  In this paper, we obtain eigenvalue estimates for a larger class of elliptic
differential operators in divergence form on a bounded domain in a complete
Riemannian manifold isometrically immersed in Euclidean space. As an
application, we give eigenvalue estimates in the Gaussian shrinking soliton,
and we find a domain that makes the behavior of these estimates similar to the
estimates for the case of the Laplacian. Moreover, we also give an answer to
the generalized conjecture of P\'olya.
