Researches Of Gradient Estimates On Graphs

There are two main research objects in this paper.One is gradient estimation under standard curvature dimension condition and the other is gradient estimation under nonlinear curvature dimension condition.The second chapter mainly reviews the methods used in gradient estimation,one is maximum principle,the other is semigroup method.In Chapter 3,we use the maximum principle to prove the global and local Li-Yau type estimates under the nonlinear curvature dimension CDΨ(d,-K)of the graph,and get some good results.The inferences we get when we take a specific function are in agreement with the results obtained by predecessors.In Chapter 4,we obtain the equivalent condition of the nonlinear curvature dimension condition CDΨ{∞,-K)on the graph by using the semigroup method.Similarly,the inferences we get when we take a particular function agree with the results obtained by our predecessor.In Chapter 5,we obtain the long-term existence and monotonicity of the modified heat equation generated by p-Laplace under the standard curvature dimension condition CD(n,0),and the most important is the Li-Yau type estimation and Hamilton type estimation of the equation,from Li-Yau type estimation which Harnack inequality is generated.