Estimates for eigenvalues of weighted Laplacian and weighted $p$-Laplacian

Abstract

A bstract . In this paper, we study two eigenvalue problems of the weighted Laplacian and get the Reilly-type bounds and isoperimetric type bounds for the first nonzero n eigenvalues on hypersurfaces of the Euclidean space. Besides, we give lower bounds for the first eigenvalue of weighted p -Laplacian on submanifolds with locally bounded weighted mean curvature. Meanwhile, several applications of these estimates have also been given.