A gradient estimate on graphs with the C D E (m, K)-condition

作者：Wang, Lin Feng^*; Zhou, Yu Jie

来源：ARCHIV DER MATHEMATIK, 2017, 109(4): 383-391.

DOI：10.1007/s00013-017-1078-4

• Graph
• CDE(m, K)-condition
• Gradient estimate
• Harnack inequality
• Liouville theorem

Let G(V, E) be an infinite (locally finite) graph. In this paper we prove that if G satisfies the CDE(m,-K)-condition for some m > 0, K >= 0, and u : V -> R is a positive solution to the following equation
Delta(mu)u = -lambda u,
then lambda <= mK/2 and u satisfies a gradient estimate, which is parallel to the results on complete noncompact Riemannian manifolds established by Li (Geometric analysis. Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 2012), and on smooth metric measure spaces established by Wang (Ann Glob Anal Geom 37: 393-402, 2010). As byproducts, we also get two Liouville theorems and a Harnack inequality.

• 出版日期2017-10
• 单位南通大学