HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1751-1758.

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

林勇¹, 宋宏业^1,2

1 Department of Mathematics, Renmin University of China, Beijing 100872, China;
2 School of General Education, Beijing International Studies University, Beijing 100024, China

收稿日期:2017-08-10 修回日期:2018-01-26 出版日期:2018-12-25 发布日期:2018-12-28
通讯作者: Hongye SONG E-mail:songhongye@bisu.edu.cn
作者简介:Yong LIN,E-mail:linyong01@ruc.edu.cn
基金资助:
The authors were supported by the National Science Foundation of China (11671401).

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

Yong LIN¹, Hongye SONG^1,2

1 Department of Mathematics, Renmin University of China, Beijing 100872, China;
2 School of General Education, Beijing International Studies University, Beijing 100024, China

Received:2017-08-10 Revised:2018-01-26 Online:2018-12-25 Published:2018-12-28
Contact: Hongye SONG E-mail:songhongye@bisu.edu.cn
Supported by:
The authors were supported by the National Science Foundation of China (11671401).

摘要/Abstract

摘要： We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

关键词: harmonic function, subharmonic function, Harnack inequality, mean value inequality, graph

Abstract: We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

Key words: harmonic function, subharmonic function, Harnack inequality, mean value inequality, graph

林勇, 宋宏业. HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS[J]. 数学物理学报(英文版), 2018, 38(6): 1751-1758.

Yong LIN, Hongye SONG. HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1751-1758.

参考文献

[1] Yau S T. Harmonic functions on complete Riemannian manifolds. Comm Pure Appl Math, 1975, 28:201-228
[2] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 27:333-354
[3] Li P, Yau S T. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 156:153-201
[4] Li J. Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J Funct Anal, 1991, 100:233-256
[5] Negrin E. Gradient estimates and a Liouville type theorem for the Schrödinger operator. J Funct Anal, 1995, 127:198-203
[6] Ma L. Gradient estimates for a simple elliptic equation on non-compact Riemannian manifolds. J Funct Anal, 2006, 241:374-382
[7] Yang Y. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc Amer Math Soc, 2008, 136:4095-4102
[8] Yang Y. Gradient estimates for the equation △u + cu^-α=0 on Riemannian manifolds. Acta Math Sinica, English Series, 2010, 26:1177-1182
[9] Bauer F, Horn P, Lin Y, Lippner G, Mangoubi D, Yau S T. Li-Yau inequality on graphs. J Differential Geom, 2015, 99:359-405
[10] Lin Y, Liu S, Yang Y. Global gradient estimate on graph and its applications. Acta Math Sin, Engl Ser, 2016, 32:1350-1356
[11] Lin Y, Liu S, Yang Y. A gradient estimate for positive functions on graphs. J Geom Anal, 2017, 27:1667-1679
[12] Li P, Schoen R. L^p and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math, 1984, 153:279-301
[13] Holopainen I, Soardi P M. A strong Liouville theorem for p-harmonic functions on graphs. Ann Acad Sci Fenn Math, 1997, 22:205-226
[14] Holopainen I, Soardi P M. p-harmonic functions on graphs and manifolds. Manuscripta Math, 1997, 94:95-110
[15] Rigoli M, Salvatori M, Vignati M. Subharmonic functions on graphs. Israel J Math, 1997, 99:1-27
[16] Lin Y, Xi L. Lipschitz property of harmonic function on graphs. J Math Anal Appl, 2010, 366:673-678
[17] Hua B, Jost J. L^q harmonic functions on graphs. Israel J Math, 2014, 202:475-490
[18] Karp L. Subharmonic functions on real and complex manifolds. Math Z, 1982, 179:535-554
[19] Lin Y, Yau S T. Ricci curvature and eigenvalue estimate on locally finite graphs. Math Res Lett, 2010, 17:343-356
[20] Grigor'yan A. Analysis on Graphs, Lecture Notes. University Bielefeld, 2009

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	林勇, 吴艺婷. BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS[J]. 数学物理学报(英文版), 2018, 38(3): 843-856.
[2]	周少波, 薛明皋. EXPONENTIAL STABILITY FOR NONLINEAR HYBRID STOCHASTIC PANTOGRAPH EQUATIONS AND NUMERICAL APPROXIMATION[J]. 数学物理学报(英文版), 2014, 34(4): 1254-1270.
[3]	任广斌, Uwe KAHLER. PSEUDOHYPERBOLIC METRIC AND UNIFORMLY DISCRETE SEQUENCES IN THE REAL UNIT BALL[J]. 数学物理学报(英文版), 2014, 34(3): 629-638.
[4]	谷龙飞, 张忠祥. GAUSS-MEAN VALUE FORMULA FOR TRIHARMONIC FUNCTIONS AND ITS APPLICATIONS IN CLIFFORD ANALYSIS[J]. 数学物理学报(英文版), 2013, 33(5): 1419-1430.
[5]	K. BALACHANDRAN, S. KIRUTHIKA, J. J. TRUJILLO. EXISTENCE OF SOLUTIONS OF NONLINEAR FRACTIONAL PANTOGRAPH EQUATIONS[J]. 数学物理学报(英文版), 2013, 33(3): 712-720.
[6]	Muhammad IMRAN, Syed Ahtsham ul Haq BOKHARY, Ali AHMAD, Andrea SEMANICOVá-FE?OVCíKOVá. ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION[J]. 数学物理学报(英文版), 2013, 33(1): 187-206.
[7]	曹丽. SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES[J]. 数学物理学报(英文版), 2012, 32(6): 2096-2104.
[8]	姜乐, 杜金元. RIEMANN BOUNDARY VALUE PROBLEMS FOR SOME K-REGULAR FUNCTIONS IN CLIFFORD ANALYSIS[J]. 数学物理学报(英文版), 2012, 32(5): 2029-2049.
[9]	张艳慧, 邓冠铁, 高洁欣. ON THE LOWER BOUND FOR A CLASS OF HARMONIC FUNCTIONS IN THE HALF SPACE[J]. 数学物理学报(英文版), 2012, 32(4): 1487-1494.
[10]	程志云, 高红铸. SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY[J]. 数学物理学报(英文版), 2012, 32(2): 663-671.
[11]	Sarp Adali. VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM[J]. 数学物理学报(英文版), 2012, 32(1): 325-338.
[12]	A. Cheer, Harry A. Dwyer, T. Kim. NEW METHOD FOR IMPROVED CALCULATIONS OF UNSTEADY COMPLEX FLOWS IN LARGE ARTERIES[J]. 数学物理学报(英文版), 2011, 31(6): 2247-2264.
[13]	陈恕行. A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION[J]. 数学物理学报(英文版), 2011, 31(6): 2378-2388.
[14]	陈爱莲, 张福基, 李皓. THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS[J]. 数学物理学报(英文版), 2011, 31(3): 1155-1166.
[15]	M.I.Ostrovskii . MINIMUM CONGESTION SPANNING TREES IN BIPARTITE AND RANDOM GRAPHS[J]. 数学物理学报(英文版), 2011, 31(2): 634-640.