta }_{2}+{\sigma }_{2}\right)}{1-\epsilon }\right)}^{\frac{1}{4}}.$ (3.11)

$\begin{array}{c}0\ge \frac{1}{2n}{\left(ph+q-G\right)}^{2}-\frac{{|\nabla f|}^{2}G}{n}+\left[p-\left(n-1\right)K\right]G-h|\nabla p|{G}^{\frac{1}{2}}-|\nabla q|{G}^{\frac{1}{2}}\\ \ge -\frac{1}{n}\left(ph+q\right)G+\frac{{G}^{2}}{2n}-\frac{{|\nabla f|}^{2}}{n}G-|p|G-\left(n-1\right)KG-\frac{|h||\nabla p|}{2}\left(1+G\right)-\frac{|\nabla q|}{2}\left(1+G\right)\\ =\frac{{G}^{2}}{2n}-\left[\frac{|ph+q|}{n}+\frac{{|\nabla f|}^{2}}{n}+|p|+\left(n-1\right)K+\frac{|h||\nabla p|}{2}+\frac{|\nabla q|}{2}\right]G-\frac{|h||\nabla p|+|\nabla q|}{2}.\end{array}$ (3.12)

$\begin{array}{c}{G}^{2}\left({x}_{1}\right)\le 2n\left[\frac{|ph+q|}{n}+\frac{{|\nabla f|}^{2}}{n}+|p|+\left(n-1\right)K+\frac{|h||\nabla p|}{2}+\frac{|\nabla q|}{2}\right]G+n\left(|h||\nabla p|+|\nabla q|\right)\\ \le 2n\left[\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}\right]G\left({x}_{1}\right)+n\left(D{\theta }_{2}+{\sigma }_{2}\right).\end{array}$ (3.13)

$\frac{|\nabla u|}{u}\le \sqrt{2n}{\left[\frac{D{\theta }_{1}+{\sigma }_{1}}{n}+\frac{{\eta }_{1}^{2}}{n}+{\theta }_{1}+\left(n-1\right)K+\frac{D{\theta }_{2}}{2}+\frac{{\sigma }_{2}}{2}\right]}^{\frac{1}{2}}+{\left[n\left(D{\theta }_{2}+{\sigma }_{2}\right)\right]}^{\frac{1}{4}}.$ (3.14)

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