Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaHongye SongSchool of General Education, Beijing International Studies University, Beijing 100024, China; Department of Mathematics, Renmin University of China, Beijing 100872, China
Analysis of PDEsDifferential Geometrymathscidoc:2207.03005
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.
Yong LinDepartment of Mathematics, Renmin University of ChinaShuang LiuDepartment of Mathematics, Renmin University of ChinaHongye SongDepartment of Mathematics, Renmin University of China; Beijing International Studies University
Analysis of PDEsmathscidoc:2207.03004
Mathematical Physics and Computer Simulation, 20, (3), 99-110, 2017.9
Based on a global estimate of the heat kernel, some important inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain exponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.
We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.
Alexander Grigor’yanDepartment of Mathematics, University of Bielefeld, Bielefeld, 33501, Germany; School of Mathematical Sciences and Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin, 300071, ChinaYong LinDepartment of Mathematics, Renmin University of China, Beijing, 100872, ChinaYunYan YangDepartment of Mathematics, Renmin University of China, Beijing, 100872, China
Let G = (V, E) be a locally finite graph, whose measure μ(x) has positive lower bound, and Δ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz (1973), we establish existence results for some nonlinear equations, namely Δu + hu = f(x, u), x ∈ V. In particular, we prove that if h and f satisfy certain assumptions, then the above-mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation Δu + hu = f(x, u) + ϵg. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.
Alexander Grigor’yanDepartment of Mathematics, University of Bielefeld, 33501, Bielefeld, GermanyYong LinDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of ChinaYunyan YangDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China
Analysis of PDEsmathscidoc:2207.03001
Calculus of Variations and Partial Differential Equations, 55, (92), 2016.7
Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).