Research On Log-Harnack Inequality And Gradient Estimate Of P-laplace Operator

Log-Harnack inequality and gradient estimate are important research contents in stochas-tic analysis and geometric analysis.On Riemannian manifolds,the Log-Harnack inequality and gradient estimate for elliptic and parabolic equations are concerned and studied by a large number of scholars;In recent years,people have defined new curvature reasonably on the graph,using the improved curvature dimension condition and the maximum princi-ple,the gradient estimate of the positive solution of the linear equation has also made new progress on graph.The natural thought is studying the Log-Harnack inequality and gradient estimate of the nonlinear equation,especially the positive solution of the equation related to the p-Laplace operator on Riemannian manifold and graphs,In this article,the specific studies include:(1)On weighted Riemannian manifolds with m-Bakry-émery Ricci curva-ture bounded below by-K,we prove the Log-Harnack inequality for L~p-Log-Sobolev func-tion;(2)Under the assumption of non-negative m-Bakry-émery Ricci curvature,we obtain a global Li-Yau type gradient estimate and a Hamilton type estimate for the positive solutions to the parabolic p-Laplace equation with logarithmic nonlinearity,as applications,the corre-sponding Harnack inequalities are derived.(3)Using the curvature dimension condition and the maximum principle,we prove the Li-Yau type gradient estimate for nonlinear equation on finite graph.The establishment of the Log-Harnack inequality provides a direct method for esti-mating the lower bound of the Log-Sobolev constant;Gradient estimate plays a key role in studying some properties of partial differential equation solutions.