Eigenvalues of the generalized Laplacian and the G-dynamics of type I

Abstract

In this paper, we consider the generalized Laplace operator equipped with the G-dynamics operator of type I, the Dirichlet and Neumann eigenvalue problems are extended to associate with the G-dynamics of type I, it is proved that the G-dynamics of type I satisfies an integral identity. The G-dynamics of type II for generalized Laplacian is studied as well. Using the general method related to Dirichlet eigenvalue problem, an estimate analysis for the generalized Laplacian with some conditions is made.