TY - JOUR
T1 - A Gradient Estimate for Positive Functions on Graphs
AU - Lin, Yong
AU - Liu, Shuang
AU - Yang, Yunyan
N1 - Publisher Copyright:
© 2016, Mathematica Josephina, Inc.
PY - 2017/4/1
Y1 - 2017/4/1
N2 - We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.
AB - We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.
KW - Gradient estimate
KW - Harnack inequality
KW - Locally finite graph
KW - Spectral graph
UR - http://www.scopus.com/inward/record.url?scp=84987615763&partnerID=8YFLogxK
U2 - 10.1007/s12220-016-9735-6
DO - 10.1007/s12220-016-9735-6
M3 - Article
AN - SCOPUS:84987615763
SN - 1050-6926
VL - 27
SP - 1667
EP - 1679
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
IS - 2
ER -