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
DOI: 10.12677/AAM.2019.84073, PDF,   
作者: 王子君：中国地质大学(武汉)，数学与物理学院，湖北 武汉
关键词: 梯度估计；椭圆方程；完备黎曼流形；Gradient Estimate； Elliptic Equation； Complete Riemannian Manifold
摘要: 本文的主要目的是在一个完备黎曼流形上推导出一个二阶椭圆方程Δfu=plogu+qu的局部梯度估计，该方程具有光滑函数f、p和q。
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.
文章引用：王子君. 完备黎曼流形上椭圆方程的局部梯度估计[J]. 应用数学进展, 2019, 8(4): 657-663. https://doi.org/10.12677/AAM.2019.84073

参考文献

[1] Yau, S.T. (1975) Harmonic Functions on Complete Riemannian Manifolds. Communications on Pure and Applied Mathematics, 28, 201-228. [Google Scholar] [CrossRef] 
[2] Ma, L. (2006) Gradient Estimates for a Simple Elliptic Equation on Complete Non-Compact Riemannian Manifolds. Journal of Functional Analysis, 241, 374-382. [Google Scholar] [CrossRef] 
[3] Ma, B., Huang, G. and Luo, Y. (2018) Gradient Estimates for a Nonlinear Elliptic Equation on Complete Riemannian Manifolds. Proceedings of American Mathematical Society, 146, 4993-5002. [Google Scholar] [CrossRef] 
[4] Brighton, K. (2013) A Liouville-Type Theorem for Smooth Metric Measure Spaces. Journal of Geometric Analysis, 23, 562-570. [Google Scholar] [CrossRef] 
[5] Ge, H. and Zhang, S. (2018) Liouville-Type Theorems on the Complete Gradient Shrinking Ricci Solitons. Differential Geometry and its Applications, 56, 42-53. [Google Scholar] [CrossRef] 
[6] Calabi, E. (1958) An Extension of E. Hopf’s Maximum Principle with Application to Riemannian Geometry. Duke Mathematical Journal, 25, 45-56. [Google Scholar] [CrossRef] 
[7] Cheng, S.Y. and Yau, S.T. (1975) Differential Equations on Riemannian Manifolds and Their Geometric Applications. Communications on Pure and Applied Mathematics, 28, 333-354. [Google Scholar] [CrossRef] 
[8] Wei, G.F. and Wylie, W. (2009) Comparison Geometry for the Bakry-Émery Ricci Tensor. Journal of Differential Geometry, 83, 377-405. [Google Scholar] [CrossRef] 
[9] Wu, J.Y. (2017) Elliptic Gradient Estimates for a Nonlinear Heat Equation and Applications. Nonlinear Analysis: Theory, Methods & Applications, 1511-1517. [Google Scholar] [CrossRef] 

为你推荐