GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

doi:10.1016/S0252-9602(13)60055-7

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (4): 963-974.doi: 10.1016/S0252-9602(13)60055-7

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

王宇钊¹|杨杰²|陈文艺¹

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China；
2. College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

收稿日期:2012-06-11 修回日期:2012-10-26 出版日期:2013-07-20 发布日期:2013-07-20

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

WANG Yu-Zhao¹, YANG Jie², CHEN Wen-Yi¹

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China
2. College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

Received:2012-06-11 Revised:2012-10-26 Online:2013-07-20 Published:2013-07-20

摘要/Abstract

摘要：

Let (M, g, e^−fdv) be a smooth metric measure space. In this paper, we con-sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equation

∂u/∂t= efdiv(e−f |∇u|p−2∇u)
on M × [0, ∞), where 1 < p < ∞ and f is a smooth function on M under the assumption that the m-dimensional nonnegative Bakry-Émery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Émery Ricci curva-ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

关键词: gradient estimates, weighted p-heat equation, entropy monotonicity formula, m-Bakry-Émery Ricci curvature

Abstract:

Key words: gradient estimates, weighted p-heat equation, entropy monotonicity formula, m-Bakry-Émery Ricci curvature

中图分类号:

58J35

王宇钊, 杨杰, 陈文艺. GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES[J]. 数学物理学报(英文版), 2013, 33(4): 963-974.

WANG Yu-Zhao, YANG Jie, CHEN Wen-Yi. GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES[J]. Acta mathematica scientia,Series B, 2013, 33(4): 963-974.

参考文献

[1] Bakry D, ´Emery M. Diffusion hypercontractivitives//S´eminaire de Probabilit´es XIX, 1983/1984. Lecture Notes in Math, Vol 1123. Berlin: Springer-Verlag, 1985: 177–206

[2] Chen Li, Chen Wenyi. Gradient estimates for positive smooth f-harmonic functions. Acta Math Sci, 2010, 30B(5): 1614–1618

[3] Chow B, Lu P, Ni L. Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics 3. Science Press and American Mathematical Society, 2006

[4] Huang Guangyue, Ma Bingqing. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch Math, 2010, 94(3): 265–275

[5] Li P, Yau S T. On the parabolic kernel of the Schr¨odinger operator. Acta Math, 1986, 156: 153–201

[6] Li Xiangdong. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J Math Pures Appl, 2005, 84(9): 1295–1361

[7] Li Xiangdong. Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature. Math Ann, 2012, 353(2): 403–437

[8] Li Xiangdong. Perelman’sW-entropy for the Forkker-Plank equation over complete Riemannian manifolds. Bull Sci Math, 2001, 125: 871–882

[9] Kotschwar B, Ni L. Gradient estimate for p-harmonic functions, 1/H flow and an entropy formula. Ann Sci Éc Norm Sup´er, 2009, 42(4): 1–36

[10] Munteanu O, Wang Jiaping. Smooth metric measure spaces with non-negative curvature. Comm Anal Geom, 2011, 19(3): 451–486

[11] Munteanu O, Wang Jiaping. Analysis of weighted Laplacian and applications to Ricci solitons. Comm Anal Geom, 2012, 20(1): 55–94

[12] Ni Lei. Monotonicity and Li-Yau-Hamilton inequalities//Surv Differ Geom 12: Geometric flows. International Press, 2008: 251–301

[13] Ni Lei. The entropy formula for linear heat equation. J Geom Anal, 2004, 14: 87–100

[14] Perelman G. The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/abs/maths0211159

[15] Wei Guofang, Wylie Will. Comparison geometry for the Bakry-Émery Ricci tensor. J Differ Geom, 2009, 83(2): 377–405

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

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