  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.
