Eigenvalue Estimates Of The P-laplacian And The Drifting Laplacian On Riemannian Manifolds

One of important research topics in geometry and analysis on manifolds is to estimate eigenvalues of differential operators on Riemanian manifolds.The research of this issue not only makes people have a deeper understanding of manifolds,but also promotes the research and applications of related problems in other branches of mathematics.In recent years,the p-Laplacian and drifting Laplaeian-have-become-the preface research objects in the study of eigenvalue estimate problems.As general generalizations of the Laplacian,the p-Laplacian and drifting Laplacian have a wide range of functions in many fields,such as mathematics,physics,quantum mechanics,image processing and so on.Estimates for the first eigenvalue and universal inequalities are two important research directions of eigenvalue estimates.In view of the above analysis,this paper studies the eigenvalue estimate problems of the p-Laplacian and universal inequalities of the drifting Laplacian on Riemnnian manifolds.Firstly,this paper reviewed some research progress of the first eigenvalue estimates of the Laplacian and the p-Laplacian on Riemannian manifolds,including some classic conclusions of S.Y.Cheng,S.T.Yau and other mathematicians.Secondly,this paper introduces the research progess of universal inequalities of the Laplacian on the Riemannian manifolds and universal inequalities of the drifting Laplacian on the H-type group.Based on the previous conclusions,we consider the p-Laplacian on submanifold of Riemannian manifold with sectional curvature bounded above by a nonpositive constant,obtain a estimate for the lower bound of its first eigenvalue.This result generalizes the one obtained by Lin for the Laplacian(Nonlinear Anal.,2017,148:126-137).Moreover,we investigate the Dirichlet eigenvalue problem of the drifting Laplacian ?G+(?G?,?G(·)>on the H-type group G with the weighted measure d?=-e-?dv.We establish an Levitin-Parnovski-type universal inequality for eigenvalues of this problem.This result implies some inequalities for some related problems.In special,our results generalize the one derived by Ilias and Makhoul for the Kohn Laplacian on the Heisenberg group(J.Geom.Anal.,2012,22(1):206-222).